Spatial asymptotic expansion of the Euler equation

Peter Topalov

Spatial asymptotic expansion of the Euler equation

Spatial asymptotic expansion of the Euler equation by Saif Sultan BE Electrical Engineering, NUST Pakistan MS Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 6, 2019 Dissertation directed by Prof. Peter Topalov Professor of Mathematics 1 Dedication Dedicated to my parents Sher Sultan and Tahira Batool and my siblings for all their love, support and patience. 2 Acknowledgments I want to sincerely thank my advisor Prof. Peter Topalov for the continuous support of my Ph.D. study and for his patience, immense knowledge and for being an amazing teacher. His guidance helped me throughout my research and during writing this thesis. I could not have imagined having a better advisor and mentor for my Ph.D. study. 3 Abstract of Dissertation In this dissertation we study the Euler equation in dimension 2 and higher. We show that the Euler equation is locally well-posed in the space of spatially asymptotic functions. Furthermore, we prove that the 2d Euler equation is globally well-posed in the corresponding asymptotic space. We discuss how the Euler equation preserves certain function spaces that have asymptotic expansions. These asymptotic function spaces are known to have asymptotic terms that include logarithms. However, we show that structurally simpler spaces without log terms are also preserved by the Euler equation. In 2 dimensional fluid flow the complex structure can be used to find such smaller function spaces without log terms. 4 Table of Contents Dedication 2 Acknowledgments 3 Abstract of Dissertation 4 Table of Contents 5 Introduction 7 Chapter 1 Asymptotic expansion in two dimensions 10 1.1 . . . . . . . . . . . . . . . . Asymptotic spaces and diffeomorphisms ZDm,p N 12 1.2 Cauchy operator in asymptotic spaces . . . . . . . . . . . . . . . . . . . . . . 20 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . The Euler equation on ZDm,p N 33 1.4 . . . . . . . . . . . . . . Global solutions of the 2d Euler equation on ZDm,p N 40 1.5 Proof of Lemma 1.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 2 Asymptotic expansion in higher dimensions 61 2.1 Asymptotic spaces and diffeomorphism groups . . . . . . . . . . . . . . . . . 64 2.2 Laplace operator in asymptotic spaces . . . . . . . . . . . . . . . . . . . . . 72 2.3 Euler vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.4 Smoothness of Euler Vector Field . . . . . . . . . . . . . . . . . . . . . . . . 78 2.5 Proof of Theorem 2.0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.6 Volume preserving asymptotic diffeomorphism . . . . . . . . . . . . . . . . . 81 5 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.8 Remarks on solutions with constant asymptotic term . . . . . . . . . . . . . 86 2.9 Asymptotics preserved by Euler equation . . . . . . . . . . . . . . . . . . . . 86 Appendices 89 References 91 6 Introduction Consider the incompressible Euler equation on Rd with d ≥ 2,    vt + v · ∇v = −∇p, div v = 0,   v|t=0 = v0 , (1) where v(x, t) is the velocity of the fluid and p(x, t) is the scalar pressure. The divergence div and the covariant derivative ∇ are computed with respect to the Euclidean metric on Rd . It was proven in [14] that the Euler equation (1) is locally well-posed in the class of asymptotic d vector fields Am,p N ;0 . By definition, a vector field v on R belongs to the class of asymptotic vector fields Am,p N ;0 with integer regularity exponent m ≥ 0, a real 1 < p < ∞, and an integer parameter N ≥ 0, if it can be written in the form v(x) = a0 (θ)+ N a01 (θ) + a11 (θ) log r a0 (θ) + ... + aN N (θ)(log r) +...+ N +f (x), r rN | x | ≥ 1/2, (2) where r ≡ r(x) := | x |, θ ≡ θ(x) := x /| x | is a point on the unit sphere S d−1 , the coefficients ajk for 0 ≤ k ≤ N and 0 ≤ j ≤ k are continuous vector-valued functions on the sphere ajk : S d−1 → Rd that have some additional regularity, and f : Rd → Rd is the remainder that belongs to a weighted Sobolev space on Rd so that it is continuous, and f (x) = o(1/rN ) as r → ∞. We refer to [14, Section 2] for detail. In this sense, any vector field in Am,p N ;0 has a partial asymptotic expansion at infinity of order N . There are two points that have to be mentioned: The first is that the class Am,p N ;0 of asymptotic vector fields is natural for the Euler equation for the following reason: even if we take a generic smooth initial data with compactly supported components v0j ∈ Cc∞ (Rd ) , the corresponding 7 local solution of the Euler equation will develop non-trivial asymptotics at infinity of the form (2) with non-vanishing leading asymptotic term ad+1 (θ)/rd+1 . The second point is that if we take a generic initial data of the form (2) with vanishing coefficients in front of the log terms, then the local solution of the Euler equation will develop non-trivial asymptotics with non-vanishing log terms. The reason for this is that the space Am,p N ;0 with vanishing log terms requires no restrictions on the spherical Fourier modes of the coefficients ajk , 0 ≤ k ≤ N and 0 ≤ j ≤ k. We refer to [14, Appendix B] for detail. The goal of this work is to study well-posedness of the Euler equation in simpler asymptotic spaces which, for example, do not involve log term. To this end we will be working with asymptotic spaces Am,p n,N as defined below (cf. [13, Section 3]) n o X ak (θ) m,p m+1+N −k,p d−1 := u Am,p u = χ(r) + f such that f ∈ W and a ∈ H (S ) , k γN n,N k r n≤k≤N (3) where χ : R → R is a C ∞ -smooth cut-off function with bounded derivatives such that χ(ρ) = 1 for ρ ≥ 2 and χ(ρ) = 0 for 0 ≤ ρ ≤ 1, equipped with the norm kukAm,p := n,N X n≤k≤N kak kH m+1+N −k (S d−1 ) + kf kWγm,p . N (4) Here Wδm,p denotes weighted Sobolev space with weight δ ∈ R and smoothness m ≥ 0 defined as m,p Wδm,p := f ∈ Hloc (C, C) hxiδ+|α| ∂ α f ∈ Lp for |α| ≤ m , supplied with the norm kf kWδm,p := P |α|≤m hxiδ+|α| ∂ α f Lp where hxi = (1 + | x |2 )1/2 , and m,p Hloc is space of functions that are locally in Sobolev spaces. We will denote the weight γN := N + γ0 where 0 < γ0 + p2 < 1 is given a priori and does not depend on the choice of N . This choice of weights γN ensures that the remainder terms f ∈ Wγm,p satisfy f = o(1/rN ) as N m,p lim r → ∞. Furthermore, we will generally denote Am,p N := A0,N . For m > d/p these spaces are Banach algebras. V. Arnold in [2] viewed the fluid flow as a solution to an ODE on an infinite dimensional 8 group of diffeomorphisms of the underlying space. This approach was used in [7] and [3] to show the well-posedness of Euler equation. The well-posedness of the incompressible Euler equation in the weighted Sobolev spaces Wδm,p was then studied [5] while the well-posedness in asymptotics spaces Am,p N ;0 is established in [14]. 9 Chapter 1 Asymptotic expansion in two dimensions In this chapter we restrict our attention to the two dimensional case (d = 2) and answer the following questions: 1) Can one find a subspace of the asymptotic space Am,p N ;0 with non-trivial asymptotic part and such that it is invariant with respect to the Euler equation and does not have log terms in its asymptotic part? 2) Are the solutions of the Euler equation in such a space global in time? We will show that the answer to the both questions is positive. An important feature of our analysis is that we identify R2 with the complex plane by setting z = x + iy for (x, y) ∈ R2 , √ i = −1 and rewrite the Euler equation (1) in complex form as    ut + uuz + ūuz̄ = −2∂z̄ p, div u = uz + ūz̄ = 0, (1.1)   u|t=0 = u0 , where u : C → C is the holomorphic component of the fluid velocity (i.e. u(z, z̄) = v1 (z, z̄) + v2 (z, z̄) where v1 , v2 are components of the velocity vector v) and the subscripts z and z̄ denote 10 the partial differentiations with respect to the Cauchy operators ∂z and ∂z̄ respectively. Also we will inter-changeably denote r = |z| depending on the context. In particular, this allows us to define the following asymptotic space for integer m > 2/p and N ≥ 0, n ZNm,p := u u = χ(r) X 0≤k+l≤N o akl m,p + f such that f ∈ W and a ∈ C , kl γN z k z̄ l (1.2) where k, l ≥ 0 and χ : R → R is a C ∞ -smooth cut-off function with bounded derivatives such that χ(ρ) = 1 for ρ ≥ 2 and χ(ρ) = 0 for 0 ≤ ρ ≤ 1, equipped with the norm kukZ m,p := X N |akl | + kf kWγm,p . (1.3) N 0≤k+l≤N Here Wδm,p denotes Weighted Sobolev space with weight δ ∈ R and smoothness m ≥ 0 defined as m,p (C, C) hziδ+|α| ∂ α f ∈ Lp for |α| ≤ m , Wδm,p := f ∈ Hloc supplied with the norm kf kWδm,p := P |α|≤m hziδ+|α| ∂ α f Lp ∂ α ≡ ∂zα1 ∂z̄α2 , where hzi = (1 + |z|2 )1/2 and m,p is space of functions that are locally in Sobolev spaces. We will denote the weight Hloc γN := N + γ0 where 0 < γ0 + p2 < 1 is given a priori and does not depend on the choice of N . This choice of weights γN ensures that the remainder terms f ∈ Wγm,p satisfy f = o(1/rN ) N m,p as as lim r → ∞. For 0 ≤ n ≤ N we will define following subspace of Zn,N n X m,p := u u = χ(r) Zn,N n≤k+l≤N o akl m,p + f such that f ∈ W and a ∈ C kl γN z k z̄ l (1.4) m,p where we omit the summation term if n = N + 1 and set ZNm,p +1,N ≡ WγN . Take ρ > 0 and denote by BZNm,p (ρ) ∈ ZNm,p the open ball of radius ρ centered at zero . The main result of this chapter is the following Theorem proved in Section 1.4. Theorem 1.0.1. Assume m > 3 + 2/p where 1 < p < ∞. Then for any u0 ∈ ZNm,p the Euler equation (1.1) has a unique global solution in time u ∈ C [0, ∞), ZNm,p ∩ C 1 [0, ∞), ZNm−1,p m+1,p for any t ≥ 0. The solution depends continuously such that the pressure p(t) lies in Z1,N 11 on the initial data u0 ∈ ZNm,p in the sense that for any given T > 0 the data-to-solution map u0 7→ u, ZNm,p → C [0, T ], ZNm,p ∩ C 1 [0, T ], ZNm−1,p is continuous. 1.1 Asymptotic spaces and diffeomorphisms ZDm,p N We can define the space of asymptotic diffeomorphisms associated with the asymptotic space ZNm,p as o n m,p 1 2 , ϕ = id +u, u ∈ Z := ϕ ∈ Diff (R ) ZDm,p + N N (1.5) where Diff+1 (R2 )1 denotes the group of orientation preserving C 1 -diffeomorphisms of R2 and id : R2 → R2 is the identity map. Note that the asymptotic space ZNm,p is a closed subspace m,p m,p in Am,p N and therefore ZD N is a subset of the asymptotic diffeomorphisms AD N considered in (cf. [14, Section 2]) n o m,p 1 2 := ϕ ∈ Diff (R ) ADm,p ϕ = id +u, u ∈ A . + N N m,p In this section we show that ZDm,p N is a topological subgroup of AD N (see Theorem 1.1.12). and we will need to show is a submanifold in ADm,p For this we use the fact that ZDm,p N N that: 1. ZDm,p N is closed under composition of diffeomorphisms. 2. ZDm,p N is closed under inversion of diffeomorphisms. The following two propositions summarize the properties of the asymptotic spaces ZNm,p and their reminder spaces Wδm,p . For the proof we refer to Proposition 1.5.1, Proposition 1.5.3, and Remark 15 in Section 1.5 (cf. [13, § 2]). 1 We will denote the identity diffeomorphism on Rd by id. I will denote the identity matrix on Rd . 12 Proposition 1.1.1. (i) For any 0 ≤ m1 ≤ m2 and for any real weights δ1 ≤ δ2 we have that the inclusion map Wδm2 2 ,p → Wδm1 1 ,p is bounded. (ii) For any m ≥ 0, δ ∈ R, and for any k ∈ Z the map f 7→ f · χ , zk m,p Wδm,p → Wδ+k is bounded. The same holds with z replaced by z̄. Here χ(z, z̄) ≡ χ(|z|) is a C ∞ -smooth cut-off function such that χ : R → R and χ|(−R,R) ≡ 1 for some R > 0. (iii) For any regularity exponent m ≥ 0, δ ∈ R, and a multi-index α ∈ Z2≥0 such that m−|α|,p |α| ≤ m the map ∂ α : Wδm,p → Wδ+|α| is bounded. → × Wδm,p (iv) For m > 2/p and for any weights δ1 , δ2 ∈ R the map (f, g) 7→ f g, Wδm,p 2 1 Wδm,p is bounded. In particular, for δ ≥ 0 the space Wδm,p is a Banach algebra. 1 +δ2 +(2/p) Proposition 1.1.1 implies Proposition 1.1.2. Assume that m > 2/p. Then is → Znm,p (i) For any 0 ≤ n1 ≤ n2 ≤ N1 ≤ N2 we have that the inclusion map Znm,p 1 ,N1 2 ,N2 bounded. (ii) For any 0 ≤ n ≤ N , an integer k ≥ 0, a regularity exponent m > k + (2/p), and a m−|α|,p m,p → Zn+|α|,N +|α| is bounded. multi-index |α| ≤ k the map ∂ α : Zn,N (iii) For m > 2/p and for any 0 ≤ n1 ≤ N1 and 0 ≤ n2 ≤ N2 the map (f, g) 7→ f g, m,p , → Zn,N × Znm,p Znm,p 2 ,N2 1 ,N1 where n := n1 + n2 and N := min(n1 + N2 , n2 + N1 ) is bounded. In particular, ZNm,p is a Banach algebra for any N ≥ 0. For a given R > 0, we will also consider the space ZNm,p (BRc ) that is the image of the restriction map of u ∈ ZNm,p to BRc = x ∈ R2 | x | > R . An equivalent norm we will use with ZNm,p (BRc ) is |u|ZNm,p (BRc ) = X 0≤k+l≤N |akl | + kf kWγm,p (B c ) . R N Rk+l 13 Remark 1. This norm is equivalent to the norm it inherits from ZNm,p . This follows from the fact that any two norms (see (1.3)) on RM for M ≥ 1 are equivalent. We will later show that ZNm,p (BRc ) is a Banach algebra with this norm and more importantly there is a constant CZ independent of the radius R such that |a · b|ZNm,p (BRc ) ≤ CZ |a|ZNm,p (BRc ) |b|ZNm,p (BRc ) . Following Lemma allows us to work between the two spaces ZNm,p and ZNm,p (BRc ). m,p (C, C) we have that u ∈ ZNm,p if and Lemma 1.1.3. Assume that R > 0. Then, for u ∈ Hloc only if u ∈ H m,p (BR+1 ) and u ∈ ZNm,p (BRc ). The norms k·kZNm,p and k·kH m,p (BR+1 ) +|·|ZNm,p (BRc ) on ZNm,p are equivalent. We begin with following technical lemma. Its proof involves several technical steps and is defered to Section 1.5. The constant CZ appearing in below is independent of radius R > 0 of the ball BR . Lemma 1.1.4. Let m > 2/p then there exists a constant CZ independent of radius R > 0 such that for u1 , u2 ∈ ZNm,p (BRc ) |u1 · u2 |Z m,p (B c ) ≤ CZ |u1 |Z m,p (B c ) |u2 |Z m,p (B c ) . N R N R N R Lemma 1.1.5. Let m > 2/p and let u ∈ ZNm,p such that |1 + u| > 0 and u has no constant asymptotics (a00 = 0), then 1 1+u ∈ ZNm,p and the map u 7→ 1 1+u is analytic at such u. Proof. Let us denote C∗ = min(1/2, CZ−1 /2). We first show that 1 1+u ∈ ZNm,p . Note that by Proposition 1.5.1 u is a continuous function. Also by Corollary 1.5.5, we can take R > 2 large enough such that, |u|ZNm,p (BRc ) < C∗ so that supz∈BRc |u| ≤ |u|ZNm,p (BRc ) by Corollary 1.5.5. By expanding in geometric series, we see that (1 + u)−1 = X uk k≥0 converges both pointwise and also in ZNm,p (BRc ). Continuity of u shows that |1+u| is bounded below in BR+1 by a positive number and so (1 + u)−1 is bounded. By directly differentiating 14 (1 + u)−1 we see that (1 + u)−1 ∈ H m,p (BR+1 ). Hence 1 1+u ∈ ZNm,p by Lemma 1.1.3. Now let ψ = 1 + u, by continuity of multiplication in ZNm,p and the embedding ZNm,p ֒→ L∞ given by Corollary 1.5.5, we can find a open neighborhood V of 0 in ZNm,p such that for v ∈ V v· 1 ψ m,p ZN < C∗ and v· 1 ψ < C∗ . L∞ Then we can expand as 1 1 1 = ψ+v ψ1+ v ψ j ∞ 1 X −v = ψ j=0 ψ which shows that the map is analytic at u. Remark 2. The proof of this Lemma shows that this is also true for ZNm,p (BRc ) for R > 2 instead of ZNm,p . That is if u ∈ ZNm,p (BRc ) satisfies conditions of the above Lemma then (1 + u)−1 ∈ ZNm,p (BRc ). We will need following space when working with the Cauchy operator in the next section. (See Remark 10) n χ ZeNm,p := u|u = 2 z o akl m,p and a ∈ C + f such that f ∈ W kl γN z k z̄ l 0≤k+l≤N −2 X (1.6) where χ ≡ χ(|z|) and we set that Ze1m,p ≡ Wγm,p . Note that ZeNm,p is a closed subspace in ZNm,p . 1 m,p we have Proposition 1.1.6. Let m > 1 + 2/p and ϕ ∈ ZDm,p N . Then for any u ∈ ZN u ◦ ϕ ∈ ZNm,p . Similarly, for u ∈ ZeNm,p we have u ◦ ϕ ∈ ZeNm,p . Proof. We can take ϕ = z + v such that v ∈ ZNm,p . First we show that if u = χ(z,z̄) z then u ◦ ϕ ∈ ZNm,p . We can write u◦ϕ= χ ◦ ϕ(z, z̄) 1 χ ◦ ϕ(z, z̄) . = z + v(z, z̄) z 1 + vz (1.7) For large enough R > 2 we have χ ◦ ϕ ≡ 1 in BRc since v is bounded by Proposition 1.5.1(ii). For |z| ≥ R, v/z ∈ ZNm,p (BRc ) and since v/z has zero constant term therefore by Remark 2 15 following Lemma 1.1.5, we have (1 + v/z)−1 ∈ ZNm,p (BRc ). Since ϕ is a C 1 -diffeomorphism of the complex plane C = R2 , the expression ϕ = z + v does not vanish on the support of χ ◦ ϕ. This shows that u ◦ ϕ = χ ◦ ϕ/(z + v) belongs to H m,p (BR+1 ) and so u ◦ ϕ ∈ ZNm,p by Lemma 1.1.3. Similarly it holds that if u = χ(z,z̄) z̄ that u ◦ ϕ ∈ ZNm,p and since ZNm,p is an algebra so this holds for any asymptotic part. Finally we show that if f ∈ Wγm,p then f ◦ ϕ ∈ Wγm,p . N N This follows directly from Corollary 6.1 in [13]. The case when u belongs to the auxiliary space ZeNm,p follows easily by the same arguments. we have that Remark 3. Formula (1.7) implies that for u = χ/z and for any ϕ ∈ ZDm,p N u◦ϕ= χ 1+f , z m,p f ∈ Z1,N +1 , (1.8) and a similar formula for u = χ/z̄. This together with Proposition 1.1.2 and Lemma 6.5 in [13] then shows that the following stronger statement holds: for any ϕ ∈ ZDm,p N and for any m,p m,p . , 0 ≤ n ≤ N + 1 we have that u ◦ ϕ ∈ Zn,N u ∈ Zn,N Following Proposition uses the previous result and the fact that ADm,p N is a topological group with same smoothness property as below. Proposition 1.1.7. Let m > 1 + 2/p then ZDm,p N is closed under composition of diffeomorm,p m,p phisms. Also the composition map (ϕ, ψ) 7→ ϕ ◦ ψ, ZDm,p N × ZD N → ZD N , is continuous and the associated map (ϕ, ψ) 7→ ϕ ◦ ψ, m,p × ZDm,p ZDm+1,p N → ZD N , N is C 1 -smooth. Remark 4. Similarly, for m > 1 + (2/p) the composition map (u, ψ) 7→ u ◦ ψ, ZNm,p × m,p ZDm,p N → ZN , is continuous and the map (u, ψ) 7→ u ◦ ψ, m,p ZNm+1,p × ZDm,p N → ZN , is C 1 -smooth. 16 Proof. Let ϕ = z +v and ψ = z +ω and ϕ, ψ ∈ ZDm,p N then ϕ◦ψ = ψ +v ◦ψ = z +ω +v ◦ψ = z + u where u = w + v ◦ ψ and last result shows that u is in ZNm,p . Since ϕ ◦ ψ ∈ Diff+1 (R2 ) we then conclude that ϕ ◦ ψ ∈ ZNm,p . The last statement of the proposition then follows from the analogous result for the asympm,p totic group ADm,p N (see [13, Proposition 5.1]) and the fact that ZD N is a closed submanifold in ADm,p N . is closed under inversion.This uses results that In rest of this section we show that ZDm,p N follow directly from related results in [13] and are added here. This and Proposition 1.1.7 m,p allow us to concldue that ZDm,p N is a topological subgroup of AD N . (Theorem 1.1.12 below). Proposition 1.1.8. Assume m > 2 + 2/p then for any ǫ > 0 small enough there exists a such that for any ϕ ∈ Uǫ we have ϕ−1 ∈ ZDm,p Uǫ ǫ-neighborhood of id in ZDm,p N . N can be connected by a Proposition 1.1.9. Assume m > 1 + 2/p, then any ϕ ∈ ZDm,p N such that ϕ0 − id has compact to a ϕ0 := γ(0) ∈ ZDm,p continuous path γ : [0, 1] → ZDm,p N N support and such that γ(1) = ϕ. then there exists a neighProposition 1.1.10. Assume m > 2 + 2/p and let ϕ ∈ ZDm,p N such that the left translation ψ 7→ Lϕ (ψ) := ϕ ◦ ψ is a C 1 borhood U of id ∈ ZDm−1,p N . diffeomorphism U → Lϕ (U ) ⊂ ZDm−1,p N we have that ϕ−1 ∈ ZDm,p Proposition 1.1.11. Let m > 3 + 2/p then for any ϕ ∈ ZDm,p N N m,p −1 → → ZDm,p and the map ϕ 7→ ϕ−1 , ZDm,p N , is continuous and the map ϕ 7→ ϕ , ZD N N is a C 1 -map. ZDm−1,p N m,p Proof. Let ϕ ∈ ZDm,p N . By Proposition 1.1.9, ϕ is connected by a path γ : [0, 1] → ZD N to γ(0) = ϕ0 = id +f such that f has compact support and γ(1) = ϕ. Note that γ . In particular, we see that f ∈ H m,p (C, C), and hence is also continuous into ZDm−1,p N ϕ0 ∈ Dm,p (R2 ) where Dm,p (R2 ) is the group of Sobolev type diffeomorphisms of R2 considered in [8]. Hence, ϕ−1 ∈ Dm,p (R2 ). Since, f has compact support we obtain that 0 17 m,p m,p ϕ−1 (C, C) has compact support. This implies that ϕ−1 0 = id +g where g ∈ H 0 ∈ ZD N with vanishing asymptotic part. an open neighAlong this path for each ϕt , by Proposition 1.1.8, there exists Vt ⊂ ZDm−1,p N and in which every diffeomorphism is invertible within ZDm−1,p borhood of id ∈ ZDm−1,p N N . Evsuch that by Proposition 1.1.10, Ut = Lϕt (Vt ) is open neighborhood of ϕt in ZDm−1,p N since we can write ϕ∗ = ϕ0 ◦ ψ with ψ ∈ Vt , so ery ϕ∗ in U0 is invertible in ZDm−1,p N . Now we can cover the path connecting ϕ0 to ϕ = ϕ1 by ϕ−1 = ψ −1 ◦ ϕ−1 ∈ ZDm−1,p 0 ∗ N open sets U0 , Ut1 , Ut2 , ..., U1 (ordered by sequence in which they intersect consecutively). By . For k = 1 choose induction on k, we show that any ϕ∗ ∈ Utk is invertible in ZDm−1,p N −1 ◦ ϕ̃−1 hence it is ϕ̃ ∈ U0 ∩ Ut1 then ϕ̃ = ϕt1 ◦ ψ for some ψ ∈ Vt1 , so we can write ϕ−1 t1 = ψ . . Continuing in this way we see that ϕ−1 ∈ ZDm−1,p invertible in ZDm−1,p N N −1 = id +v and Now we use a bootstrapping argument to show that ϕ−1 ∈ ZDm,p N . Let ϕ ϕ = id +u, then we can write the differentials of ϕ−1 ◦ ϕ = id as (Dv ◦ ϕ)(I + Du) = −Du. If det(I + Du)−1 ∈ ZNm−1,p then by writing Du ◦ ϕ = −Du · (I + Du)−1 and by ϕ−1 ∈ ZDm−1,p N m,p and so ϕ−1 ∈ ZDm,p it follows that Dv ∈ ZDm−1,p N . Now det(I + Du) = 1 + w, w ∈ ZN , N then by Lemma 1.1.5 we get that det(I + Du)−1 ∈ ZNm−1,p . The last statement of the proposition follows from the analogous results for the asymptotic m,p is a closed submanifold group ADm,p N (see [13, Proposition 5.2], [16]) and the fact that ZN in ADm,p N . Theorem 1.1.12. For integers m > 3 + 2/p and N ≥ 0, ZDm,p N is a topological group under m−1,p and → ZDm−1,p composition. Further the composition (ψ, ϕ) 7→ ψ ◦ ϕ, ZDm,p N N × ZD N m−1,p are C 1 -maps. the inverse map ϕ 7→ ϕ−1 , ZDm,p N → ZD N is a C 1 -map. → ZDm−1,p Remark 5. Similarly the map (u, ϕ) 7→ u ◦ ϕ, ZNm,p × ZDm−1,p N N Lemma 1.1.13. Let u ∈ C ([0, T ], ZNm,p ) for some T > 0, N ≥ 0 and m > 2 + 2/p. Then 18 there exists a unique solution ϕ ∈ C 1 ([0, T ], ZDm,p N ) of the equation ϕ̇ = u ◦ ϕ, ϕ t=0 = id . (1.9) → ZNm−1,p . By Theorem 1.1.12, Proof. Let us denote F (t, ϕ) = u(t) ◦ ϕ, [0, T ] × ZDm−1,p N composition is C 1 -map and so the second partial derivative → L(ZNm−1,p , ZNm−1,p ) D2 F : [0, T ] × ZDm−1,p N is continuous. Hence F is locally Lipschitz in ϕ for any t ∈ [0, T ], therefore for any t0 ∈ [0, T ], there exists ǫ0 > 0 such that there is a unique solution ϕ ∈ C 1 [t0 − ǫ0 , t0 + ǫ0 ], ZDm−1,p N of the equation ϕ̇ = u ◦ ϕ, ϕ t=t0 = id . , then since right translation with fixed ψ0 , ϕ 7→ ϕ ◦ ψ0 is C ∞ smooth so Now if ψ0 ∈ ZDm−1,p N for any t0 ∈ [0, T ], we have same ǫ0 > 0 such that ψ = ϕ◦ψ0 ∈ C 1 [t0 − ǫ0 , t0 + ǫ0 ], ZDm−1,p N is the unique solution of ψ̇ = u ◦ ψ, ψ t=t0 = ψ0 . we have a solution ϕ ∈ C 1 [0, T ], ZNm−1,p of By compactness of [0, T ] in [0, T ] × ZDm−1,p N equation (1.9). It remains to show that ϕ(t) ∈ ZDm,p N . Applying differential to equation (1.9) we get following equation. (dϕ)˙ = du ◦ ϕ · dϕ. Consider the system Ẏ = A(t)Y , Y t=0 = I, where2 A(t) = du(t)◦ϕ. Since A(t) is continuous by remark following Proposition 1.1.7, this linear system has a unique solution Y (t) ∈ ZNm−1,p +1 . m−1,p then ϕz = 1 + ∂z w gives Hence ϕz ∈ ZNm−1,p +1 . Now if we write ϕ = z + w, w ∈ ZN m−2,p m−1,p m−1,p we have ∂z w ∈ Z̃ N +1 . Therefore ∂z w ∈ Z̃ N +1 . ∂z w ∈ ZNm−1,p +1 . A priori from ϕ ∈ ZD N Now by Theorem 1.2.7 w ∈ ZNm,p hence ϕ ∈ C 1 ([0, T ], ZNm,p ). 2 I denotes the d × d identity matrix. 19 1.2 Cauchy operator in asymptotic spaces In [11] behavior of the Laplace operator on the weighted Sobolev spaces Wδm,p is studied. Depending on the weight, δ, the Laplace operator is shown to be injective or surjective with well described kernel and closed image. This essentially depends on the form of the fundamental solution kernel of the Laplacian and how it relates to the weights δ. This same approach can be applied to the Cauchy operator to obtain similar properties. To show injectivity of Cauchy operator, we will construct a bounded inverse that agrees with fundamental solution kernel on its image. In particular, we show that as in the case of the Laplace operator (cf. [11]), the Cauchy operator is an injective or surjective Fredholm operator for all but a discrete values of the weight δ ∈ R. In the Fredholm case we describe explicitly the kernel and the (closed) image of ∂z . Note that the fundamental solution of the Cauchy operator ∂z is K := 1/πz̄. The following estimate will play an important role in our analysis. Lemma 1.2.1. Let Ψ be a C ∞ -smooth complex valued function such that Ψ(z, z̄) = 1/z̄ for |z| ≥ 2 and Ψ(z, z̄) ≤ 1 for z ∈ C. For z, w ∈ C and any given l ∈ Z≥0 define, 1 K(z̄, w̄) := , z̄ − w̄ e l (z, z̄; w, w̄) := Ψ(z, z̄) K l X w̄k Ψ(z, z̄)k . k=0 Then, there exists a constant C ≡ Cl > 0 such that for any z, w ∈ C, e l (z, z̄; w, w̄) ≤ Cl K(z̄, w̄) − K where hwi = (1 + |w|2 ) 1/2 hwil+1 |z − w|hzil+1 (1.10) . Proof of Lemma 1.2.1. We will prove (1.10) with hzil+1 and hwil+1 replaced by 1+|z|l+1 and 1 + |w|l+1 respectively. These are equivalent by the equivalence of norms on Rd+1 . Assume 20 that w z ≥ 12 . If |z| ≥ 2 then l X l+1 l+1 e w̄k Ψ(z, z̄)k K − Kl (z̄ − w̄)(1 + |z| ) = (1 + |z| ) 1 − (z̄ − w̄)Ψ(z, z̄) (1.11) k=0 w̄ X w̄k = (1 + |z| ) 1 − 1 − z̄ k=0 z̄ k w̄ l+1 1 ≤ 1 + l+1 (1 + |w|l+1 ) = (1 + |z|l+1 ) z̄ |z| ≤ 2 1 + |w|l+1 l+1 l which proves (1.10) in the considered case. If |z| ≤ 2 then the right hand side of (1.11) is P estimated by 1+2l+1 1+3|w| lk=0 |wk | since Ψ(z, z̄) ≤ 1 and |z̄ − w̄| ≤ |z|+|w| ≤ 3|w|. This together with the estimate |w| l X k=0 |w|k = |w| 1 + |w|l+1 ≤ 1 + |w|l+1 1 + |w| then implies that the right hand side of (1.11) is bounded by 4 1 + 2l+1 1 + |w|l+1 . This proves (1.10) in the case when w z Now, assume that w z ≥ 21 . ≤ 12 . Then, K(z̄, w̄) = 1 z̄ P∞ w̄ k k=0 ( z̄ ) . If |z| ≥ 2 we have ∞ 1 X w̄ k 2 |w|l+1 e K(z̄, w̄) − Kl (z, z̄; w, w̄) = ≤ z̄ k=l+1 z̄ |z| |z|l+1 ≤ 3 2 |w|l+1 1 + |w|l+1 ≤ 3 2 |z − w| |z|l+1 |z − w|(1 + |z|l+1 ) where we used that 3 |z − w| ≤ |z| + |w| ≤ |z| and 2 1 + |w|l+1 |w|l+1 ≤ . |z|l+1 1 + |z|l+1 The second inequality in (1.12) follows from the estimate (in the case when (1.12) w z ≤ 12 ) 1 |w|l+1 1 + |z|l+1 − |z|l+1 1 + |w|l+1 = |w|l+1 − |z|l+1 = l+1 − 1 |z|l+1 ≤ 0. 2 21 w z If |z| ≤ 2 then we obtain from Ψ(z, z̄) ≤ 1 and e l (z, z̄; w, w̄) K(z̄, w̄) − K = ≤ 1 2 that ∞ l 1 X w̄ k 1 X w̄ k + 1 − z̄ k+1 Ψ(z, z̄)k+1 z̄ k=l+1 z̄ z̄ k=0 z̄ ∞ 1 X w ≤ |z| k=l+1 z k 1 X 1 k + 1 + 2k+1 |z| k=0 2 l 1 2 |w|l+1 + (4 + 2l) ≤ l+1 |z| |z| |z| (1 + |w|l+1 ) ≤ C(l) |z − w|(1 + |z|l+1 ) where, in order to conclude the final estimate, we used (1.12) and the fact that 1 ≤ 1 + 2l+1 1 + |w|l+1 1 + |z|l+1 for any |z| ≤ 2 and for any w ∈ C. This completes the proof of the lemma. Now we can directly use this estimate to construct a bounded map that inverts the Cauchy operator on its image. Also the image is identified as the null space of certain functionals on the range Lpδ+1 . We will denote by p′ the exponent conjugate to p that is 1 = 1 p + p1′ . In the ′ following (· , ·) : Lpδ × Lp−δ → C is the functional pairing Z (u, v) = u(x)v(x)d x . R2 ′ 1,p Using this pairing, we can extend ∂z to Lpδ into the dual (W−δ−1 )∗ by defining ∂z u(v) = −(u, ∂z v) ′ 1,p when v ∈ W−δ−1 and u ∈ Lpδ . By the weighted Hölder Inequality (Lemma 1.5.4) we can ′ 1,p write | − (u, ∂z v)| ≤ kukLp k∂z vkLp′ ≤ C kukLp kvkW 1,p′ which shows that ∂z u ∈ (W−δ−1 )∗ . δ −δ δ −δ−1 Following lemma shows that this map extends ∂z . Lemma 1.2.2. Assume that δ ∈ R, 1 < p < ∞ and p′ be exponent conjugate to p. If ′ 1,p u ∈ Wδ1,p and v ∈ W−δ−1 then (∂z u, v) = −(u, ∂z v). 22 (1.13) Proof. When u, v ∈ Cc∞ we can use integration by parts to see these bilinear forms on ′ 1,p Wδ1,p × W−δ−1 agree on a dense subset. Lemma 1.2.3. If δ + 2 p > 0 then ∂z : Wδ1,p → Lpδ+1 is injective. Remark 6. In fact, one can easily see by using the Fourier transform that the kernel of the Cauchy operator ∂z : S ′ → S ′ consists of all polynomials of z̄ with complex coefficients, ker ∂z = C[z̄]. Then, Lemma 1.2.3 follows since for δ + (2/p) > 0 we have that z̄ k ∈ / Lpδ for any k ≥ 0. Proof. Assume that u ∈ Lpδ and ∂z u = 0. Take an arbitrary test function ϕ ∈ Cc∞ . Since δ + (2/p) > 0 we conclude that (δ + 2)p′ > 2, p′ is conjugate exponent to p. This together ′ with the estimate K ∗ ϕ = O 1/hzi implies that K ∗ ϕ ∈ Lp−δ−1 . By using that K is the fundamental solution of ∂z we obtain from (1.13) that ϕ, u = ∂z (K ∗ ϕ) (u) = − K ∗ ϕ, ∂z u = 0 since ∂z u = 0. Since this holds for any ϕ ∈ Cc∞ we conclude that u = 0. Lemma 1.2.4. For any weight δ ∈ R we have that u ∈ Lpδ and ∂z u ∈ Lpδ+1 imply that u ∈ Wδ1,p . Remark 7. For α ∈ R, if u ∈ Lpδ then hziα u ∈ Lpδ−α follows directly from definition. Also ∂z (hziα u) = z̄ hziα−1 u hzi + hziα ∂z u. Then as z̄ hzi is bounded we get that hziα u ∈ Lpδ−α+1 . Proof of Lemma 1.2.4. By above Remark 7 above we can reduce to the case when 0 < δ + 2/p < 1. Therefore assume < δ + 2/p < 1 and let u ∈ Lpδ and f := ∂z u ∈ Lpδ+1 . We can take a sequence with compact support, (fk )k≥1 in Cc∞ such that fk → f in Lpδ+1 as k → ∞. Denote by uk the C ∞ functions defined as uk := K ∗ fk . 23 (1.14) Since K ∈ S ′ is the fundamental solution of the Cauchy operator ∂z we have that for any k ≥ 1, ∂ z uk = f k and ∂z̄ uk = (∂z̄ K) ∗ fk (1.15) in distributional sense in S ′ . We will first show by a direct computation that ∂z̄ K = −p.v. 1 ∈ S′ 2 πz̄ (1.16) where p.v. denotes the Cauchy principal value. To see this let ϕ ∈ Cc∞ be any test function and let R > 0 sufficiently large, then we have Z Z 1 1 1 1 ∂z̄ K (ϕ) = −(K, ∂z̄ ϕ) = ϕz̄ dz ∧ dz̄ = lim ϕz̄ dz ∧ dz̄ 2πi C z̄ 2πi r→0+ r≤|z|≤R z̄ Z Z 1 1 1 1 ∂z̄ ϕ dz ∧ dz̄ lim ϕ dz ∧ dz̄ + lim = 2πi r→0+ r≤|z|≤R z̄ 2πi r→0+ r≤|z|≤R z̄ 2 Z 1 1 = p.v. ϕ dz ∧ dz̄. 2 2i C πz̄ Here we used Stokes’ theorem to obtain lim r→0+ Z Z I 1 1 1 d ϕ dz = lim ∂z̄ ϕ dz ∧ dz̄ = − lim ϕ dz r→0+ r≤|z|≤R r→0+ |z|=r z̄ z̄ z̄ r≤|z|≤R Z 2π = i lim e2iθ ϕ(reiθ ) dθ = 0. r→0+ 0 By Lemma 1 in [11] (cf. Theorem B* in [20] and Lemma 2.4 in [10]) for 0 < δ + (2/p) < 1 the convolution with the fundamental solution K of the Cauchy operator ∂z extends to a bounded linear map Lpδ+1 → Lpδ . Moreover, by Theorem 3 in [21, Ch. II, §4], Theorem 1 in [19], and (1.16), for 0 < δ + (2/p) < 2 the convolution with ∂z̄ K extends to a bounded linear map Lpδ+1 → Lpδ+1 . (Note that the cancellation condition required in Theorem 3 in [21, Ch. R 2π II,§4] holds since 1/z̄ 2 = e2iθ /r2 and 0 e2iθ dθ = 0.) Therefore in (1.14) since fk → f in Lpδ+1 we can pass to the limit to get ũ := K ∗ f = lim K ∗ fk ∈ Lpδ , k→∞ 24 which is to say that uk → ũ in Lpδ and hence in S ′ (see Remark 6) where the limit exists in Lpδ . Similarly, by passing to the limit in S ′ in (1.15) we obtain that ∂z ũ = f ∈ Lpδ+1 and ∂z̄ ũ = (∂z̄ K) ∗ f ∈ Lpδ+1 . This implies that ũ ∈ Wδ1,p . Since ∂z ũ = f = ∂z u we obtain from the Lemma 1.2.3 that u = ũ. Hence, u ∈ Wδ1,p . This completes the proof of the lemma. 1,p Remark 8. From the proof it also follows that ∂z u := f ∈ Hδ+1 ⊂ Lpδ+1 then ∂z̄ ∂z−1 f = ∂z−1 ∂z̄ f that is ∂z̄ , ∂z−1 commute. Proposition 1.2.5. For 0 < δ + (2/p) < 1 the map ∂z : Wδ1,p → Lpδ+1 is an isomorphism of Banach spaces. Proof of Proposition 1.2.5. By Lemma 1.2.4 this map is injective so it only remains to show that it is surjective. For this let us take any f ∈ Lpδ+1 and define u := K ∗ f . Since for 0 < δ + (2/p) < 1 the convolution with K extends to a bounded map Lpδ+1 → Lpδ we obtain that u ∈ Lpδ . This follows by Lemma 1 in [11] (cf. Theorem B* in [20] and Lemma 2.4 in [10]). Furthermore since K is the fundamental solution of ∂z , we have ∂z u = f ∈ Lpδ+1 . To see this let (fk )k≥1 ∈ Cc∞ be a sequence such that fk → f in Lpδ+1 then uk := K ∗ fk gives ∂z uk = fk , finally since fk → f implies uk → u and as u 7→ ∂z u is a bounded map so ∂z uk → ∂z u. But also ∂z u := fk → f . Hence ∂z u = f and so we have that the map is surjective and therefore by open mapping theorem it is also an isomorphism. Proposition 1.2.6. For l + 1 < δ + (2/p) < l + 2 and l ∈ Z≥0 the map ∂z : Wδ1,p → Lpδ+1 is injective with closed image in Lpδ+1 given by Rl := f ∈ Lpδ+1 f, z̄ k = 0 for 0 ≤ k ≤ l . Rl has (complex) co-dimension l + 1. Proof of Proposition 1.2.6. Let us fix a given l ∈ Z≥0 . For l ≥ 0 injectivity of this map follows from Lemma 1.2.3. Next For z, w ∈ C denote e l (z, z̄; w, w̄). Fl (z, z̄; w, w̄) := K(z̄, w̄) − K 25 (1.17) By Lemma 1.2.1 we have that Fl (z, z̄; w, w̄) ≤ Cl hwil+1 . |z − w|hzil+1 (1.18) Fl (z, z̄; w, w̄)u(w, w̄) dw ∧ dw̄, (1.19) We will show that the following map Fl : Cc∞ ∞ →C , 1 Fl u (z, z̄) := − 2πi Z C associated to the kernel Fl /π extends to a bounded linear map Lpδ+1 → Lpδ . In order to see this note that (1.19) can be decomposed as Fl = (M−l−1 ) ◦ F ◦ Ml+1 (1.20) where for any given weight µ ∈ R, Ml+1 : Lpµ → Lpµ−(l+1) , f 7→ hzil+1 f, is a linear isomorphism and F : Cc∞ → C ∞ is of the form (1.19) with kernel F (z, z̄; w, w̄) := F−1 (z, z̄; w, w̄) satisfying, by the estimate (1.18), the inequality F (z, z̄; w, w̄) ≤ C |z − w| with a constant C > 0 depending only on l ≥ 0. Since δ satisfies 0 < δ − (l + 1) + (2/p) < 1 then the above estimate for F (z, z̄; w, w̄) implies that the map F : Cc∞ → C ∞ extends to a bounded linear map Lp[δ−(l+1)]+1 → Lpδ−(l+1) as discussed in the proof of Lemma 1.2.4. Therefore from the decomposition (1.20) it follows that the map (1.19) extends to a bounded linear map Fl : Lpδ+1 → Lpδ . (1.21) Now for any f ∈ Lpδ+1 we can write from equation (1.17) that e l (f ) + Fl (f ), K∗f =K 26 (1.22) e l : Lp → C ∞ is the integral operator with kernel K e l /π, where K δ+1 l X e l (f ) := Ψ f, z̄ k Ψk . K π k=0 Note that for l + 1 < δ + (2/p) < l + 2 we have that (δ + 1 − l)p′ > 2 where (1.23) 1 p + p1′ = 1. This implies that ′ ′ z̄ k ∈ Lp−(δ+1)+(l−k) ⊆ Lp−(δ+1) (1.24) for any 0 ≤ k ≤ l. Thus for any 0 ≤ k ≤ l, the map u 7→ u, z̄ k , Lpδ+1 → C appearing in (1.23) is well defined and continuous map. Now we are ready to show that ∂z (Wδ1,p := Rl ⊂ Lpδ . Take f ∈ Lpδ+1 ∩ Rl . By (1.23) and e ) = 0. This, together with (1.22) and (1.21) then the fact that f ∈ Rl we obtain that K(f implies that K ∗ f = Fl (f ) ∈ Lpδ . Since K is the fundamental solution of ∂z we also have that ∂z K ∗ f = f ∈ Lpδ+1 . Then, by Lemma 1.2.4 we obtain that K ∗ f ∈ Wδ1,p . This proves that Rl is contained in the image of ∂z : Wδ1,p → Lpδ+1 . On the other side, it follows from (1.24) that for u ∈ Wδ1,p and for any 0 ≤ k ≤ l, ∂z u, z̄ k = − u, ∂z (z̄ k ) = 0. Hence, Rl is the image of ∂z : Wδ1,p → Lpδ+1 . This completes the proof of the proposition. Theorem 1.2.7. Assume that m ∈ Z≥0 . Then m,p is an isomorphism of Banach (i) For 0 < δ + (2/p) < 1 the map ∂z : Wδm+1,p → Wδ+1 spaces. (ii) For l + 1 < δ + (2/p) < l + 2, l ∈ Z≥0 , and any regularity exponent m ≥ 0 the map m,p m,p m,p f, z̄ k = Rl := f ∈ Wδ+1 is injective with closed image in Wδ+1 ∂z : Wδm+1,p → Wδ+1 0 for 0 ≤ k ≤ l . Rl has (complex) co-dimension l + 1. 27 Remark 9. Theorem 1.2.7 also holds with ∂z replaced by ∂z̄ and Rl replaced by m,p Rl := f ∈ Wδ+1 f, z k = 0 for 0 ≤ k ≤ l . Proof Theorem 1.2.7. We will prove the theorem by induction in the regularity exponent m ≥ 0. For m = 0 the statement follows from Proposition 1.2.5. Now, take m ≥ 1 and m,p m−1,p assume that the statement holds for m replaced by m − 1 ≥ 0. Take f ∈ Wδ+1 ⊆ Wδ+1 ⊆ Lpδ+1 . By the induction hypothesis there exists u ∈ Wδm,p such that ∂z u = f . Then for any multi-index α ∈ Z2≥0 with |α| ≤ m we have that ∂ α f ∈ Lpδ+|α|+1 . Hence, ∂ α u ∈ Lpδ+|α| and ∂z ∂ α u = ∂ α ∂z u = ∂zα f ∈ Lpδ+|α|+1 . 1,p for any |α| ≤ m. This Then, we apply Lemma 1.2.4 to conclude that that ∂ α u ∈ Wδ+|α| m,p is onto. The injectivity of this map implies that u ∈ Wδm+1,p . Hence, ∂z : Wδm+1,p → Wδ+1 follows from Lemma 1.2.3. The first statement of the theorem then follows from the open mapping theorem. The same arguments together with Proposition 1.2.6 prove the second statement of theorem. Recall that for m, N ≥ 0 the weight γN of the remainder space Wγm,p of the asymptotic space N Zγm,p (cf. (1.2)) are taken of the form γN = N + γ0 where 0 < γ0 + (2/p) < 1. In particular, N we see that N < γN + (2/p) < N + 1. (1.25) Then, by Theorem 1.2.7 above, the map ∂z : Wγm+1,p → Wγm,p is injective with closed image N +1 N in Wγm,p N +1 = f ∈ Wγm,p ∂z Wγm+1,p N +1 N f, z̄ k = 0 for 0 ≤ k ≤ N − 1 where we assume that for N = 0 the set on the right hand side is equal to Wγm,p , and hence, N +1 the map is an isomorphism when N = 0. In fact, our method allows to construct a right inverse of the map ∂z : Wγm+1,p → Wγm,p that is defined on the whole of Wγm,p and takes N +1 N +1 N values in the asymptotic space ZNm+1,p . More specifically, we have the following important proposition. 28 Proposition 1.2.8. For any integers m, N ≥ 0 there exists a bounded map L : Wγm,p → N +1 L(f ) = f and we have that ∂ ZNm+1,p such that for any f ∈ Wγm,p z +1 N N 1 χX L(f ) = f, z̄ k−1 k + R(f ) π k=1 z̄ → Wγm+1,p where R : Wγm,p is a bounded linear map and for N = 0 we omit the summation N +1 N term in the formula. Proof of Proposition 1.2.8. The case when N = 0 is trivial since the map ∂z : Wγm+1,p → N Wγm,p is an isomorphism. For N ≥ 1 consider the bounded linear map P : Wγm,p → Wγm,p , N +1 N +1 N +1 N χz 1X f, z̄ k−1 k , P(f ) := f − π k=1 z̄ (1.26) where χz ≡ ∂z χ ∈ Cc∞ . This maps Wγm,p into RN ⊂ Wγm,p , the image of Wγm+1,p under ∂z . N +1 N +1 N To see this note that for any k ∈ Z we have by Stokes’ theorem that Z I χ 1 1 dz̄ k χz , 1/z̄ = − d k dz̄ = − = πδk1 2i |z|≤R z̄ 2i |z|=R z̄ k (1.27) where R > 0 is chosen sufficiently large. This, together with (1.26) then implies that P(f ), z̄ k = 0 for any 0 ≤ k ≤ N − 1. Hence since P(f ) = f for f ∈ RN := ∂z Wγm+1,p by N Theorem 1.2.7, we get m+1,p P Wγm,p = ∂ . W z +1 γ N N → Wγm,p is injective with closed image we Also by Theorem 1.2.7, the map ∂z : Wγm+1,p N +1 N → obtain from the open mapping theorem that there is a bounded linear map ı : ∂z Wγm+1,p N . In particular, we have that Wγm+1,p such that ∂z ı(f ) = f for any f ∈ ∂z Wγm+1,p N N . This, together with (1.26) then gives that ∂z ı ◦ P (f ) = P(f ) for any f ∈ Wγm,p N +1 ! N χX k−1 1 =f f, z̄ ∂z ı ◦ P (f ) + π k=1 z̄ k for any f ∈ Wγm,p . By setting R(f ) := ı ◦ P (f ) and N +1 N 1 χX L(f ) := f, z̄ k−1 k + ı ◦ P (f ), π k=1 z̄ we conclude the proof of the proposition. 29 f ∈ Wγm,p , N +1 Recall that (see (1.6), (1.4)) and n χ m,p e ZN ≡ u|u = 2 z o akl m,p + f such that, f ∈ WγN and akl ∈ C z k z̄ l 0≤k+l≤N −2 X n X m,p ≡ χ Zn,N n≤k+l≤N o akl m,p f ∈ W and a ∈ C + f kl γN z k z̄ l (1.28) m,p em,p ≡ W m,p . We will next where 0 ≤ n ≤ N + 1 and we set that ZNm,p γ1 +1,N ≡ WγN and Z1 study the properties of ∂z acting on scale of spaces ZNm,p . m+1,p → ZeNm,p Theorem 1.2.9. For any m > 1 + (2/p) and N ≥ 0 we have that ∂z : Z1,N +1 is an isomorphism of Banach spaces. Remark 10. Note that we have ∂z̄ = τ ◦ ∂z ◦ τ where τ : u 7→ ū is the operation of taking the complex conjugate of a function. Using this we can obtain similar statement for the Cauchy operator ∂z̄ if we replace ZeNm,p +1 by the space nχ m,p e ZN +1 = z̄ 2 o akl m,p + f f ∈ WγN +1 and akl ∈ C . z k z̄ l 0≤k+l≤N −1 X Furthermore we have that ∂z̄−1 = τ ◦ ∂z−1 ◦ τ , and similarly ∂z−1 = τ ◦ ∂z̄−1 ◦ τ . Also note that ∂z−1 and ∂z̄−1 commute since the Cauchy operators ∂z and ∂z̄ commute. m+1,p Proof of Theorem 1.2.9. First we show that ∂z maps Z1,N into ZeNm,p +1 . For any k ≥ 1 and l ≥ 0 we have χ 1 χ = −k +g (1.29) z k z̄ l z 2 z k−1 z̄ l . Also since where g := χz /z k z̄ l ∈ Cc∞ ⊆ Wγm,p . Similarly, ∂z χ/z̄ l = χz /z̄ l ∈ Cc∞ ⊆ Wγm,p N +1 N +1 m+1,p ⊆ ZeNm,p , we have ∂z Z1,N ∂z : Wγm+1,p → Wγm,p +1 . Hence the map N +1 N ∂z m+1,p → ZeNm,p ∂z : Z1,N +1 (1.30) m+1,p ⊆ Wγ1,p where γ0 + (2/p) > 0. Hence, the map is well defined. Further, note that Z1,N 0 (1.30) is injective by Lemma 1.2.3. Let us now prove that the map is onto. Take u ∈ ZeNm,p +1 , 30 then we can write u in general form as u= χ z2 X akl + f, z k z̄ l 0≤k+l≤N −1 f ∈ Wγm,p . N +1 From (1.29) we obtain that u = ∂z X 0≤k+l≤N −1 = ∂z X 0≤k+l≤N −1 akl χ − k + 1 z k+1 z̄ l ! + f − g̃ akl χ + L f − g̃ − k + 1 z k+1 z̄ l ! m+1,p is the map in Proposition 1.2.8. Note where g̃ ∈ Cc∞ ⊆ Wγm,p and L : Wγm,p → Z1,N N +1 N +1 P m+1,p χ akl . This shows that (1.30) is a bijective map onto ∈ Z1,N that 0≤k+l≤N −1 − k+1 z k+1 z̄ l its image. The statement of the theorem then follows from the open mapping theorem. Proposition 1.2.10. For m > 2 + 2/p, the map m−1,p m,p → Z̃ N +1 (ϕ, v) 7→ Rϕ ◦ ∂z ◦ Rϕ−1 (v), ZDm,p N × ZN (1.31) is real analytic. Proof. Let v ∈ ZNm,p then we can write, with ψ = ϕ−1 . First observe that Rϕ ◦ ∂z ◦ Rϕ−1 (v) ∈ m−1,p Z̃ N +1 follows from Theorem 1.2.9 and Lemma 1.1.6. By a direct computation we show that this operator does not involve composition with ϕ. ∂z ◦ Rϕ−1 (v) = vz ◦ ψ · ψz + vz̄ ◦ ψ · ψ̄z Rϕ ◦ ∂z ◦ Rϕ−1 (v) = vz · ψz ◦ ϕ + vz̄ · ψ̄z ◦ ϕ By differentiating with z and z̄ the expression ψ ◦ ϕ = z we get ψz ◦ ϕ · ϕz + ψz̄ ◦ ϕ · ϕ̄z = 1 ψz ◦ ϕ · ϕz̄ + ψz̄ ◦ ϕ · ϕ̄z̄ = 0. 31 Solving this system we get ϕ̄z̄ D ϕ̄z ψ̄z ◦ ϕ = − D ψz ◦ ϕ = where D = ϕz ϕ̄z̄ − ϕz̄ ϕ̄z . This gives Rϕ ◦ ∂z ◦ Rϕ−1 (v) = vz ϕ̄z̄ − vz̄ ϕ̄z . D We can write D = 1 + uz ūz̄ − uz̄ ūz + uz + ūz̄ := 1 + w, w ∈ ZNm−1,p . Since |D| > 0 and w has no constant term in asymptotic expansion so by Lemma 1.1.5, ϕ 7→ 1 , D m,p ZDm,p N → ZN is real analytic in ϕ. Since v 7→ vz is analytic so Rϕ ◦ ∂z ◦ Rϕ−1 (v) is analytic in (ϕ, v). Proposition 1.2.11. For m > 3 + (2/p) the map (ϕ, v) 7→ Rϕ ◦ ∂z−1 ◦ Rϕ−1 (v), m,p em−1,p ZDm,p N × ZN +1 → Z1,N , (1.32) is real analytic. Here ∂z−1 denotes the inverse of the Cauchy operator given be Theorem 1.2.9. Proof of Proposition 1.2.11. First, note that by Lemma 1.1.6 the map v 7→ Rϕ ◦ ∂z ◦ m,p → ZeNm−1,p Rϕ−1 (v), Z1,N +1 , is a bounded linear map. By Theorem 1.2.9, this map is a linear isomorphism with inverse Rϕ ◦ ∂z ◦ Rϕ−1 −1 = Rϕ ◦ ∂z−1 ◦ Rϕ−1 . (1.33) m,p em−1,p where GL(X, Y ) denotes the Banach , ZN +1 In particular, Rϕ ◦ ∂z ◦ Rϕ−1 ∈ GL Z1,N manifold of linear isomorphisms between two Banach spaces X and Y . On the other side, since the map (1.31) in Lemma 1.2.10 is real analytic, we obtain that its first partial derivative G ≡ D2 F with respect to the second argument m,p em−1,p G : ZDm,p , N → GL Z1,N , ZN +1 ϕ 7→ Rϕ ◦ ∂z ◦ Rϕ−1 , is also real analytic. Note in addition that the map m,p m,p em−1,p , ZN +1 → GL ZeNm−1,p ı : GL Z1,N +1 , Z1,N , 32 L 7→ L−1 , (1.34) is analytic. It then follows from (1.33) and (1.34) that the map em−1,p m,p ZDm,p N → GL ZN +1 , Z1,N , ı ◦ G : ϕ 7→ Rϕ ◦ ∂z−1 ◦ Rϕ−1 , is real analytic as it is a composition of real analytic maps. This implies that the map (1.32) is also real analytic. m−1,p In next section we will see that another operator Q : ZNm,p → Z̃ N +1 defined by Q(a) = (az )2 + az̄ āz plays important role in studying the well-posedness of Euler Equation in asymptotic spaces ZNm,p . m,p → Proposition 1.2.12. For m > 2+2/p, the map (ϕ, v) 7→ Rϕ ◦Q◦Rϕ−1 (v), ZDm,p N ×ZN m−1,p Z̃ N +1 is real analytic. m−1,p m−1,p is an Proof. First we show that Q maps into Z̃ N +1 . However we have Z̃ N +1 ⊂ ZNm−1,p +1 m−1,p m−1,p is closed under conjugation so az and āz ∈ Z̃ N +1 hence Q(a) ∈ Z̃ N +1 . ideal and ZNm−1,p +1 To show analyticity, as in previous proposition we see that first term of Rϕ ◦ Q ◦ Rϕ−1 (v) is Rϕ ∂z (v ◦ ϕ−1 ) 2 = (Rϕ ◦ ∂z ◦ Rϕ−1 (v))2 = (vz ϕ̄z̄ − vz̄ ϕ̄z )2 . D2 and this is real analytic by last proposition. Similarly the second term is Rϕ ∂z̄ (v ◦ ϕ−1 ) ∂z (v̄ ◦ ϕ−1 ) = (Rϕ ∂z Rϕ−1 (v̄)) (Rϕ ∂z Rϕ−1 (v̄)) which is again real analytic by last proposition. 1.3 The Euler equation on ZDm,p N In this section we convert the Euler equation from ZNm,p to a dynamical system on the smooth m,p Banach manifold ZDm,p N × ZN . In Appendix 2.9 it is shown that the Euler’s equation in complex plane can be written using Cauchy operator as u̇ + (uuz + ūuz̄ ) = −2∂z̄ p, u t=0 = u0 33 div u = uz + ūz̄ = 0 (1.35) where p : R2 → R is the scalar pressure. For uniqueness of solutions of (1.35) we will require m+1,p . The operator ∂z−1 in equation (1.36) below is the operator defined in that p ∈ Z1,N Theorem 1.2.9. This operator is defined on specific spaces, namely ZeNm−1,p +1 . Therefore to see that in (1.36) below the right hand side is well defined note that for u ∈ ZNm,p we have that m−1,p m−1,p em−1,p uz , ūz ∈ ZeNm−1,p +1 and uz̄ ∈ ZN +1 . Since ZN +1 is an ideal in the Banach algebra ZN +1 we conclude that (see Proposition 1.2.12) Q(u) ≡ uz uz + ūz uz̄ ∈ ZeNm−1,p +1 . Proposition 1.3.1. Assume that m > 2 + 2/p and u ∈ C ([0, T ], ZNm,p ) ∩ C 1 [0, T ], ZNm−1,p . m+1,p , t ∈ [0, T ], if and only if u satisfies Then u satisfies (1.35) for some real-valued p(t) ∈ Z1,N u̇ + (uuz + ūuz̄ ) = ∂z−1 (uz )2 + |uz̄ |2 with divergence free initial condition u0 = u t=0 (1.36) , div u0 = 0. m+1,p . Apply divergence Proof. Assume u satisfies equation (1.35) for some real-valued p ∈ Z1,N ˙ to equation (1.35) and use div u̇ = 0 and (since m > 2 + 2/p we can use div u̇ = (div u)) div (−2∂z̄ p) = −2(∂z ∂z̄ p + ∂z̄ ∂z̄ p) = −4∂z ∂z̄ p. Combine this with expression div (uuz + ūuz̄ ) = 2(uz )2 + 2|uz̄ |2 = 2Q(u) (1.37) (see equation (47)) we get equation −2∂z (pz̄ ) = (uz )2 + |uz̄ |2 . m,p m,p m+1,p we have that pz̄ ∈ Z2,N Since p ∈ Z1,N +1 ⊂ Z1,N and therefore by Theorem 1.2.9 we get that −2pz̄ = ∂z−1 (uz )2 + |uz̄ |2 showing that u satisfies equation (1.36). For the converse statement assume that u ∈ C [0, T ], ZNm,p ∩ C 1 [0, T ], ZNm−1,p satisfies 34 equation (1.36) with divergence free initial data div u0 = 0. We apply divergence to equation (1.36) and after simplification we get (div u)˙ + u ∂z (div u) + ū ∂z̄ (div u) = 0 (1.38) Let ϕ be the solution of equation (1.9), which exists by Lemma 1.1.13 since m > 2 + 2/p. m−1,p ∩ C 1 [0, T ], ZNm−2,p we conclude Since ϕ ∈ C 1 ([0, T ], ZDm,p N ) and div u ∈ C [0, T ], ZN that div u ∈ C 1 [0, T ] × C, R and ϕ ∈ C 1 ([0, T ] × C, C). This together with (1.38) implies that for any given (x, y) ∈ R2 and for any t ∈ [0, T ], (div u) ◦ ϕ ˙ = 0, (1.39) Hence for any t ∈ [0, T ], div u(t) = div u0 = 0 for any t ∈ [0, T ]. By Lemma 1.3.2 below m+1,p such that −2pz̄ = ∂z−1 (uz )2 + |uz̄ |2 for any t ∈ [0, T ]. there exists p ∈ C [0, T ], Z1,N This completes the proof of the proposition. Remark 11. Since u is divergence free we have that div u = uz + ūz̄ = 0 and by formula (48) in Appendix 2.9, 1 (1.40) uz − ūz̄ = uz . 2 . By applying the Cauchy operator ∂z to (1.36) we obtain that ω = − uuz + ūuz̄ z + (uz )2 + uz̄ ūz = − u ωz + ū ωz̄ that is the 2d Euler equation in vorticity form, ω := i curl ~v ≡ . ω + u ωz + ū ωz̄ = 0. (1.41) This and the arguments used to prove the analogous formula (1.39) then implies that ∂t ω(t)◦ ϕ(t) = 0 for any t ∈ [0, T ]. In particular, for any t ∈ [0, T ] we have that ω(t) = ω0 ◦ ψ(t) (1.42) , and ω0 = ∂z u0 ∈ ZeNm−1,p where ψ(t) := ϕ(t)−1 , ϕ ∈ C 1 [0, T ], ZDm,p +1 . We will use this N preservation of vorticity in next section to prove global existence of solutions. 35 In the proof of Proposition 1.3.1 we used the following lemma. Lemma 1.3.2. Assume that m > 2 + (2/p). For any divergence free u ∈ ZNm,p there m+1,p such that −2pzz̄ = (uz )2 + |uz̄ |2 . Moreover, we have that exists a real valued p ∈ Z1,N −2pz̄ = ∂z−1 (uz )2 + |uz̄ |2 and −2p = ∂z̄−1 ∂z−1 (uz )2 + |uz̄ |2 . Proof of Lemma 1.3.2. Take u ∈ ZNm,p such that div u = uz + ūz̄ = 0. Then we have Q(u) ≡ (uz )2 + uz̄ ūz = −uz (uz ) + ūz (ūz ). (1.43) We will first consider the case when N ≥ 1. Since uz , ūz ∈ ZeNm−1,p +1 we obtain from Proposition m−1,p 1.1.1, Proposition 1.1.2 and (1.43) that Q(u) ∈ Z4,N +3 and Q(u) = χ 2 z z̄ 2 X akl + f, z k z̄ l 0≤k+l≤N −1 f ∈ Wγm−1,p . N +3 (1.44) m,p In particular, we see that Q(u) ∈ Wγm−1,p . Then, by Proposition 1.2.8, ∂z−1 Q(u) ∈ Z1,2 and 3 ∂z−1 Q(u) = χ χi 1h Q(u), 1 + Q(u), z̄ 2 + g, π z̄ z̄ g ∈ Wγm,p . 2 Since Q(u), 1 = Q(u), z̄ = 0 by Lemma 1.3.3 below we conclude that . ∂z−1 Q(u) ∈ Wγm,p 2 (1.45) Note that when considering ∂ −1 Q(u), we have to view Q(u) ∈ Wγm−1,p instead of being in 3 m,p Z1,N +2 , since in computing the momenta Q(u), 1 and Q(u), z̄ the function Q(u) has to be in the remainder spaces Wγm,p (see Theorem 1.2.9). On the other side, by Theorem 1.2.9 N m,p m−1,p −1 and the fact that Q(u) ∈ Z4,N +3 we have that ∂z Q(u) ∈ Z1,N +2 . Moreover, it follows from (1.44) and Proposition 1.2.8 (cf. the proof of Theorem 1.2.9) that ! X X b b b χ χ b 1 kl k 2 ∂z−1 Q(u) = χ + χ 2 + + + h, h ∈ Wγm,p N +2 z̄ z̄ z z̄ 2 0≤k+l≤N −1 z k z̄ l z̄ 3 3≤k≤N +2 z̄ k−3 where bk := π1 f, z̄ k−1 . By comparing this with (1.45) we obtain that b1 = b2 = 0, and hence ∂z−1 Q(u) = χ z z̄ 2 X bkl χ + z k z̄ l z̄ 3 0≤k+l≤N −1 X bk z̄ k−3 3≤k≤N +2 36 ! + h, h ∈ Wγm,p . N +2 (1.46) In particular, this implies that ∂z−1 Q(u) m,p e ∈ ZN +2 . Hence, by Theorem 1.2.9 and Remark 10, we obtain that 1 m+1,p m+1,p p := − ∂z̄−1 ∂z−1 Q(u) ∈ Z1,N . +1 ⊆ Z1,N 2 Let us now consider the case when N = 0. Then, u ∈ Z0m,p and u = a00 + f , f ∈ Wγm,p . This 0 implies that uz , ūz ∈ Wγm−1,p and hence by Proposition 1.1.1 and (1.43) we obtain that 1 m−1,p ⊆ Wγm−1,p Q(u) ∈ W2γ 2 1 +(2/p) where we used that 1 < γ1 + (2/p) < 2. This, together with Proposition 1.2.8 and Lemma 1.3.3 below then implies that . ∂z−1 Q(u) ∈ Wγm,p 1 Finally, by applying Theorem 1.2.7 (i) we obtain that 1 m+1,p p := − ∂z̄−1 ∂z−1 Q(u) ∈ Wγm+1,p ≡ Z1,0 . 0 2 Let us now prove that p is real valued. Recall that τ : u 7→ ū is the operation of taking the complex conjugate of a function. By Remark 10 we then have −2p = τ ◦ ∂z̄−1 ∂z−1 Q(u) = τ ◦ ∂z̄−1 ◦ τ ◦ τ ◦ ∂z−1 τ ◦ τ Q(u) = ∂z̄−1 ∂z−1 Q(u) = −2p where we used that Q(u) is real valued by (1.43) and that ∂z−1 = τ ◦ ∂z̄−1 ◦ τ and similarly ∂z̄−1 = τ ◦ ∂z−1 ◦ τ . This completes the proof of the proposition. Lemma 1.3.3. Assume that m > 2 + (2/p) and N ≥ 1. If u ∈ ZNm,p is divergence free then Q(u), 1 = 0 and Q(u), z̄ = 0. Proof of Lemma 1.3.3. Since Q(u) does not involve the constant asymptotic term of u ∈ m,p . Then, the lemma follows ZNm,p we will assume without loss of generality that u ∈ Z1,N 37 easily from the Stokes’ theorem and the relation (1.37). In fact, with f := uuz + ūuz̄ we obtain from (1.37) that 1 2 Q(u), 1 = − 2i Z C div f dz ∧ dz̄ = − Im Z C fz dz ∧ dz̄ = − Im lim R→∞ I f dz̄ |z|=R where we used that f = O 1/|z|3 and Q(u) = O 1/|z|4 as |z| → ∞. Similarly, 1 2 Q(u), z̄ = − 2i Z 1 z̄ div f dz ∧ dz̄ = − 2i C Z 1 z̄(fz + f¯z̄ ) dz ∧ dz̄ = 2i C Z C f¯ dz ∧ dz̄ =0 (1.47) where we used as above the Stokes’ theorem to conclude that Z and Z C C z̄fz dz ∧ dz̄ = z̄ f¯z̄ dz ∧ dz̄ = Z C Z z̄f C z dz ∧ dz̄ = 0 (z̄ f¯)z̄ − f¯ dz ∧ dz̄ = − Z C f¯ dz ∧ dz̄. On the other side, since div u = uz + ūū = 0, we have 1 1 f = uuz + ūuz̄ = (u2 )z + (uū)z̄ − uūz̄ = (u2 )z + (uū)z̄ + uuz = (u2 )z + (uū)z̄ . 2 2 Since u2 , uū = O 1/|z|2 as |z| → ∞ we then conclude again from Stokes’ theorem that R f¯ dz ∧ dz̄ = 0. This together with (1.47) then implies that Q(u), z̄ = 0. C Proposition 1.3.4. Let m > 2 + 2/p and u0 ∈ ZNm,p . There is a bijection between solutions of the dynamical system   . .  (ϕ, v) = v, Rϕ ◦ ∂z−1 ◦ Q ◦ Rϕ−1 (v)   (ϕ, v)|t=0 = (id, u0 ) (1.48) m,p m,p and solutions of (1.36). More specifically, (ϕ, v) ∈ C 1 ([0, T ], ZDm,p on ZDm,p N ×ZN ) N ×ZN is a solution of (1.48) if and only if u = v ◦ ϕ−1 ∈ C ([0, T ], ZNm,p ) ∩ C 1 [0, T ], ZNm−1,p is a solution of (1.36). Proof. Assume that (ϕ, v) is a solution of the dynamical system (1.48). By Theorem 1.1.12, is continuous and since v(t) and ϕ(t) is continuous so u = → ZDm,p ϕ 7→ ϕ−1 , ZDm,p N N 38 m−1,p v ◦ ϕ−1 ∈ C([0, T ], ZNm,p ). Similarly by Theorem 1.1.12, since ϕ 7→ ϕ−1 , ZDm,p N → ZD N is C 1 -map, and v(t) and ϕ(t) are C 1 -maps so by Theorem 1.1.12, u ∈ C 1 ([0, T ], ZNm−1,p ). A direct computation shows that with v = u ◦ ϕ v̇ = (u ◦ ϕ)˙ = u̇ ◦ ϕ + uz ◦ ϕϕ̇ + uz̄ ◦ ϕϕ̄˙ = (u̇ + uz u + uz̄ ū) ◦ ϕ. (1.49) Now from (1.49) u̇ + uz u + uz̄ ū = Rϕ−1 (v̇) = Rϕ−1 Rϕ ◦ ∂z−1 ◦ Q ◦ Rϕ−1 (v) = ∂z−1 Q(u). Conversely if u is a solution of (1.36) then by Proposition 1.1.13, we have a solution ϕ of equation (1.9). By Sobolev embedding ZNm−1,p ⊂ C 1 so u ∈ C 1 [0, T ] × C, C and ϕ ∈ C 1 (C, C). Then pointwise we can write, with v := u ◦ ϕ, v̇ = (u̇ + uz u + uz̄ ū) ◦ ϕ = Rϕ (∂z−1 Q(u)) = Rϕ ◦ ∂z−1 ◦ Q ◦ Rϕ−1 (v) = E2 (ϕ, v). By pointwise integration gives v(t, (z, z̄)) = u0 (z, z̄) + Z t 0 E2 (ϕ, v) (s,(z,z̄)) ds . Now as E2 is real analytic map, s 7→ E2 (ϕ(s), v(s)), [0, T ] → ZNm,p +1 is continuous, so this . integral converges in ZNm,p . This shows that v ∈ C 1 [0, T ], ZNm,p satisfies v = E2 (ϕ, v) where . satisfies ϕ = u ◦ ϕ = v, ϕ|t=0 = id. This shows that the curve ϕ ∈ C 1 [0, T ], ZDm,p N m,p satisfies (1.48). (ϕ, v) ∈ C 1 [0, T ], ZDm,p N × ZN For any ρ > 0 denote by BZNm,p (ρ) the ball of radius ρ in ZNm,p centered at zero. Theorem 1.3.5. Assume m > 3 + 2/p and 1 < p < ∞ and N ≥ 0. Then for any ρ > 0, there exists T > 0 such that for any u0 ∈ BZNm,p (ρ) the Euler equation (1.35) has a unique m+1,p for solution u ∈ C([0, T ], ZNm,p ) ∩ C 1 ([0, T ], ZNm−1,p ) such that the pressure p(t) ∈ Z1,N t ∈ [0, T ]. This solution depends continuously on the initial data u0 ∈ BZNm,p (ρ) in the sense that the data-to-solution map u0 7→ u, BZNm,p (ρ) → C([0, T ], ZNm,p ) ∩ C 1 ([0, T ], ZNm−1,p ) is continuous. 39 Proof. By Proposition 1.3.1 and Proposition 1.3.4 such solutions of the Euler equation are given by solutions of the dynamical system (1.48). By the existence and uniqueness theorem of solutions of an ODE in a Banach space (cf. [9]) there exist ρ0 > 0 and T0 > 0 such that m,p for any u0 ∈ BZNm,p (ρ0 ) there exists a unique solution (ϕ, v) ∈ C 1 ([0, T ], ZDm,p N × ZN ) and therefore by Proposition 1.3.4 u = v ◦ ϕ−1 ∈ C([0, T ], ZNm,p ) ∩ C 1 ([0, T ], ZNm−1,p ). Now by scaling by a constant c > 0, uc = cu(ct), one can check that uc is a solution of (1.35) with initial condition uc 1.4 t=0 = cu0 . Now for any ρ > 0, we can take T = ρ0 T. ρ 0 Global solutions of the 2d Euler equation on ZDm,p N In this section we will show that the solutions of the Euler equation given by Theorem 1.3.5 exist for all time t ≥ 0 within asymptotic space ZNm,p . For this we will use the known global existence of solutions in the little Hölder spaces cm,γ and the corresponding diffeomorphism b groups discussed below. The little Hölder space cm,γ is the closure of Cb∞ in the Hölder b spaces Cbm,γ ≡ Cbm,γ (R2 , R2 ), 0 ≤ γ ≤ 1. Recall that the Hölder space Cbm,γ (R2 , R) consists of functions in f ∈ Cbm (R2 , R) with finite Hölder semi-norms [Dα f ]γ for any multi-index α ∈ Z2≥0 such that |α| = m. The Hölder semi-norm is defined as [f ]γ := sup x6=y f (x) − f (y) |x − y|γ and the norm in Cbm,γ (R2 , R) is |f |m,γ := |f |m + max|α|=m [Dα f ]γ where |f |m is the norm in the space Cbm (R2 , R) of C m -smooth functions on R2 with bounded derivatives of order ≤ m, P that is |f |m = |α|≤m sup |Dα f |. For γ = 0 we have that Cbm ≡ Cbm,0 = cm,0 b . Consider the group of diffeomorphisms of R2 of class cm,γ defined as, b diff m,γ (R2 ) := ϕ = id +u u ∈ cm,γ and ∃ ε > 0 such that det(I + du) > ε . b For m ≥ 1 and 0 < γ < 1, by Theorem 3.1 in [22] (cf. [15]) diff m,γ (R2 ) is a topological group with regularity properties of the composition and the inversion of diffeomorphisms similar 40 to the ones in Theorem 1.1.12. We will show that if there is a global solution of the Euler equation in cm,γ with initial velocity u0 ∈ ZNm,p b |d−2 then in fact the solution stays in the space ZNm,p |d−2 for all time t ≥ 0. We begin with the following. m,p Lemma 1.4.1. Assume that m > 1 + 2/p. Then for any u ∈ ZNm,p +2 and for any ϕ ∈ ZD N m,p m,p m,p we have that u◦ϕ ∈ ZNm,p +2 . Also this composition map (u, ϕ) 7→ u◦ϕ, ZN +2 ×ZD N → ZN +2 , em,p is continuous. If, in addition, u ∈ ZeNm,p +2 then u ◦ ϕ ∈ ZN +2 . The first and last statement of the lemma follow from Remark 3 (see formula (1.8)) while continuity of the map follows from Proposition 1.1.2 (iii), and Lemma 6.5 in [13]. As a consequence of Lemma 1.4.1 we obtain the following proposition. Proposition 1.4.2. Assume that m > 3+(2/p). Let u ∈ C [0, T ], ZNm,p ∩C 1 [0, T ], ZNm−1,p for T > 0 be a solution of the 2d Euler equation (1.35) with u|t=0 ∈ ZNm,p +1 . Then, u ∈ m−1,p 1 . C [0, T ], ZNm,p +1 ∩ C [0, T ], ZN +1 Proof of Proposition 1.4.2. Assume that u ∈ C [0, T ], ZNm,p ∩C 1 [0, T ], ZNm−1,p is a solution of the Euler equation (1.35) with initial velocity u0 := u|t=0 ∈ ZNm,p +1 . Then, for any t ∈ [0, T ] we have by the preservation of vorticity (see (1.41)) that ω(t) = ∂z u(t) is such that ω(t) = ω0 ◦ ψ(t), ω0 ∈ ZeNm−1,p +2 , (1.50) is the Lagrangian coordinate of the solution where ψ(t) := ϕ(t)−1 and ϕ ∈ C 1 [0, T ], ZDm,p N . This, together with ∩ C 1 [0, T ], ZDm−1,p u. By Theorem 1.1.12, ψ ∈ C [0, T ], ZDm,p N N (1.50) and Lemma 1.4.1 above, implies that for any t ∈ [0, T ] we have that ∂z u(t) = ω(t) = ω0 ◦ ψ(t) ∈ ZeNm−1,p +2 . and ∂z u ∈ C [0, T ], ZeNm−1,p +2 Hence, we conclude from Theorem 1.2.9 that u ∈ C [0, T ], ZNm,p +1 . This implies that the local m,p solution of the 2d Euler equation in ZNm,p +1 with initial data u0 ∈ ZN +1 given by Theorem 1.3.5 has a finite ZNm,p +1 -norm on the intersection of its maximal interval of existence with 41 [0, T ]. Thus, the maximal interval of existence of this solution contains [0, T ] otherwise by local existence of solutions we can extend beyond the maximal interval of existence hence m−1,p 1 completing the proof of the proposition. u ∈ C [0, T ], ZNm,p ∩ C [0, T ], Z N +1 +1 Let us recall that for m ≥ 0, 1 < p < ∞, and any weight δ ∈ R, Hδm,p (R2 , R) := f ∈ Lploc (R2 , R) hxiδ Dα f ∈ Lp for |α| ≤ m , Dα ≡ ∂xα11 ∂xα22 . These spaces appear in Proposition 1.4.4 below. In the proposition we start with initial vorticity ω0 ∈ Z14,p and show that since the vorticity ω is preserved, that is, ω = ω0 ◦ ϕ−1 where ϕ ∈ diff 3,γ (R2 ) therefore ω ∈ Z14,p as well. To this end we differentiate ω four times and show that every time we differentiate the decay also increases. But since ϕ ∈ diff 3,γ (R2 ), ϕ − id does not necessarily decay at infinity. So apriori we only get that ω ∈ Hγ4,p . Note 0 +1 that for m > 2/p and and real weights δ the space Hδm,p (R2 , R) is a Banach algebra with respect to pointwise multiplication of functions (see Proposition 2.1 in [13]). We have the following lemma. Lemma 1.4.3. Assume that 1 ≤ p ≤ ∞. Then for 0 ≤ γ < 1 and any weight δ ∈ R we have that the composition map (u, ϕ) 7→ u ◦ ϕ, Lpδ × diff 1,γ (R2 ) → Lpδ , is continuous. More generally, for any integers m ≥ 1 and 0 ≤ k ≤ m we have that the composition map (u, ϕ) 7→ u ◦ ϕ, Hδk,p × diff m,γ (R2 ) → Hδk,p , is continuous. Proof of Lemma 1.4.3. It is enough to prove the lemma only for the case when γ = 0. Then the general statement follows since diff m,γ (R2 ) is continuously embedded in diff m (R2 ) ≡ diff m,0 (R2 ). Also the second statement of the lemma follows from the first one and the fact that ϕ has bounded derivatives. So we will prove the first statement. Assume that 0 ≤ γ < 1, 1 ≤ p < ∞ and δ ∈ R and take u1 , u2 ∈ Lpδ ≡ Lpδ (R2 , R) and ϕ ∈ diff 1 (R2 ). By change of variables y = ϕ(x) in the corresponding integral below we obtain that ku2 ◦ ϕ − u1 ◦ ϕkLp = k(u2 − u1 ) ◦ ϕkLp ≤ C ku2 − u1 kLp δ δ 42 δ (1.51) where C ≡ C(ϕ) > 0 depends continuously on |ϕ−1 |1 . Since ϕ ∈ diff 1 (R2 ) is a topological group (see Theorem 3.1 in [22]) so the map ϕ 7→ ϕ−1 is continuous and therefore the constant C > 0 in (1.51) can be chosen locally uniformly in ϕ ∈ diff 1 (R2 ), i.e. the same constant C can be chosen for any ψ in a small neighborhood U (ϕ) of ϕ such that (1.51) holds for ψ in place of ϕ. Now for any u ∈ Cc∞ and ϕ1 , ϕ2 ∈ diff 1 (R2 ) we can obtain from the mean-value theorem that for any x ∈ R2 , Z (u ◦ ϕ2 )(x) − (u ◦ ϕ1 )(x) = 1 0 (∇u)|ζt (x) dt · ϕ2 (x) − ϕ1 (x) where ζt (x) := ϕ1 (x) + t(ϕ2 (x) − ϕ1 (x)). By applying first the Cauchy-Schwartz inequality and then Jensen’s inequality we obtain (u ◦ ϕ2 )(x) − (u ◦ ϕ1 )(x) p ≤ C ϕ2 − p ϕ1 1 Z 1 0 p ∇u(ζt (x)) dt where the constant C > 0 depends only on the choice of 1 ≤ p < ∞. By multiplying this inequality by hxipδ and then by integrating it over x ∈ R2 we obtain that Z 1 p p p |∇u| ◦ ζt Lp dt u ◦ ϕ2 − u ◦ ϕ1 Lp ≤ C ϕ2 − ϕ1 1 δ (1.52) δ 0 with a constant C > 0 that depends only on the choice of 1 ≤ p < ∞. It follows from (1.51) that for ϕ2 chosen in an open neighborhood U (ϕ1 ) of ϕ1 in diff 1 (R2 ) we have that ∇u ◦ ζt Lpδ ≤ C k∇ukLpδ . This gives us that for any u ∈ Hδ1,p and for any ϕ1 ∈ diff 1 (R2 ) and any ϕ2 in a small neighborhood U (ϕ1 ) we have that u ◦ ϕ2 − u ◦ ϕ1 Lpδ ≤ C(p) ϕ2 − ϕ1 1 ∇u Lpδ ≤ C(p) kukH 1,p ϕ2 − ϕ1 1 . δ (1.53) Finally, it follows from (1.51) and (1.53) that there exist an open neighborhood U (ϕ1 ) of ϕ1 ∈ diff 1 (R2 ) and constants C1 > 0 and C2 > 0 such that for any ϕ2 ∈ U (ϕ1 ) and for any u1 , u2 ∈ Lpδ and u e ∈ Cc∞ we have u2 ◦ ϕ 2 − u1 ◦ ϕ 1 Lpδ ≤ u2 ◦ ϕ2 − u e ◦ ϕ2 Lpδ + u e ◦ ϕ2 − u e ◦ ϕ1 Lpδ + u e ◦ ϕ 1 − u1 ◦ ϕ 1 ≤ C1 ku2 − u ekLpδ + C2 ke ukH 1,p ϕ2 − ϕ1 1 + C1 ke u − u1 kLpδ . δ 43 Lpδ Now to show continuity at point (u1 , ϕ1 ) ∈ Hδ1,p × diff 1 (R2 ) we take ǫ > 0. For any u2 ∈ Lpδ and u e ∈ Cc∞ , u e 6≡ 0, inside the open ball in Lpδ of radius ρ > 0 centered at u1 ∈ Lpδ the inequality above then implies that u2 ◦ ϕ 2 − u 1 ◦ ϕ 1 Lpδ ≤ 3C1 ρ + C2 ke ukH 1,p ϕ2 − ϕ1 1 . δ Finally, by choosing 0 < ρ < ǫ/(6C1 ) and ϕ2 ∈ U (ϕ1 ) such that |ϕ2 − ϕ1 we conclude that u2 ◦ ϕ 2 − u1 ◦ ϕ 1 Lpδ 1 < ǫ/ 2C2 ke ukH 1,p δ < ǫ. Finally the case p = ∞ follows similarly completing the proof of the lemma. Remark 12. By Proposition 1.5.1 (iii) in section 1.5 for m > 2/p, δ + (2/p) > 0, and 0 < γ < 1 chosen sufficiently small, we have the continuous embedding Wδm,p ⊆ Cbk,γ for any integer 0 ≤ k < m − 2/p. Since the elements of Wδm,p are approximated by functions in Cc∞ , we then conclude that Wδm,p is continuously embedded in the little Hölder space cm,γ b . The same arguments applied to the asymptotic coefficients and the definition of the norm in the asymptotic space ZNm,p show that ZNm,p is continuously embedded in ck,γ b . Since 1 < p < ∞ we conclude that 0 < 2/p < 2, and hence for m ≥ 3 we have the continuous embedding . In what follows we will assume that 0 < γ < 1 is chosen so that these ZNm,p ⊆ cm−2,γ b embeddings hold. In order to show global persistence of solutions to the Euler equation we will use the fact m,γ (R2 ) is a that global solutions exist in spaces cm,γ b . Since, for m ≥ 1, 0 < γ < 1, diff we will say that ϕ ∈ C 1 ([0, T ], diff m,γ (R2 )) topological group modeled on Banach space cm,γ b is a solution of the 2d Euler equation if ϕ ∈ C 2 ([0, T ], diff m,γ (R2 )) and it satisfies following second order ODE   . . −1  (ϕ, v) = v, Rϕ ◦ ∂z ◦ Q ◦ Rϕ−1 (v)   (ϕ, v)|t=0 = (id, u0 ) (1.54) Here the Cauchy operator is defined as acting on the tempered distributions S ′ and ∂z−1 is the extension of the operator in Proposition 1.2.5 to the space S ′ . Furthermore since diff m,γ (R2 ) 44 is a topological group (cf. [22, Theorem 3.1]) therefore we have that 1 [0, T ], cm−1,γ u := ϕ̇ ◦ ϕ−1 ∈ C ([0, T ], cm,γ b b )∩C (1.55) Now as discussed in Proposition 1.3.4 if ϕ ∈ C 1 ([0, T ], diff m,γ (R2 )) is a solution of Euler equation in Lagrangian coordinates then u := ϕ̇ ◦ ϕ−1 satisfies the 2d Euler equation (1.36). Also it follows by similar arguments that u and ω = ∂z u satisfies (1.41) and therefore we have that ω(t) = ω0 ◦ ϕ−1 (t). (1.56) In the following proposition we show that if the initial velocity u0 is in asymptotic space Z05,p then it evolves in the same space. We use that the vorticity ω(t) is preserved (see above equation) and show in particular that it follows from (1.56) that ω ∈ Z14,p . there exists a unique solution of the Proposition 1.4.4. Assume that for u0 ∈ Z05,p ⊆ c3,γ b 2d Euler equation in Lagrangian coordinates ϕ̃ ∈ C 1 [0, T ], diff 3,γ (R2 ) for some T > 0 with ˙ t=0 = u0 . Then the solution of the 2d Euler equation with u|t=0 = u0 given by Theorem ϕ̃| 1.3.5 extends to the whole interval [0, T ] and u ∈ C [0, T ], Z05,p ∩ C 1 [0, T ], Z04,p . Proof of Proposition 1.4.4. Let us take u0 ∈ Z05,p so that we can write u0 = c + f0 where 1 3,γ 2 c is a constant and f0 ∈ Wγ5,p . Assume that for T > 0, ϕ̃ ∈ C [0, T ], diff (R ) is the 0 solution of the 2d Euler equation in Lagrangian coordinates (see (1.54)). Assume that the local solution given by Theorem 1.3.5 does not extend to all of the interval [0, T ] then there exists 0 < τ < T such that [0, τ ) is the maximal interval of existence of the solution u ∈ C [0, τ ), Z05,p ∩ C 1 [0, τ ), Z04,p given by Theorem 1.3.5. Then, it follows easily from Theorem 1.3.5 that ku(t)kZ 5,p → ∞ as t → τ − 0, 0 (1.57) since otherwise we can again apply Theorem 1.3.5 to extend the solution beyond t = τ . We will show that (1.57) cannot happen. We start by considering the curve ω e (t) := ω0 ◦ ψ̃(t), 45 t ∈ [0, T ], (1.58) . Also as diff 3,γ (R2 ) is a topological where ψ̃(t) := ϕ̃(t)−1 and ω0 := ∂z u0 ∈ Wγ4,p ⊆ Hγ4,p 0 +1 0 +1 group we get that ψ̃ ∈ C ([0, T ], diff 3,γ (R2 )). Since in particular ω0 ∈ Hγ3,p , it follows from 0 +1 Lemma 1.4.3 that ω̃ ∈ C [0, T ], Hγ3,p . 0 +1 (1.59) Note that for t ∈ [0, τ ) we have by the conservation of the vorticity ω ≡ ∂z u on [0, τ ) (see (1.42) in Remark 11) that ω(t) = ω̃(t) and u(t) = c + ∂z−1 ω̃(t), t ∈ [0, τ ). (1.60) Let us denote r̃ := ∂z−1 ω̃(t), t ∈ [0, T ]. Note that by (1.59) we have that ∂ α ω̃ ∈ C [0, T ], Lpγ0 +1 , |α| ≤ 3. (1.61) Since γ0 satisfies 0 < γ0 + (2/p) < 1 we obtain from Theorem 1.2.7 (i) and Remark 10 that , |α| ≤ 3 (See Remark 8). Therefore if we for any |α| ≤ 3 we have that ∂ α r̃ ∈ C [0, T ], Wγ1,p 0 define the curve ũ := c + r̃ (1.62) then (1.61) implies that ũ − c ∈ C [0, T ], Wγ1,p 0 and dũ ∈ C [0, T ], Hγ3,p . +1 0 (1.63) Note that by (1.55) it follows that ϕ̃ ∈ C 1 [0, T ], diff 3,γ (R2 ) satisfies the equation . ϕ̃ = ũ ◦ ϕ̃, ϕ̃|t=0 = id . This implies that Y := dϕ̃ = I + df , where ϕ̃ = id +f and f ∈ c3,γ b , satisfies the equation . Y = A(t)Y , Y |t=0 = I, where A := dũ ◦ ϕ̃ ∈ C [0, T ], Hγ3,p by Lemma 1.4.3. We can 0 +1 rearrange this equation in terms of df as (df ). = A(t)(df ) + A(t), df |t=0 = 0, (1.64) and by the Banach algebra property in Hγ3,p we can consider it as a linear ODE on df in the 0 +1 space of 2 × 2 matrices with elements in Hγ3,p . Therefore by the existence and uniqueness 0 +1 46 of solutions of ODEs in Banach spaces, we obtain that dϕ̃ − I ∈ C [0, T ], Hγ3,p . Now since 0 +1 ψ̃ = ϕ̃−1 so we can write dψ̃ = Adj(dϕ̃)/ det(dϕ̃) ◦ ψ̃, (1.65) where Adj(dϕ̃) is the transpose of the cofactor matrix of dϕ̃, we conclude from the Banach algebra property in Hγ3,p , the arguments used to prove Lemma 1.1.5, and Lemma 1.4.3, 0 +1 that dψ̃ − I ∈ C [0, T ], Hγ3,p . 0 +1 (1.66) ⊆ Lpγ0 +1 . Hence, by Theorem Remark 13. In particular, this implies that ∂z ψ̃ −z ∈ Hγ3,p 0 +1 1.2.7 (i), we obtain that ψ̃−z −c0 ∈ Wγ1,p for some constant c0 ∈ C, and therefore ψ̃−z −c0 ∈ 0 Hγ4,p . Similarly for ϕ̃ and ψ̃ we get that 0 ϕ̃, ψ̃ ∈ C [0, T ], AD04,p (1.67) m,p with N ≥ 0 denotes the group of orientation preserving C 1 -smooth diffeomorwhere ADN phisms associated to the weighted Sobolev space Hγ4,p and defined in a similar fashion as N in (1.5) (see Definition 5.1 in [13, Section 5] for more detail). By Corollary 6.1 in ZDm,p N [13], the composition map (f, ν) → f ◦ ν, Hγ4,p × AD04,p → Hγ4,p , 0 0 is continuous. . A simple approximation arguThis remark allows us to conclude that ω̃ ∈ C [0, T ], Hγ4,p 0 ment involving Remark 13 also shows that we can differentiate the equality (1.58) four times 47 to obtain the following expressions for the differentials of ω̃ of higher order, dω̃ = (dω0 ) ◦ ψ̃ (dψ̃), d2 ω̃ = (d2 ω0 ) ◦ ψ̃ (dψ̃, dψ̃) + (dω0 ) ◦ ψ̃ (d2 ψ̃), d3 ω̃ = (d3 ω0 ) ◦ ψ̃ (dψ̃, dψ̃, dψ̃) + 3 (d2 ω0 ) ◦ ψ̃ (d2 ψ̃, dψ̃) + (dω0 ) ◦ ψ̃ (d3 ψ̃), (1.68) d4 ω̃ = (d4 ω0 ) ◦ ψ̃ (dψ̃, dψ̃, dψ̃, dψ̃) + 6 (d3 ω0 ) ◦ ψ̃ (d2 ψ̃, dψ̃, dψ̃) + 4 (d2 ω0 ) ◦ ψ̃ (d3 ψ̃, dψ̃) + 3 (d2 ω0 ) ◦ ψ̃ (d2 ψ̃, d2 ψ̃) + (dω0 ) ◦ ψ̃ (d4 ψ̃). , as well as formula , ψ̃ ∈ C [0, T ], c3,γ These formulas together with the fact that ω̃0 ∈ Wγ4,p b 0 +1 (1.66), Lemma 1.4.3, and Proposition 1.5.1 (ii) imply that ω̃ ∈ C [0, T ], Wγ2,p 0 +1 and d3 ω̃, d4 ω̃ ∈ C [0, T ], Lpγ0 +3 . (1.69) To see this note that first dk ψ̃ is a bounded function for 1 ≤ k ≤ 2. Furthermore dk ω0 ◦ ψ̃ ∈ Lpγ0 +1+k for 0 ≤ k ≤ 4. The only non-trivial part is to show that (dω0 ) ◦ ψ̃ (dk ψ̃) ∈ Lpγ0 +3 , k ≥ 2. This will follow from (1.66) and the observation that |(dω0 )◦ ψ̃(x) = O(1/hxiγ0 +2+2/p ) and using 0 < γ0 + 2/p < 1. Now by Lemma 2.1 (d) in [13] and (1.67) we have that |ϕ̃ − id | is bounded and therefore for some constant Cϕ̃ we have that hϕ̃(x)i ≤ Cϕ̃ hxi. Now as dω0 ∈ Wγ3,p then by Proposition 1.5.1 (ii), with δ = γ0 + 2 and y = ψ̃(x) we have that 0 +2 sup hxiγ0 +2+2/p |dω0 ◦ ψ̃(x)| = sup hϕ̃(y)iγ0 +2+2/p |dω0 (y)| (1.70) y∈R2 x∈R2 γ +2+2/p ≤ Cϕ̃0 sup h yiγ0 +2+2/p |dω0 (y)| ≤ C kdω0 kW 3,p γ0 +2 y∈R2 By combining (1.69) with Theorem 1.2.7 (i), Remark 10, and (1.62) we obtain that ũ − c ∈ C [0, T ], Wγ3,p 0 and d4 ũ, d5 ũ ∈ C [0, T ], Lpγ0 +3 . (1.71) . and from following equations that A = dũ ◦ ϕ̃ ∈ Wγ2,p It follows from (1.71) that dũ ∈ Wγ3,p 0 0 dA = (d2 ũ) ◦ ϕ̃ (dϕ̃), d2 A = (d3 ũ) ◦ ϕ̃ (dϕ̃, dϕ̃) + (d2 ũ) ◦ ϕ̃ (d2 ϕ̃). 48 (1.72) Here we use that d2 ϕ̃ ∈ Lpγ0 +1 , d2 ũ ∈ Wγ2,p and similarly to (1.70) (d2 ũ)◦ϕ̃ = O(1/hxiγ0 +1+2/p ). 0 +1 Since Wγ2,p is a Banach algebra therefore from (1.64) it follows that dϕ̃ − I ∈ Wγ2,p . Now 0 0 and (d2 ũ) ◦ ϕ̃ ∈ Wγ1,p gives that d2 A ∈ Lpγ0 +3 . Proposition 2.2 in [13] with d2 ϕ̃ ∈ Wγ1,p 0 +1 0 +2 ) so that from (1.64) we have This together with (1.71) gives that A ∈ C([0, T ], Wγ2,p 0 +1 dϕ̃ − I ∈ Wγ2,p . This along with (1.65) gives that d3 ψ̃ ∈ Lpγ0 +2 . Now from (1.68) we 0 +1 get that ω̃ ∈ C [0, T ], Wγ3,p . By differentiating A ≡ dũ ◦ ϕ̃ and so ũ − c ∈ C [0, T ], Wγ4,p 0 +1 0 we obtain following equation d3 A = (d4 ũ) ◦ ϕ̃ (dϕ̃, dϕ̃, dϕ̃) + 3 (d3 ũ) ◦ ϕ̃ (d2 ϕ̃, dϕ̃) + (d2 ũ) ◦ ϕ̃ (d3 ϕ̃) (1.73) which shows that d3 A ∈ C([0, T ], Lpγ0 +4 ). Here we use that Proposition 2.2 in [13] and 2 (d ũ) ◦ ϕ̃ = O(1/hxiγ0 +2+2/p ). Again A ∈ C([0, T ], Wγ3,p ) so that from (1.64) we have dϕ̃ − 0 +1 4,p I ∈ Wγ3,p . Finally using (1.65) the last equation in (1.68) gives that ω̃ ∈ C [0, T ], W γ0 +1 0 +1 . By (1.60), this contradicts the blow-up of the Z05,p -norm in and so ũ − c ∈ C [0, T ], Wγ5,p 0 (1.57). Therefore, τ > T . This completes the proof of the theorem. Now we can finally prove the main result about the global existence of solutions to the Euler equation (see introduction). The above proposition considers the case when m ≥ 5. However one case occurs when m = 4 and p > 2 so that we still have m > 3 + 2/p. The above proof can directly be adopted to this special case. Proof of Theorem 1.0.1. We will consider the case when m ≥ 5. The special case when m = 4 and p > 2 is similar. Assume therefore that m > 5 and let uo ∈ ZNm,p ⊂ Z05,p . for some 0 < γ < Now Z05,p is continuously embedded into the little Hölder space cm,γ b 1. Therefore from the results in [17, Theorem 2] and [15] it follows that there exists a solution ϕ ∈ C ([0, ∞), diff 3,γ (R2 )) of the Euler equation in Lagrangian coordinates with initial velocity ϕ̇|t=0 = u0 . By Proposition 1.4.4 we conclude that the local solution of the Euler equation in Z05,p given by Theorem 1.3.5 extends to a global solution u ∈ C [0, ∞), Z05,p ∩ C 1 [0, ∞), Z04,p , 49 ϕ ∈ C 1 [0, ∞), ZD5,p . 0 Inductively we can conclude that in fact u ∈ C ([0, ∞), Z0m,p ) ∩ C 1 [0, ∞), Z0m−1,p . (1.74) To see this let m = 6. Since u0 ∈ Z06,p so ω0 = ∂z u0 ∈ Z15,p so by Lemma 1.4.1 we have ω(t) = ω0 ◦ ϕ−1 ∈ C [0, ∞), Z15,p . And therefore we have that u = ∂z−1 ω ∈ C [0, ∞), Z06,p . This shows that the Z06,p -norm of the curve t 7→ u(t) is bounded and therefore by the local existence theorem 1.3.5 we see that u ∈ C [0, ∞), Z06,p ∩ C 1 [0, ∞), Z05,p . Using this inductive argument we show that infact (1.74) holds for any m > 5. New we can increase the asymptotic expansion of u from 0 to N using Proposition 1.4.2 directly. Finally the solution is unique and depends continuously on initial data by local existence theorem 1.3.5. 1.5 Proof of Lemma 1.1.4 In this section we prove several technical results used in the main body of this chapter. First we study properties of auxiliary space ZNm,p (BRc ). To this end we study the remainder space Wδm,p (BRc ) where BRc = x ∈ Rd |x| > R . The key point in following proposition is to show that the constant C is independent of the ball radius R (Lemma 1.1.4). We will need this later to show that composition is closed in ZDm,p N . For a non-negative real number β ∈ R denote by ⌊β⌋ its integer part i.e. the largest integer number that is less or equal than β. Similarly, let {β} := β − ⌊β⌋ be the fractional part of β ≥ 0. We have the following weighted version of the Sobolev embedding theorem. Proposition 1.5.1. Assume that 1 < p < ∞, δ ∈ R, and R ≥ 1 then i h m,p q dp c one has Wδ− (BRc ). Moreover, (i) If 0 ≤ m < d/p then for any q ∈ p, d−mp d (BR ) ⊂ L δ− d p p there exists a constant C ≡ C(d, p, q, m, δ) > 0 independent of the choice of R ≥ 1 such 50 m,p c that for any f ∈ Wδ− d (BR ) p kf kLq d δ− p c) (BR ≤ C kf kW m,p (B c ) . δ− d p R If mp = d the above statement holds for q ∈ [p, ∞). (ii) If mp > d, the for any integer 0 ≤ k < m − d p m,p c k c one has that Wδ− d (BR ) ⊂ C (BR ) and p for any multi-index α such that |α| ≤ k, sup hxiδ+α |Dα f (x)| ≤ C kf kW m,p (B c ) , d δ− p c x∈BR R Dα ≡ ∂xα11 · · · ∂xαdd , with a constant C ≡ C(d, p, q, m, δ) > 0 independent of the choice of R ≥ 1. (iii) Assume that m > d/p, 0 ≤ k < m − (d/p), and δ + (d/p) > 0. Choose a real number γ such that 0 < γ < m − k − (d/p) for m − k − (d/p) 6= 0 or 0 < γ < 1 for m − k − (d/p) = 0. Then, Wδm,p (Rd ) ⊆ Cbk,γ (Rd ) and the embedding is bounded. Remark 14. The statement of this lemma also holds with BRc replaced by Rd (see Lemma 1.2 in [13]). An important technical part of this proposition is that the constant C > 0 is independent of the radius R ≥ 1. Proof. Let us start with (i). Assume 0 ≤ m < d/p. For a given open set U ⊂ Rd denote |||f |||pm,p,δ;U = X Z 0≤α≤m U |x|δ+|α| Dα f p dx For R ≥ 1 consider the annulus AR = B2R \ BR where BR denotes the open ball of radius R in Rd centered at zero and BR is its closure. By changing variables in the corresponding integrals one easily sees that for any f ∈ Cc∞ (AR ), d |||fR |||m,p,δ;A1 = R−δ− p |||f |||m,p,δ;AR , 51 (1.75) where fR (x) = f (R x) for x ∈ Rd . By using that R ≤ | x | ≤ 2R on AR X Z 0≤α≤m AR d d | x |δ− p +|α| R p |Dα f | p dx ≤ |||f |||pm,p,δ;AR X Z ≤ 0≤α≤m AR d d | x |δ− p +|α| (2R) p |Dα f | p dx, which impleis d d R p |||f |||m,p,δ− d ;AR ≤ |||f |||m,p,δ;AR ≤ (2R) p |||f |||m,p,δ− d ;AR . p (1.76) p This together with (1.75) shows that for any 0 ≤ m < d/p there exist constants 0 < M1 < M2 independent of R ≥ 1 such that for any f ∈ Cc∞ (AR ), M1 R−δ |||f |||m,p,δ− d ;AR ≤ |||fR |||m,p,δ;A1 ≤ M2 R−δ |||f |||m,p,δ− d ;AR . p (1.77) p h By the Sobolev embedding theorem for any given q ∈ p, e ≡ C(d, e q) such that for any g ∈ W m,p (A1 ), C dp d−mp i there exists a constant e kgk m,p kgkLq (A1 ) ≤ C W (A1 ) , (1.78) where we have used following notation kgkqq;A1 = kgkm,p;A1 = Z A1 |g(x)|q dx X 0≤α≤m kDα gkLp (A1 ) . By applying (1.78) to | x |δ g(x) with g ∈ Cc∞ (A1 ) and then using Dα | x |δ ≤ Cδ | x |δ−|α| we obtain kgkLq (A1 ) = |x|δ g δ ≤ C2 ≤ C3 Lq (A1 ) X X |α|≤m β≤α X |α|≤m e ≤C X |α|≤m Dα (| x |δ g(x)) | x |δ−|α|+|β| Dβ g | x |δ+|α| Dα g Lp (A1 ) Lp (A 1) Lp (A1 ) ≤ C2 X X X X |α|≤m β≤α |α|≤m |β|≤|α| ≤ C4 |||g|||m,p,δ;A1 . 52 ≤ C1 Dα−β | x |δ Dβ g | x |δ+|β| Dβ g Lp (A1 ) Lp (A1 ) By taking g = fR with f ∈ Cc∞ (AR ) we get |||fR |||q,δ;A1 ≤ C4 |||fR |||p,m,δ;A1 which together with (1.77) and the fact that |||·|||Lq (U ) ≡ |||·|||0,q,δ;U for any δ ∈ R and an open set U ⊂ Rd implies, δ R−δ |||f |||q,δ− d ;AR ≤ |||fR |||q,δ;A1 ≤ C0 |||fR |||m,p,δ;A1 ≤ CC0 R−δ |||f |||m,p,δ− d ;AR q p hence for any f ∈ Cc∞ (AR ), |||f |||Lq δ− d q (AR ) ≤ C|||f |||m,p,δ− d ;AR ., (1.79) p with C > 0 independent of R ≥ 1. Finally we write BRc = AR ∪ A2 R ∪ ... ∪ A2k R ∪ ... and using (1.79) obtain that for any f ∈ Cc∞ (BRc ), kf kLq δ− d q c) (BR = ∞ Z X k=0 ≤C A 2k R ∞ X k=0 |x| δ− dq |f | |||f |||qm,p,δ− d ;A p q 2k R ! 1q ! 1q = ∞ X k=0 ≤C |||f |||qq,δ− d ;A q ∞ X k=0 2k R ! 1q |||f |||pm,p,δ− d ;A p 2k R ! p1 = C |||f |||m,p,δ− d ;B c . p R Here we used Lemma 1.5.2 below. The case when m = d/p is treated the same way. This proves item (i). To prove (ii) i.e. mp > d, applying Sobolev inequality to A1 and using Dβ | x |δ+|α| ≤ (δ + |α|)| x |δ+|α|−|β| gives X sup | x |δ+|α| |Dα g| ≤ C1 x∈A1 |β|≤m ≤ C2 ≤ C2 Dβ (| x |δ+|α| Dα g) X X |β|≤m |γ|≤|β| X X |β|≤m |γ|≤|β| Lp (A1 ) | x |δ+|α|+|γ|−|β| Dα+γ g | x |δ+|α+γ| Dα+γ g (1.80) Lp (A1 ) Lp (A1 ) ≤ C3 |||g|||m,p,δ;A1 Now with g = fR and Dα fR = R|α| (Dα f )R we get using (1.77) R−δ sup |y|δ+|α| |Dα f (y)| ≤ C3 |||fR |||m,p,δ;A1 ≤ C4 R−δ |||f |||m,p,δ− d ;AR p y∈AR 53 giving sup |y|δ+|α| |Dα f (y)| ≤ C4 |||f |||m,p,δ− d ;AR ≤ C4 |||f |||m,p,δ− d p y∈AR p and taking sup over BRc gives the result. Let us now prove item (iii). Assume that m > d/p, 0 ≤ k + (d/p) < m, δ + (d/p) > 0, and choose γ ∈ R as described in the proposition. It follows from item (ii) that Wδm,p (B1c ) ⊆ C k (B1c ) and that for any |α| ≤ k and f ∈ Wδm,p (B1c ) we have sup hxi(δ+(d/p))+|α| |Dα f (x)| ≤ C kf kW m,p (B c ) < ∞. δ x∈B1c 1 Since δ + (d/p) > 0 we then obtain that Wδm,p (Rd ) ⊆ Cbk (Rd ). Let us now estimate the Hölder semi-norm of Dα f with |α| ≤ k. For R ≥ 1 consider the open annulus AR in Rd defined above and let | · |0,γ;AR and [·]γ;ĀR be the Hölder norm and semi-norm in the closure ĀR of AR in Rd . For any δ ∈ R and for any g ∈ C k (Ā1 ) we have [g]γ;Ā1 = |x|−δ |x|δ g γ;Ā1 ≤ 2|δ| |x|δ g γ;Ā1 + Kδ |x|δ g L∞ (Ā1 ) ≤ Cδ |x|δ g γ;Ā1 (1.81) with constants depending only on the choice of δ ∈ R. Here we used that for any non-empty U ⊆ Rd we have that [f g]γ ≤ |f |L∞ (U ) [g]γ;U + |g|L∞ (U ) [f ]γ;U for any f, g ∈ C 0,γ (U ). By the Sobolev embedding theorem in the domain A1 and by the assumptions on the regularity exponents m, k, and γ we obtain that for any |α| ≤ k there exists a constants C > 0 as well as constants C1 , C2 , C3 > 0, such that for any g ∈ H m,p (Ā1 ) we have that |x|δ Dα g γ;Ā1 ≤C X |β|≤m−|α| ≤ C2 X |α+γ|≤m Dβ |x|δ Dα g Lp (A |x|δ+|α+γ| Dα+γ g 1) ≤ C1 Lp (A1 ) X X |β|≤m−|α| γ≤β |x|δ+|γ|−|β| Dα+γ g ≤ C3 |||g|||m,p,δ;A1 Lp (A1 ) (1.82) where we argued as in (1.80). It follows from (1.81) and (1.82) that there exists a constant C > 0 such that for any g ∈ H m,p (Ā1 ) and for any |α| ≤ k we have that [Dα g]γ;Ā1 ≤ C |||g|||m,p,δ;A1 . 54 (1.83) Now, take R ≥ 1 and let f ∈ H m,p (ĀR ). Then, we have [Dα fR ]γ;Ā1 = sup x,y∈Ā1 ;x6=y = R|α| (Dα fR )(x) − (Dα fR )(y) |x − y|γ sup x,y∈Ā1 ;x6=y (Dα f )(Rx) − (Dα f )(Ry) |x − y|γ = R|α|+γ [Dα f ]γ;ĀR . (1.84) By combining (1.75) and (1.84) with (1.83) we obtain that there exists a constant C > 0 such that for any R ≥ 1, |α| ≤ k, and for any f ∈ H m,p (ĀR ), [Dα f ]γ;ĀR ≤ C kf kWδm,p (AR ) Rµ (1.85) where µ := δ + (d/p) + γ + |α| > 0. In addition, by the Sobolev embedding theorem in the unit ball B1 we also have that there exists a constant, denoted again by C > 0, such that for any f ∈ H m,p (B̄ 1 ), [Dα f ]γ;B̄ 1 ≤ C kf kW m,p (B1 ) (1.86) where B̄ 1 denotes the closure of B1 in Rd . Note that Rd = B̄ 1 ∪ [ k≥0 Ā2k . Now, take f ∈ Cb∞ (Rd ) and two points x, y ∈ Rd such that x 6= y. The closed segment [x, y] := x + s(y − x) 0 ≤ s ≤ 1 intersects any given annulus Ā2k with k ≥ 0 as well as the closed ball B̄ 1 at no more than two straight segments Ā2k ∩ [x, y] = Ik′ ∪ Ik′′ , B̄ 1 ∩ [x, y] = I−1 where Ik′ and Ik′′ with k ≥ 0, and I−1 , can be empty. If such a segment is not empty we set Ik′ = [x′k , yk′ ], Ik′′ = [x′k , yk′ ], I−1 = [x−1 , y−1 ]. Then, [x, y] = I−1 ∪ [ k≥0 55 Ik′ ∪ Ik′′ . This, together with (1.85) and (1.86) then implies that for any f ∈ Wδm,p (Rd ) we have that X (Dα f )(x′k ) − (Dα f )(yk′ ) (Dα f )(x−1 ) − (Dα f )(y−1 ) (Dα f )(x) − (Dα f )(y) ≤ + |x − y|γ |x−1 − y−1 |γ |x′k − yk′ |γ k≥0 X (Dα f )(x′′k ) − (Dα f )(yk′′ ) X ≤ [Dα f ]γ;B̄ 1 + 2 [Dα f ]γ;Ā2k ′′ ′′ γ |x − y | k k k≥0 k≥0 X 1 kf kWδm,p ≤C 1+2 µk 2 k≥0 + where we omit the terms corresponding to empty intervals in the second estimate above and use that for any k ≥ 0 we have that kf kWδm,p (A2k ) ≤ kf kWδm,p (Rd ) and a similar inequality for the ball B1 . Hence, there exists a positive constant C ≡ Cµ < ∞ such that for any f ∈ Wδm,p (Rd ) and for any |α| ≤ k we have that [Dα f ]γ ≤ C kf kWδm,p . Since we already proved that Wδm,p (Rd ) ⊆ Cbk (Rd ), this completes the proof of item (iii). In the above Proposition 1.5.1 we used following simple Lemma about norms on lp , p ≥ 1 spaces. lp is a Banach space of complex-valued sequences a = (ak )k≥1 with finite lp norm P p 1/p ||a||lp = . k≥1 |ak | Lemma 1.5.2. For 1 ≤ p ≤ q < ∞ one has that lp ⊂ lq so that for any a ∈ lp we have ||a||lq ≤ ||a||lp . Proof. Without loss of generality we can take ||a||lp = 1. Then in particular |ak | ≤ 1 and therefore since q ≥ p we have ||a||qlq = X k≥1 |ak |q ≤ X k≥1 |ak |p = ||a||plp = 1. Next Proposition give the multiplicative property of the weighted Sobolev spaces. Proposition 1.5.3. For any real δ1 , δ2 ∈ R and integers 0 ≤ k ≤ l ≤ m with m+l −k > d/p there exists a constant C ≡ C(d, p, l, k, m, δ1 , δ2 ) > 0 independent of the choice of R ≥ 1 such 56 that for any f ∈ Wδm,p (BRc ) and for any g ∈ Wδl,p− d (BRc ) we have that f g ∈ Wδk,p+δ −d 1 2 p 1 p d 2− p (BRc ) and kf gkW k,p c) (BR δ1 +δ2 − d p ≤ C kf kW m,p c d (BR ) δ1 − p kgkW l,p c) (BR δ2 − d p . (1.87) In particular, for m > d/p and δ ∈ R the weighted Sobolev space Wδm,p (BRc ) is a Banach algebra. Remark 15. Note that inequality (1.87) as well as items (i) and (ii) of Proposition 1.5.1 hold with BRc replaced by Rd (see Lemma 1.2 in [13]). This easily follows from Proposition 1.5.1 (i), (ii), Proposition 1.5.3, and Lemma 2.1.6. Proof. By the generalized Holder inequality with weights (see Lemma below) we can write kf gkLp d δ− p where δ = δ1 + δ2 and 1 p = 1 q1 c) (BR 1 . q2 + ≤ kf kLq1 δ1 − qd 1 c) (BR kgkLq2 δ2 − qd 2 p kf gkLp 1 p δ− d p = c) (BR (1.88) First consider the case when 0 ≤ m ≤ d/p. Then we m,p q have by Proposition 1.5.1 (i) that Wδ− for p ≤ q ≤ d ⊂ L δ− d We need to find q1 , q2 such that c) (BR 1 q1 + q 1 q2 dp d−mp is continuous embedding. then from (1.88) above ≤ kf kW m,p c d (BR ) δ1 − p kgkLm,p c) (BR δ2 − d p . (1.89) We can find such q1 , q2 if d − mp d − lp 1 + ≤ dp dp p d p which reduces to to satisfy 1 p = 1 q1 + ≤ m + l. If mp = d then we can again select q1 , q2 in range p ≤ q < ∞ 1 q2 and we obtain (1.89) in the same way. Finally if mp > d take q1 = ∞ and q2 = p to get kf gkLp δ− d p c) (BR ≤ kf kL∞ (B c ) kgkLp δ1 R c d (BR ) δ2 − p . (1.90) Now apply Proposition 1.5.1 (ii) to f and (i) or (ii) to g depending if l ≤ d/p or l > d/p we obtain (1.89) again. 57 To complete the proof we want to show that | x ||α| Dα (f g) ∈ Lpδ d 1 +δ2 −|α|− p k ≤ l ≤ m. |α| α | x | D (f g) = X α β≤α | x ||β| Dβ f β m−|β|,p d 1− p d 1 +δ2 − p | x ||α−β| Dα−β g l−|γ|,p d 2− p with γ = α − β we have | x ||β| Dβ f ∈ Wδ belongs to Lpδ (BRc ) for all |α| ≤ and | x ||γ| Dγ g ∈ Wδ so the product (BRc ) if (m − |β| + l − |γ|)p = (m + l − |α|)p ≥ (m + l − k)p > d. In the Proposition above we used the following generalized weighted Hölder inequality Lemma 1.5.4. For any real δ1 , δ2 ∈ R, 1 < p ≤ q1 , q2 ≤ ∞ with 1 p = 1 q1 + 1 q2 and R > 0 we have that kf gkLp d δ− p c) (BR ≤ kf kLq1 δ1 − qd 1 kgkLq2 c) (BR δ2 − qd 2 c) (BR for any f ∈ Lqδ1 − d (BRc ) and g ∈ Lqδ2 − d (BRc ). 1 2 q1 q2 1 p Proof. First consider case < p ≤ q1 , q2 < ∞. Then 1 q1 = + q12 implies that 1 = 1 + (q21/p) . (q1 /p) Then by using (standard) Holder inequaltiy we get ||f g||pLp d δ1 +δ2 − p = = Z c BR Z f g| x | c BR δ1 +δ2 − pd f| x | δ1 − qd 1 p p dx = dx Z !p/q1 f| x | c BR Z c BR δ1 − qd g| x | p 1 δ2 − qd 2 g| x | p dx δ2 − qd p 2 !p/q2 dx . This gives the weighted Holder inequality in this case. When q1 = p and q2 = ∞ it is east to see this inequality holds. Corollary 1.5.5. Assume m > 2/p then for any u ∈ ZNm,p we have (i) |u|ZNm,p (BRc ) → a00 as R → ∞. (ii) For R ≥ 2 we have that supz∈BRc |u| ≤ |u|ZNm,p (BRc ) . (iii) There is a constant Cχ depending on the cut-off function used in asymptotic expansion of u ∈ ZNm,p such that ||u||L∞ ≤ Cχ kukZ m,p . N 58 Proof. Take u ∈ ZNm,p such that u = P akl χ 0≤k+l≤N z k z̄ l + f . By Proposition 1.5.1 it follows that kf kWγm,p (B c ) → 0 proving 1. To prove 2 we can write R N sup |u| ≤ c z∈BR X 0≤k+l≤N X |akl | |akl | + sup + kf kWγm,p (B c ) = |u|ZNm,p (BRc ) . |f | ≤ k+l R N Rk+l z∈BRc R 0≤k+l≤N Finally since mp > 2, u is continuous so ||u||L∞ = sup |u| so with Cχ = max(1, sup χ) ! X ||u||L∞ = sup |u| ≤ Cχ |a|ij + sup |f | . 0≤i+j≤N Finally use sup |f | = limR→0 supz∈BRc |f | ≤ limR→0 kf kWγm,p (B c ) = kf kWγm,p . N R N Proof of Lemma 1.1.4. Assume that m > 2/p and that R ≥ 1. In the first case if u1 , u2 ∈ Wγm,p (BRc ) then by Proposition 1.5.3 N |u1 u2 |ZNm,p (BRc ) ≤ C|u1 |ZNm,p (BRc ) |u2 |ZNm,p (BRc ) . If u1 = a zk and u2 ∈ Wγm,p (BRc ) then u1 .u2 ∈ Wγm,p (BRc ) so that N N |u1 u2 |pZ m,p (B c ) N R = X Z 0≤|α|≤m c BR |z|γN +|α| Dα a p u2 dx dy zk p 1 α |z|γN +|α| Dα−β ( k ) Dβ u2  dx dy ≤ |a|p c z β BR 0≤|β|≤|α| |α| !p Z X X 1 |z|γN +|α| ≤ |a|p Cm,N k+|α|−|β| Dβ u2 dx dy c |z| BR α β≤α !p p X Z X |a| ≤ dx dy |z|γN +|β| Dβ u2 C0 c Rk BR α β≤α p |a| ≤ C1p |u2 |pZ m,p (B c ) . k N R R X Z  X We have obtained |u1 u2 |Z m,p (B c ) ≤ C1 |u1 |Z m,p (B c ) |u2 |Z m,p (B c ) . N R N 59 R N R (1.91) Finally we have the trivial relation when 0 ≤ (k1 + l1 ) + (k2 + l2 ) ≤ N a b z k1 z̄ l1 z k2 z̄ l2 = m,p c) ZN (BR a z k1 z̄ l1 ZNm,p (BRc ) b z k2 z̄ l2 . m,p c) ZN (BR Also when (k1 + l1 ) + (k2 + l2 ) ≥ N + 1 we obtain (1.91) by direct integration with C1 independent of R. 60 Chapter 2 Asymptotic expansion in higher dimensions In the last chapter we considered in the 2-dimensional case the smaller spaces of the asymptotic Am,p n,N that are preserved by Euler equation. In particular to obtain these spaces we used the complex structure of R2 . Furthermore as we saw the Fourier modes in the numerator of the asymptotic terms were limited to finitely many. In dimensions d ≥ 3 we do not have a similar complex structure in Rd therefore we consider limiting the number of Fourier modes in the asymptotics again. We will show by carefully selecting certain Fourier modes we can obtain a Banach subalgebra of Am,p n,N such that the Euler equation is preserved. We will also see that the method of selecting/limiting the Fourier modes in the asymptotic terms used in this chapter works for any dimension d ≥ 2 however the case d = 2 presents a special case in this approach and there the method used in chapter is infact needed. Consider again the incompressible Euler equation on Rd for d ≥ 3,    ut + u · ∇u = −∇p, div u = 0,   u|t=0 = u0 , (2.1) where u(x, t) is the velocity of the fluid and p(x, t) is the scalar pressure. As discussed in the main introduction it is proven in [14] that the Euler equation (2.1) is locally well-posed in the 61 class of asymptotic vector fields Am,p N ;0 (see the introduction for definitions) with log-terms as defined in (2). Spaces with asymptotic terms are natural spaces for studying Euler equation since even with smooth initial velocity u0 ∈ Cc∞ (Rd , Rd ), the solution develops asymptotics. However the log-term asymptotics appear since certain Fourier modes in a0k (θ) give log-terms when inverting the Laplacian. In this chapter we find smaller spaces with restrictions on the Fourier modes of these asymptotic coefficients ajk (θ) such that the log-terms do not appear anymore. In particular throughout this chapter we will work with the asymptotic spaces Am,p n,N (cf. [13, Section 2]) and it’s properties discusses in Section 2 to work with its Banach subalgebra ZNm,p |l . Since we will be working with restrictions on Fourier modes of the coefficients, we will use homogeneous functions in Rd for functions on S d−1 . Let us first define the space of homogeneous functions Zk|l = M ′ rl Hk−l′ (2.2) l′ ≥l where Hk is space of harmonic polynomials of homogeneous degree k in Rd . Note that this sum is finite since Hk<0 = {0}. Thus Zk|l is the space of all homogeneous functions in Rd of degree k whose restriction to unit sphere lies in the linear span of spherical harmonics of degree at most k − l. By Mk we will denote homogeneous polynomials of degree k and we have following representation (cf. [18, Theorem 22.2]). Mk = Hk M r2 Hk−2 · · · M r2l Hk−2l M ... (2.3) Remark 16. We will generally denote elements in Zk|l as Jk and polynomials in Mk as Pk . By above grading we can write Jk = Pk + rPk−1 . Furthermore given Jk ∈ Zk|l we can write Jk = rl Jk−l where Jk−l ∈ Zk−l|0 . For ease of notation, in the last factorization, we will sometimes denote J˜k := Jk−l and therefore we will have J˜k ∈ Zk−l|0 with k, l depending on L rMk−1 and Zk|l = rl Zk−l|0 . the context. Also we have the relations Zk|0 = Mk Now we can define the subspace of Am,p n,N with restricted Fourier modes in the asymptotic 62 coefficients. For integers m > d/p and N, l ≥ 0 we define ZNm,p |l o X χ(r) Jk (r, θ) m,p + f, Jk ∈ Zk|l and f ∈ WγN , = uu= rk rk 0≤k≤N n (2.4) where χ : R → R is a C ∞ -smooth cut-off function with bounded derivatives such that χ(r) ≡ 1 for r ≥ 2. Different choices of χ give same space ZNm,p |l with equivalent norms (cf. [13, Remark 3.2]). l Remark 17. The first non-zero asymptotic coefficient in ZNm,p |l that occurs is of the order 1/r with constant coefficient (only constant Fourier mode). For next term 1/rl+1 the numerator has constant and linear Fourier modes. For example in the interesting case of d = 3 and l = d − 2 = 1 (See Theorem 2.0.1 below) we have following first two terms u = χ(r) a0N (θ) a0 1 (ax + by + cz) + dr + · · · + + 2 r r r2 rN +f where x, y, z are coordinates on R3 and a0 , a, b, c, d ∈ R are constants. Therefore J2 (x, y, z) = ax + by + cz + dr. Note that this space ZNm,p |l does not contain log-terms in the asymptotic expansion and that the corresponding coefficients ak (θ) = Jk (r,θ) rk have only the first k − l Fourier modes. Furthermore we will show that ZNm,p |d−2 is preserved by the incompressible Euler equation in three and higher dimensions (d ≥ 3). This space is equipped with following norm kukZ m,p = N |l X 0≤k≤N sup |Jk (1, θ)| + kf kWγm,p N which is equivalent to the norm it inherits from Am,p n,N . This is true since all norms on the finite dimensional subspace of asymptotic terms in ZNm,p |l are equivalent. Let us also define m,p m,p (ρ) to be the space of all divergence free vector fields in ZNm,p |l as Z̊N |l . We will denote BZ̊ N |l the open ball in Z̊Nm,p |l of radius ρ > 0. The layout of this chapter is as follows. In section 2 we will study the properties of asymptotic spaces ZNm,p |l . In section 3 we prove our main result. 63 Theorem 2.0.1. Let d ≥ 3 and m > 3 + d/p where 1 < p < ∞. Then for any ρ > 0 there exists T > 0 such that for any divergence-free vector field u0 ∈ BZ̊ m,p (ρ) there exists a N |d−2 unique solution u ∈ C 0 ([0, T ], Z̊Nm,p |d−2 ) ∩ C 1 ([0, T ], Z̊Nm−1,p |d−2 ) of Euler equation (2.1) such that pressure p(t) ∈ ZNm+1, |d−2 . Furthermore the solution depends continuously on initial data in the sense that the data-to-solution map u0 7→ u, m−1,p m,p 1 0 BZNm,p (ρ) ∩ Z̊Nm,p |d−2 → C ([0, T ], Z̊N |d−2 ) ∩ C ([0, T ], Z̊N |d−2 ) |d−2 is continuous. In section 4 we give examples of initial vector fields without log terms in Am,p N ;0 that develop log terms in it’s asymptotic expansion. In the last section we discuss uniqueness of solutions of Euler equation when we allow constant asymptotic terms. 2.1 Asymptotic spaces and diffeomorphism groups m,p Let us define the space of asymptotic diffeomorphism ZDm,p N |l associated with the space ZN |l as o n m,p 1 d ZDm,p := ϕ ∈ Diff (R ) ϕ = id +u, u ∈ Z + N |l N |l , (2.5) which consists of diffeomorphisms that form a closed subgroup of the group of asymptotic d 1 d diffeomorphisms ADm,p n,N (cf. [13, Section 5]). Here id is the identity map on R and Diff+ (R ) denotes the group of orientation preserving C 1 -diffeomorphisms of Rd . We will show ZDm,p N |l is topological subgroup of ADm,p n,N when m > 2 + d/p (see Theorem 2.1.16). We begin m,p m,p with some properties of the asymptotic spaces ZNm,p |l . It is already clear that ZN |l ⊂ AD n,N (see introduction). We begin by stating properties of remainder spaces Wδm,p and asymptotic spaces ADm,p n,N that follow from [13, § 2, Lemma 3.1, 3.3] and [13, Proposition 3.1] respectively. Proposition 2.1.1. (i) For any 0 ≤ m1 ≤ m2 and for any real weights δ1 ≤ δ2 we have that the inclusion map Wδm2 2 ,p → Wδm1 1 ,p is bounded. 64 (ii) For any m ≥ 0, δ ∈ R, and for any k ∈ Z the map f 7→ f · χa(θ) , rk m,p Wδm,p → Wδ+k is bounded. Here χ(x) ≡ χ(| x |) is a C ∞ -smooth cut-off function such that χ : R → R as in (2.4) and a(θ) ∈ C ∞ (S d−1 , C), S d−1 ⊂ Rd is the unit sphere. (iii) For any regularity exponent m ≥ 0, δ ∈ R, and a multi-index α ∈ Zd≥0 such that m−|α|,p |α| ≤ m the map ∂ α : Wδm,p → Wδ+|α| is bounded. → × Wδm,p (iv) For m > d/p and for any weights δ1 , δ2 ∈ R the map (f, g) 7→ f g, Wδm,p 2 1 is bounded. In particular, for δ ≥ 0 the space Wδm,p is a Banach algebra. Wδm,p 1 +δ2 +(d/p) Proposition 2.1.2. Assume that m > d/p. Then (i) For any integers 0 ≤ n1 ≤ n2 and 0 ≤ N1 ≤ N2 we have that the inclusion map m,p Am,p n1 ,N1 → An2 ,N2 is bounded. m−1,p (ii) If m > 1 + d/pthen for 1 ≤ j ≤ d the map u 7→ ∂u/∂xj , Am,p n,N → An+1,N +1 is bounded. (iii) Let m > d/p. For any 0 ≤ ni ≤ Ni , with i = 1, 2 and ñ = n1 + n2 , Ñ = min(N1 + n2 , N2 + n1 ) the map (f, g) 7→ f g, m,p m,p Am,p n1 ,N1 × An2 ,N2 → Añ,Ñ , is bounded. In particular, Am,p n,N is a Banach algebra for any 0 ≤ n ≤ N . m,p Now we can show that in fact ZNm,p |l is a Banach subalgebra of An,N . . then uv ∈ ZNm,p , v ∈ ZNm,p Lemma 2.1.3. If m > d/p and u ∈ ZNm,p 1 +N2 |l1 +l2 1 |l1 2 |l2 Proof. By Proposition 2.1.2 (iii) we see that uv ∈ Am,p N1 +N2 . It remains to show that the product of two asymptotic terms is again in ZNm,p |l . Let Jk r2k and ∈ ZNm,p 1 |l1 ′ Jk ′ r 2k′ . ∈ ZNm,p 2 |l2 ′ Then we need to show that Jk · Jk′ ∈ Zk+k′ |l1 +l2 . This follows from rl1 Hk−l1′ · rl2 Hk′ −l2′ ⊂ ′ ′ rl1 +l2 Mk+k′ −(l1′ +l2′ ) ⊂ Zk+k′ |l1 +l2 . Here we used that ls′ ≥ ls for s = 1, 2 and the grading (2.3). 65 Remark 18. The above proof shows in particular that the product of two asymptotic terms m,p in ZNm,p |l is again in ZN |l . In the special case we get the following Corollary. m,p m,p Corollary 2.1.4. If m > d/p and u, v ∈ ZNm,p |l then uv ∈ ZN |2l ⊂ ZN |l . m,p Since ZNm,p |l consists of the remainder space WγN and finitely many asymptotic terms we see m,p that ZNm,p |l is a closed vector subspace of AN and therefore we get by Lemma 2.1.3 m,p Corollary 2.1.5. Let m > d/p then ZNm,p |l is a Banach subalgebra of AN . For any R > 2 we will define the following spaces. Here let BR = {x || x | ≤ R} be the closed ball of radius R > 0 in Rd and BRc be it’s complement. c ZNm,p |l (BR ) n X 1 Jk (r, θ) = u u= + f, rk rk 0≤k≤N Jk ∈ Zk|l and f ∈ Wγm,p (BRc ) N o . (2.6) m,p c c (BRc ) is defined similarly with BRc instead of Rd . We have ZNm,p Here Wγm,p |l (BR ) ⊂ Hloc (BR ). N This space has a norm equivalent to one restricted from ZNm,p |l and is given by c) = |u|ZNm,p (BR |l X 0≤k≤N 1 sup |Jk (1, θ)| + kf kWγm,p (B c ) . R N Rk c This space is defined analogously to Am,p n,N (BR ) [13, Section 3]. Following Lemma allows us c to work with ZNm,p |l (BR ) (cf. [14, Lemma A.4]). m,p Lemma 2.1.6. Assume that R > 0. Then, for u ∈ Hloc (C, C) we have that u ∈ ZNm,p |l if and c m,p m,p (B c) +|·|ZNm,p only if u ∈ H m,p (BR+1 ) and u ∈ ZNm,p (BR R+1 ) |l (BR ). The norms k·kZN |l and k·kH |l on ZNm,p |l are equivalent. c Lemma 2.1.7. Assume m > d/p and R > 2. If u ∈ ZNm,p |l (BR ) then xj r2 c · u ∈ ZNm,p |l (BR ). Proof. First assume that u is in the remainder space i.e. u ∈ Wγm,p (BRc ). By Proposition N 2.1.1 (ii) we have that xj r2 c · u ∈ Wγm,p (BRc ) ⊂ ZNm,p |l (BR ). Next if u is one of the asymptotic N c terms in ZNm,p |l (BR ) so that u = 1 J˜k−l (r,θ) r k rk−l where J˜k−l ∈ Zk−l|0 (See Remark 16) then xj 1 xj J˜k−l (r, θ) · u = . r2 rk+1 rk−l+1 66 Also by Remark 16 we have xj J˜k−l ∈ Zk−l+1|0 and therefore xj r2 c · u ∈ ZNm,p |l (BR ) completing the proof. Remark 19. The function xj r is bounded in BRc and when multiplied with remainder terms u ∈ Wγm,p (BRc ) the product is again in Wγm,p (BRc ). However when multiplied with asymptotic N N terms it introduces new harmonics in the numerator thus xj r · u ∈ ZNm,p |l−1 . Lemma 2.1.8. Assume m > 1 + d/p and N ≥ 0. Let ϕ ∈ ZDm,p N |l then for large enough R > 2 and r ≥ R we have 1 1 ũ 1 ◦ϕ= = + r |ϕ(x)| r r (2.7) c where ũ ∈ ZNm,p |l (BR ). x ∈ BRc (Rd ) we can write, Proof. Let us write ϕ = id +v with v ∈ ZNm,p |l . Then for ! d d 2 X X x v |v| i i |ϕ(x)|2 = r2 + |v|2 + 2 = r2 (1 + ṽ) xi vi = r 2 1 + 2 + 2 2 r r 1 1 m,p c and we have by Lemma 2.1.7 that ṽ ∈ ZNm,p |l (BR ). Since v ∈ ZN |l is bounded and ṽ has no constant term so by choosing R > 2 large enough we have that 1 + ṽ ≥ ǫ for some ǫ > 0. c Therefore by Lemma 6.1 in [13] (1 + ṽ)−1/2 ∈ Am,p N (BR ). Also as in the proof of Lemma 6.1 P j m,p c in [13] the Taylor expansion (1 + ṽ)−1/2 = 1 + N j=1 cj ṽ + RN (ṽ) where RN (ṽ) ∈ WγN (BR ) shows that by Remark 18 following Lemma 2.1.3 we have (1 + ṽ)−1/2 = 1 + ũ such that c ũ ∈ ZNm,p |l (BR ) so we can write 1 1 ũ 1 1 1 = + . ◦ϕ= = √ r |ϕ(x)| r 1 + ṽ r r Lemma 2.1.9. Assume m > 1 + d/p and N ≥ 0. Let ϕ ∈ ZDm,p N |l and θj = (2.8) xj r then for large enough R > 2 and r ≥ R we have θj ◦ ϕ = θj + c where ũ, w̃ ∈ ZNm,p |l (BR ). 67 xj ũ + w̃ r (2.9) x ∈ BRc (Rd ) we can Proof. We can write ϕ in the form ϕ = id +v with v ∈ ZNm,p |l . Then for write by Lemma 2.1.8 θj ◦ ϕ = (xj + vj ) where w̃ = vj +vj ũ r 1 ◦ϕ r = (xj + vj ) 1 ũ + r r = θj + xj ũ + w̃ r (2.10) c ∈ ZNm,p |l (BR ). m,p m,p Lemma 2.1.10. Assume m > 1 + d/p and let ϕ ∈ ZDm,p N |l and u ∈ ZN |l then u ◦ ϕ ∈ ZN |l . m,p m,p Proof. By Proposition 5.1 in [13] we know that if ϕ ∈ ZDm,p N |l and u ∈ ZN |l then u◦ϕ ∈ AN . Therefore it only remains to show that u ◦ ϕ ∈ ZNm,p |l for the asymptotic terms in u. Assume ˜k−l ϕ = id +v as before and u = χ(r) r1k Jrk−l such that J˜k ∈ Mk−l is a homogeneous polynomial of degree k − l (The case when J˜k ∈ Mk−l−1 is similar. See Remark 16). Now by Proposition 3.1 in [13] sup|x|≥R |v(x)| is bounded as R → ∞ therefore we can take R > 2 large enough so that χ ◦ ϕ ≡ 1. For given ϕ we also take R even larger if necessary so that conditions Lemma 2.1.8 and Lemma 2.1.9 are satisfied. Now since k ≥ k − l, each monomial in u has factors of the form xj r2 and therefore for r ≥ R we can write by Lemma 2.1.8 and Lemma 2.1.9 that xj ◦ϕ= r2 1 ũ + r r x xj xj xj xj j + ũ + w̃ = 2 + 2 2 ũ + w̄ = 2 + w̄j r r r r r c where w̄ ∈ ZNm,p |l (BR ) and by Lemma 2.1.7 xj ũ r2 (2.11) m,p c c ∈ ZNm,p |l (BR ) and therefore w̄j ∈ ZN |l (BR ). Therefore for r ≥ R applying this to each monomial and again applying Lemma 2.1.7 we get that 1 J˜k−l rk−l r k−l ◦ϕ= 1 J˜k−l rk−l r k−l c + w for some w ∈ ZNm,p |l (BR ) and therefore by Lemma 2.1.8 and Lemma 2.1.7 u◦ϕ= 1 ũ + r r l 1 J˜k−l +w rk−l rk−l ! =u+h c where h ∈ ZNm,p |l (BR ). Finally applying Lemma 2.1.6 completes the proof. with Lemma 2.1.11. Assume m > 1 + d/p, then for ρ > 0 small enough if ϕ ∈ ZNm+1,p |l kϕ − idkZ m+1,p < ρ, then ϕ−1 ∈ ZNm+1,p . |l N |l 68 m,p m,p Proof. The composition is a C 1 -map Am,p n,N × An,N → An,N by Proposition 5.1 in [13]. Since m,p ZDm,p N |l is a submanifold of AD n,N closed under composition by Corollary 2.1.5 and Lemma m,p 1 ×ZDm,p 2.1.10 therefore the map F (ϕ, φ) = ϕ◦φ, ZDm+1,p N |l → ZD N |l is C -map. The partial N |l derivative of F with respect to the first variable is D1 F (ϕ, φ)|h = h ◦ φ. At the identity map we have F (id, id) = id and D1 F (id, id) = idZNm,p . Therefore by inverse mapping theorem, |l there exists ρ > 0 and a C 1 -map inv : B(ρ)ZNm,p → B(ρ)ZNm,p such that F (ϕ, inv(ϕ)) = id. |l |l −1 Therefore ϕ−1 ∈ ZDm,p and so the ∈ ADm+1,p N |l N |l . However by Proposition 5.2 in [13] ϕ remainder term of ϕ−1 has smoothness m + 1 while since ϕ−1 ∈ ZDm,p N |l , the asymptotics are C ∞ -smooth. Therefore ϕ−1 ∈ ZDm,p N |l . The following two propositions are used in the proof of Proposition 2.1.14. They follow directly from similar statements for Am,p n,N as in [13, Lemmas 7.2, 7.3, 7.4]. Proposition 2.1.12. Assume m > 1+d/p, then for any ϕ ∈ ZDm,p N |l there exists a continuous path γ : [0, 1] → ZDm,p N |l such that γ(1) = ϕ and γ(0) = id +f where f has compact support. Proposition 2.1.13. Assume m > 2 + d/p and let ϕ∗ ∈ ZDm,p N |l be fixed then there exists a such that the left translation map neighborhood U of identity id ∈ ZDm−1,p N |l ψ 7→ Lϕ∗ (ψ) := ϕ∗ ◦ ψ, U → Lϕ∗ (U ) ⊂ ZDm−1,p N |l is a C 1 -diffeomorphism. −1 ∈ ZDm,p Proposition 2.1.14. Let m > 2 + d/p then for any ϕ ∈ ZDm,p N |l . N |l we have that ϕ m,p Furthermore the map ϕ 7→ ϕ−1 , ZDm,p N |l → ZD N |l is continuous and the associated map m−1,p is a C 1 -map. ϕ 7→ ϕ−1 , ZDm,p N |l → ZD N |l m,p Proof. Let ϕ ∈ ZDm,p N |l . By Proposition 2.1.12 ϕ is connected by a path γ : [0, 1] → ZD N |l to a ϕ0 := γ(0) ∈ ZDm,p N |l so that γ(1) = ϕ and ϕ0 − id has compact support. By Proposition m,p m,p −1 −1 5.1 in [13] ϕ−1 0 ∈ AN . Also ϕ0 − id has compact support and in particular ϕ0 ∈ ZN |l . Let us denote ϕt := γ(t). Now by Lemma 2.1.11, for each ϕt along this path there exists 69 open neighborhood of id in which every diffeomorphism is invertible within Ut ⊂ ZDm−1,p N |l and such that by Proposition 2.1.13, Vt = Lϕt (Ut ) is open neighborhood of ϕt . ZDm−1,p N |l since we can write ϕ∗ = ϕ0 ◦ ψ with ψ ∈ U0 , Every ϕ∗ in V0 is invertible in ZDm−1,p N |l is closed and ZDm−1,p . Since both ψ −1 , ϕ−1 ∈ ZDm−1,p so ϕ−1 = ψ −1 ◦ ϕ−1 ∈ ZDm−1,p 0 0 ∗ N |l N |l N |l under composition. Now we can cover the path connecting ϕ0 to ϕ = ϕ1 by open sets V0 , Vt1 , Vt2 , ..., V1 (ordered by sequence in which they intersect consecutively). By induction on k, we show that any ϕ∗ ∈ Vtk is invertible in ZDm−1,p . By last argument it is enough N |l to show that ϕtk is invertible. For k = 1 choose ϕ̃ ∈ V0 ∩ Vt1 then ϕ̃ = ϕt1 ◦ ψ for some −1 . Note that ϕ̃ is hence it is invertible in ZDm−1,p ψ ∈ Ut1 , so we can write ϕ−1 t1 = ψ ◦ ϕ̃ N |l by induction and that ϕ̃ ∈ V0 ∩ Vt1 . Continuing in this way we see invertible in ZDm−1,p N |l . that ϕ−1 ∈ ZDm−1,p N |l By Proposition 2.4 in [13] we infact have that ϕ ∈ Am,p N . Therefore the remainder term of ϕ−1 − id is in Wγm,p while the asymptotic coefficients ak (θ) have only finitely many Fourier N ) so in fact ak ∈ C ∞ (S d ), hence ϕ−1 ∈ ZDm,p modes (since ϕ−1 ∈ ZDm−1,p N |l . N |l Finally the last statement follows from Proposition 5.2 in [13] abd the fact that ZDm,p N |l is a closed submanifold of ADm,p N . m,p Corollary 2.1.15. For m > 2 + d/p ZDm,p N |l is a smooth subgroup of AD N . Theorem 2.1.16. Let m > 1 + d/p then the composition and inverse (u, ϕ) 7→ u ◦ ϕ, ϕ 7→ ϕ−1 , m,p m,p ZNm,p |l × ZD N |l → ZN |l m,p ZDm,p N |l → ZD N |l are continuous maps. Furthermore the following (u, ϕ) 7→ u ◦ ϕ, ϕ 7→ ϕ−1 , m,p × ZDm,p ZNm+1,p N |l → ZN |l |l ZDm+1,p → ZDm,p N |l N |l are C 1 -mappings. 70 Lemma 2.1.17. Let m > 2 + d/p and u ∈ C([0, T ], ZNm,p |l ) for some T > 0 then there exists a unique solution ϕ ∈ C 1 ([0, T ], ZDm,p N |l ) of the equation ϕ̇ = u ◦ ϕ, ϕ|t=0 = id . (2.12) → ZNm−1,p Proof. By Theorem 2.1.16 the map F (t, ϕ) = u(t) ◦ ϕ, F : [0, T ] × ZDm−1,p |l N |l is a continuous map and it’s partial derivative with respect to the second argument is a ), the space of all bounded , ZNm−1,p → L(ZNm−1,p continuous map D2 F (t, ϕ) : [0, T ] × ZDm−1,p |l |l N |l linear maps. Thus for any t0 ∈ [0, T ], F is locally Lipschitz in ϕ. Therefore by existence and uniqueness of solutions to ODEs in Banach spaces there exists a neighborhood [to −ǫ0 , to +ǫ0 ] ) of (2.12). Since the equation ϕ̇ = u ◦ ϕ is and a unique solution ϕ ∈ C 1 ([0, T ], ZDm−1,p N |l invariant under translation and since right translation by any fixed ϕ0 ∈ ZDm−1,p is a C ∞ N |l ) with initial map therefore ϕ̇ = u ◦ ϕ has a unique solution ϕ ∈ C 1 ([t0 − ǫ0 , t0 + ǫ0 ], ZDm−1,p N |l condition ϕ|t=t0 = ϕ0 . Here ǫ0 depends on t0 . Since [0, T ] is compact we see that (2.12) has ). unique solution ϕ ∈ C 1 ([0, T ], ZDm−1,p N |l It remains to show that ϕ ∈ C 1 ([0, T ], ZDm,p N |l ). By Proposition 2.1 in [14] there is a unique m,p m,p solution ϕ̃ ∈ C 1 ([0, T ], Am,p N ;0 ). Since ZN |l is a subalgebra of AN ;0 by uniqueness of solutions . Since ). So for any t ∈ [0, T ], ϕ̃(t) ∈ ZDm−1,p of ODEs ϕ̃ = ϕ ∈ C 1 ([0, T ], ZDm−1,p N |l N |l m,p the projection map Am,p N ;0 → ZN |l is continuous again by uniqueness of solution of ODEs ϕ̃ = ϕ ∈ C 1 ([0, T ], ZNm,p |l ). Before end of this section let us define a subgroup of ZDm,p N |l . This is the subgroup of volume preserving diffeomorphism. These arise as consequence of incompressibility of Euler equation (div u = 0). We will later show that it is a smooth submanifold of ZDm,p N |l preserved by the Euler equation. It is defined as m,p Z̊DN |l n = ϕ∈ ZDm,p N |l o det[dϕ] ≡ 1 . (2.13) Lemma 2.1.18. Let m > 2 + d/p and u ∈ C([0, T ], Z̊Nm,p |l ) for some T > 0 then there exists 71 m,p a unique solution ϕ ∈ C 1 ([0, T ], Z̊DN |l ) of the equation ϕ̇ = u ◦ ϕ, ϕ|t=0 = id . (2.14) Proof. Let u ∈ C([0, T ], Z̊Nm,p |l ). First we apply ∂xj , for each 1 ≤ j ≤ d, to equation (2.12) to get following ODE [dϕ]· = [du] ◦ ϕ · [dϕ], [dϕ]t=0 = I . (2.15) Let us write A := [du] ◦ ϕ then A ∈ C([0, T ], Md×d ) where Md×d is space of d × d matrices. The linear system Ẋ = A · X, X|t=0 = I has the solution X = [dϕ]. By Wronskian identity we get det[ϕ](t, x) = e Rt 0 (div u)(s,ϕ(s,x) ds . (2.16) x) ≡ 0 and from the above equation we see that Since u(t) ∈ Z̊Nm,p |l so we have div u(t, m,p det[ϕ](t, x) = 1 therefore ϕ(t) ∈ Z̊DN |l completing the proof. 2.2 Laplace operator in asymptotic spaces The Laplace operator on the scale of weighted Sobolev spaces Wδm,p with weight δ ∈ R, except for discrete values of δ, is injective operator with closed image (cf. [11]). In fact for δ + d/p > 0 the Laplace operator is injective and it is possible to define an isomorphism by extending the spaces Wγm,p to the asymptotic spaces Am,p N as done in [12]. In Theorem 2.2.2 N below we show that this isomorphism can be obtained for the smaller asymptotic spaces ZNm,p |l . Let us start with following property of asymptotic spaces. Lemma 2.2.1. If m > 1 + d/p then the map u 7→ ∂u , ∂xj m−1,p ZNm,p |l → ZN +1|l is bounded. m−1,p Proof. By Proposition 2.1.2 this map is bounded on Am,p N → AN +1 therefore by Corollary 2.1.5 it is enough to show that the asymptotics still have the same form. For an asymptotic 72 term we have ∂ Jk 1 = 2(k+1) (−2kxj Jk + r2 ∂j Jk ). 2k ∂xj r r First we show that J 7→ xj J maps Zk|l → Zk+1|l by the grading (2.3). Now xj (Hk−l′ ) ⊂ L Mk+1−l′ = µ≥0 r2µ Hk+1−l′ −2µ . Therefore xj (Zk|l ) = M l′ ≥l ′ rl xj (Hk−l′ ) ⊂ M rl l′ ≥l ′ M µ≥0 r2µ Hk+1−l′ −2µ ⊂ M ′ rl Hk+1−l′ = Zk+1|l . l′ ≥l ′ Similarly we show that J 7→ r2 ∂j J maps Zk|l → Zk+1|l . This follows directly from r2 ∂j (rl Hk−l′ ) ⊂ ′ ′ ′ ′ rl xj (Hk−l′ ) + rl +2 ∂j (Hk−l′ ) ⊂ rl Mk+1−l′ + rl +2 Hk−1−l′ ⊂ Zk+1|l . Remark 20. The spherical harmonics of degree l will be denoted as Ylj (θ) where 0 ≤ j ≤ nl − 1 where nl is the number of spherical harmonics of degree l. The Ylj ’s are the eigenvectors of the spherical Lapalcian − △S − △S Ylj (θ) = l(l + d − 2)Ylj (θ). We will denote by Yl an element in the linear span of spherical harmonics of degree l ≥ 0. Also note that there is a one to one correspondence between spherical harmonics of degree l and harmonic polynomials of degree l in Rd given by Yl = hl |S d−1 where hl ∈ Hl is a harmonic polynomial. Therefore nl is the number of linearly independent harmonic polynomials in Rd of degree l. m−2,p Theorem 2.2.2. Assume m > 2 + d/p and N ≥ 0 then △ : ZNm,p |d−2 → ZN +2|d+1 is an isomorphism. m−2,p Proof. First we show that △(ZNm,p |d−2 ) ⊂ ZN |d+1 and we only need to show this for asymptotic m−2,p m−2,p m,p terms of u ∈ ZNm,p |d−2 since for the remainder term f ∈ WγN , △f ∈ WγN +2 ⊂ ZN +2|d+1 . Pk−2−(d−2) (θ) Yl (θ) where Yl (θ) is a linear then we can write bk−2 (θ) = l=0 Now let u = bk−2 r k−2 combination of spherical harmonics of degree l. By direct computation we see that bk−2 (θ) 1 ak (θ) ∆ = k [∆S bk−2 (θ) + (k − 2)(k − d)bk−2 (θ)] = k . k−2 r r r 73 (2.17) Since Yl are spherical harmonics so − △S Yl = l(l + d − 2)Yl and therefore we can write Pk−2−(d−2) ck,l Yl (θ). As k − 2 − (d − 2) = k − d, from here we conclude that ak (θ) = l=0 △u ∈ ZNm−2,p +2|d as ak (θ) contains only first k−d spherical harmonics. However writing k−d = λ we see that ak (θ) = [∆S bk−2 (θ) + λ(λ + d − 2)bk−2 (θ)], which shows that ak does not contain the (k − d)th spherical harmonics. Hence infact △u ∈ ZNm−2,p +2|d+1 as desired. Also the operator △ is injective since there are no harmonic functions decaying asymptotically in Rd . To see △ is onto first let u ∈ Wγm−2,p be in the remainder space. By part (b) of Lemma A.3 in N +2 m−2,p [14] △−1 u ∈ ZNm,p |d−2 since when inverting the Laplacian on the remainder space WγN +2 the Yl (θ) r d−2+l only asymptotics that we get are of the form such that Yl (θ) rl is harmonic in Rd − {0}. k And these asymptotics are already in ZNm,p |d−2 Finally for asymptotic terms of form ak (θ)/r we can solve (2.17) linearly to find corresponding bk − 2(θ). This completes the proof. Remark 21. The above isomorphism gives following commutative diagram ZNm,p |d−2 △ ZNm−2,p +2|d+1 ∂j ∂j ZNm−1,p +1|d−2 △ ZNm−3,p +3|d+1 from which it follows that △−1 ◦ ∂j = ∂j ◦ △−1 on ZNm−2,p +2|d+1 . Lemma 2.2.3. Assume m > 3 + d/p and N ≥ 0 then for any ϕ ∈ ZDm,p N |d−2 , the right translation map Rϕ : v 7→ v ◦ ϕ is an isomorphism m−2,p Rϕ : ZNm−2,p +2|d+1 → ZN +2|d+1 . Proof. We need to show this for asymptotic terms only. Let ϕ = id +v, v ∈ ZNm,p |d−2 and u ∈ ZNm−2,p +2|d+1 such that u= J˜k (x) rk+d+1 rk 1 where J˜k (x) is a homogeneous polynomial upto a factor of r i.e. we can write J˜k (x) = Pk (x) + rPk−1 (x) where Pj are homogeneous polynomials of degree j (we consider the case 74 when J˜k = Pk here, the other case J˜k = rPk−1 follows similarly. Also see Remark 16). Then for large enough R > 2 as in (2.11), for r ≥ R we have Pk (x) Pk (x) ◦ ϕ = 2k + w, 2k r r and by Lemma 2.1.9 Pk (x) k+d+1 r rk 1 1 rd+1 ◦ϕ= 1 (1 r d+1 w ∈ ZNm−2,p |d−2 . + ũ), ũ ∈ ZNm,p |d−2 therefore Pk (x) 1 ◦ ϕ = k+d+1 k + d+1 r r r 1 c By Lemma 2.1.7 h ∈ ZNm,p |d−2 (BR ) and so h rd+1 Pk (x) w + wũ + 2k ũ r =u+ h rd+1 . m,p c c ∈ ZNm,p |2d−1 (BR ) ∈ ZN |d+1 (BR ) for d ≥ 2. This completes the proof. From Theorem 2.2.2 and Lemma 2.2.3 we directly get that Corollary 2.2.4. Assume m > 3 + d/p then for any ϕ ∈ ZDm,p N |d−2 , the map △ϕ = Rϕ ◦ △ ◦ Rϕ−1 is an isomorphism m−2,p △ϕ : ZNm,p |d−2 → ZN +2|d+1 with inverse map (△ϕ )−1 = Rϕ ◦ △−1 ◦ Rϕ−1 where △−1 is inverse of the map in Theorem 2.2.2. 2.3 Euler vector field We can eliminate the pressure term from the Euler equation (2.1) by applying divergence to both sides to get − △p = div(u · ∇u) = tr([du]2 ) + u · ∇(div u) = tr([du]2 ). Here we used that div u = 0. Let us define Q(u) := tr([du]2 ). Note that in the above equation the second order derivative term that was zero is u·∇(div u). We write u·∇(div u) = ∇((div u)u)−(div u)2 . From this we will define Q̃(u) = Q(u)−(div u)2 = tr([du]2 )−tr([du])2 . 75 Therefore we can write from above for the Euler equation − △p = Q(u) = Q̃(u). Substituting into (2.1) we get    ut + u · ∇u = ∇ ◦ △−1 ◦ Q̃(u),   u|t=0 = u0 , (2.18) This modification allows us to remove certain terms from Q(u) that introduce log-terms when inverting △. Following lemma combined with Theorem 2.2.2 shows that the Euler equation preserves the space ZNm,p |d−2 . m−1,p Lemma 2.3.1. Assume m > 1 + d/p and N ≥ 0. Let u ∈ ZNm,p |d−2 then Q̃(u) ∈ ZN +2|2d−2 (⊂ ZNm−1,p +2|d+1 if d ≥ 3). m−1,p m,p Proof. Since ZNm,p |d−2 ⊂ A1,N , so Q̃(u) ∈ A2,N +2 . Therefore we only need to show that the asymptotic part lies in the space ZNm−1,p +2|2d−2 . If we write the vector field as (see Remark 16) u = χ(r) X 1 J~k X rd−2 ~ J˜ + f + f = χ(r) k rk 2k k r r 0≤k≤N 0≤k≤N . We have written Jk,j , the j th component of the vector such that Jk,j ∈ Zk|d−2 and f ∈ Wγm,p N J~k , with a factor of rd−2 so that J˜k,j ∈ Zk−d+2|0 . When differentiating below we will only keep track of asymptotic terms and therefore ignore the derivatives of χ(r) since they do not contribute to the asymptotic part. Then by a direct computation we see that the asymptotic terms in div u can be written as div u ≃ X rd−2 ~˜ · ~r + r2 div J~˜ −λ J k k k r2k+2 0≤k≤N (2.19) such that λk = 2k − d + 2 and ~r is the position vector in Rd . And therefore we can write the asymptotic part of (div u)2 as r2(d−2) ~ ~ 2 ˜ ˜ (div u) ≃ λk λl (Jk · ~r)(Jl · ~r) + r Ak+l+4−2d r2(k+l+2) 0≤k+l+2≤N +1 2 X so that As ∈ Zs|0 . Similarly when computing Q(u) = tr([du]2 ) we write d−2 rd−2 r ~˜ 2 ˜ ˜ −λ x J + r ∂ J = J d k β k,α β k,α . k r2k r2k+2 α,β 76 (2.20) Then the asymptotics part of Q(u) can be written as follows. Note that in the summation below we have 0 ≤ k + l + 2 ≤ N + 1, k, l ≥ 0 and 1 ≤ α, γ ≤ d. (d is the dimension in Rd ) X rd−2 ~ rd−2 ~ J˜ · d J˜l (2.21) Q(u) ≃ tr d 2k k 2l r r k,l X r2(d−2) (−λk xγ J˜k,α + r2 ∂γ J˜k,α )(−λl xα J˜l,γ + r2 ∂α J˜l,γ ) 2k+2l+4 r k,l;γ,α X r2(d−2) 2 4 ˜ ˜ ˜ ˜ ˜ ˜ ˜ λk λl xγ xα Jk,α Jl,γ − 2r (kxγ Jk,α ∂β Jl,γ + lxα Jl,γ ∂γ Jk,α ) + r (∂γ Jk,α ∂β Jl,γ ) = r2k+2l+4 k,l;γ,α X r2(d−2) ~ ~ 2 ˜ ˜ (2.22) λk λl (Jk · ~r)(Jl · ~r) + r Bk+l+4−2d = r2(k+l+2) k,l = where Bs ∈ Zs|0 . From (2.20) and (2.22) we see that asymptotic part of Q̃(u) has numerator r2d−2 (Bk+l+4−2d − Ak+l+4−2d ) in Zk|2d−2 hence u ∈ ZNm−1,p +1|2d−2 . Lemma 2.3.2. Assume m > 3 + d/p and let l = d − 2.The curve u ∈ C([0, T ], ZNm,p |l ) ∩ ) is a solution of equation (2.18) with initial data u0 ∈ Z̊Nm,p C 1 ([0, T ], ZNm−1,p |l if and only if |l m−1,p 1 ) and it is a solution of Euler equation (2.1) for some u ∈ C([0, T ], Z̊Nm,p |l ) ∩ C ([0, T ], Z̊N |l −p ≡ p(t) ∈ ZNm+1,p |d−2 , t ∈ [0, T ]. m−1,p 1 ) be a solution of (2.18). Applying diverProof. Let u ∈ C([0, T ], ZNm,p |l ) ∩ C ([0, T ], ZN |l gence to both sides of (2.18) we get (div u)t + u · ∇(div u) = −(div u)2 . (2.23) and ∩ C 1 [0, T ], ZNm−2,p Let ϕ(t) be solution of (2.12). Since div u ∈ C [0, T ], ZNm−1,p |l |l 1 d ϕ ∈ C 1 [0, T ], ZDm,p N |l with m > 3 + (d/p) we conclude that div u ∈ C [0, T ] × R , R and ϕ ∈ C 1 (Rd , Rd ). This together with (2.23) implies that ω̃ = (div u) ◦ ϕ satisfies ω̃t = −ω̃ 2 . Solving this differential equation we can write ω̃(t) = div u0 . 1 + t div u0 (2.24) Since we assume that initial data is divergence free, i.e. div u0 = 0. This implies that for any m−1,p 1 ). Now Q̃(u) ∈ T(ZDm−1,p t, div u(t) = 0. Thus u ∈ C([0, T ], Z̊Nm,p N +2|d+1 ) so |l ) ∩ C ([0, T ], Z̊N |l 77 by Theorem 2.2.2 we can define p = △−1 Q̃(u) so u ∈ ZNm+1,p |d−2 m−1,p 1 Conversely if u ∈ C([0, T ], Z̊Nm,p |d−2 ) ∩ C ([0, T ], Z̊N |d−2 ) is a solution of Euler equation (2.1) then taking divergence on both sides of (2.1) we get − △p = div (u · ∇u) = tr([du]2 ) + u · −1 ∇ div u = Q̃(u) ∈ ZNm−1,p +2|2d−2 by Lemma 2.3.1. Now by Theorem 2.2.2 −p = △ Q̃(u) ∈ ZNm+1,p |d−2 . Substituting this in equation (2.1) shows that u satisfies equation (2.18) with div u0 = 0. Following Lemma 7.1 in [14] we write the Euler equation (2.1) in it’s equivalent form as a m,p dynamical system on the tangent bundle ZDm,p N |d−2 × ZN |d−2 as follows (ϕ̇, v̇) = (v, E2 (ϕ, v)) ≡ E(ϕ, v) (ϕ, v)|t=0 = (id, u0 ), (2.25) u0 ∈ ZNm,p |d−2 . Here E2 (ϕ, v) = (Rϕ ◦ ∆−1 ◦ Rϕ−1 ) ◦ (Rϕ ◦ ∇ ◦ Q̃ ◦ Rϕ−1 )(v) is the second component of the Euler vector field E and Rϕ (u) = u ◦ ϕ is the composition operator. m,p Lemma 2.3.3. Assume m > 3 + d/p and N ≥ 0. Let (ϕ, v) ∈ T(ZDm,p N |d−2 ) := ZD N |d−2 × m,p m,p m,p m,p ZNm,p |d−2 then E2 (ϕ, v) ∈ ZN |d−2 . So that E2 defines a map E2 : ZD N |d−2 × ZN |d−2 → ZN |d−2 . Proof. By Lemma 2.3.1 Q̃ ◦ Rϕ−1 (v) ∈ ZNm−1,p +2|d+1 . We can write by Theorem 2.2.2 that E2 (ϕ, v) = Rϕ ◦∇◦∆−1 ◦ Q̃◦Rϕ−1 (v). And by Theorem 2.2.2 now ∆−1 ◦ Q̃◦Rϕ−1 (v) ∈ ZNm+1,p |d−2 and finally Lemma 2.2.1 gives the result. 2.4 Smoothness of Euler Vector Field In this section we will show that the Euler vector field E, defined as in (2.25) is a real-analytic vector field on the tangent bundle of ZDm,p N |d−2 . Let us start with following. Lemma 2.4.1. For m > 3 + d/p the following map (ϕ, v) 7→ △ϕ (v), m−2,p m,p ZDm,p N |d−2 × ZN |d−2 → ZN +2|d+1 is real-analytic. Here △ϕ = Rϕ ◦ △ ◦ Rϕ−1 as in Corollary 2.2.4. 78 Proof. We can write following directly Rϕ ◦ ∇ ◦ Rϕ−1 (v) = [dv] · [dϕ]−1 (2.26) Rϕ ◦ div ◦ Rϕ−1 (v) = tr([dv] · [dϕ]−1 ). (2.27) By Lemma 5.2 in [14] the above maps are analytic. Finally composing these maps we see that from Corollary 2.2.4 we see that (ϕ, v) 7→ △ϕ (v) is real-analytic. Now consider the map m,p m−2,p D : ZDm,p N |d−2 × ZN +2|d+1 → ZN |d−2 , D(ϕ, v) = Rϕ ◦ △−1 ◦ Rϕ−1 (v). (2.28) Proposition 2.4.2. Assume m > 2 + d/p. Then the map (2.28) is real-analytic. m−2,p Proof. Let E = ZNm,p |d−2 and F = ZN +2|d+1 . Then the set of all invertible linear maps GL(E, F ) is a open group inside L(E, F ), the Banach space of all linear bounded maps from E to F . Furthermore the inverse map inv : GL(E, F ) → GL(F, E), G 7→ G−1 (2.29) is real-analytic using Neumann series expansion. By Corollary 2.2.4 the map ϕ 7→ △ϕ , ZDm,p N |d−2 → GL(E, F ) (2.30) is well-defined. Also △ϕ = (D2 C)(ϕ, 0) where C(ϕ, v) = △ϕ (v) and D2 denote partial derivative with respect to second argument. By Lemma 2.4.1 the map (2.30) is real-analytic. Composing this with map (2.29) we see that (2.28) is real-analytic. m,p m,p Theorem 2.4.3. The Euler vector field E : ZDm,p N |d−2 × ZN |d−2 → ZN |d−2 is real-analytic. Proof. We only need to show that E2 is real-analytic. We can write E2 = (Rϕ ◦ ∆−1 ◦ Rϕ−1 ) ◦ (Rϕ ◦ ∇ ◦ Q ◦ Rϕ−1 ) − (Rϕ ◦ div ◦Rϕ−1 )2 (v). By Theorem 6.1 in [14] we see that first term is real-analytic. Also from (2.27) (ϕ, v) 7→ (Rϕ ◦ div ◦Rϕ−1 )2 (v) is real-analytic. 79 2.5 Proof of Theorem 2.0.1 Lemma 2.5.1. Assume that m > 3 + d/p. Then the map (ϕ, v) 7→ v ◦ ϕ−1 , m−1,p m,p 1 C 1 ([0, T ], T(ZDm,p N |d−2 )) → C([0, T ], ZN |d−2 ) ∩ C ([0, T ], ZN |d−2 ) gives a bijection between the solutions of the dynamical system (2.25) and the solutions of m−1,p 1 equation (2.18) in C([0, T ], ZNm,p |d−2 ) ∩ C ([0, T ], ZN |d−2 ). m−1,p 1 Proof. Let u ∈ C([0, T ], ZNm,p |d−2 ) ∩ C ([0, T ], ZN |d−2 ) be a solution of (2.18). By Lemma 2.1.17, there exists a ϕ ∈ C 1 ([0, T ], ZDm,p N |d−2 ) satisfying (2.12). Let v = u ◦ ϕ then (ϕ, v) ∈ 1 C([0, T ], T(ZDm,p N |d−2 )) by Theorem 2.1.16. We will first show that this map is C and sat1 isfies the dynamical system (2.25). By Sobolev embedding ZNm−1,p |d−2 ⊂ C therefore u, ϕ ∈ C 1 ([0, T ] × Rd , Rd ). We can differentiate point-wise to get vt = ut ◦ ϕ + [du] ◦ ϕ · ϕ̇ = ut ◦ ϕ + [du] ◦ ϕ · u ◦ ϕ = (ut + u · ∇u) ◦ ϕ. From equation (2.25) and (2.18) we can write vt = (Rϕ ◦ ∆−1 ◦ ∇ ◦ Q̃ ◦ Rϕ−1 )(v) = E2 (ϕ, v). Integrating this equation point-wise we get v(t, x) = u0 (x) + Z t 0 E2 (ϕ(s, x), v(s, x))ds. m,p As v ∈ C([0, T ], ZNm,p |d−2 ) and ϕ ∈ C([0, T ], ZD N |d−2 ) and E2 is analytic so the integral m,p 1 converges in ZNm,p |d−2 . Therefore v ∈ C ([0, T ], ZN |d−2 ) and v̇ = E2 (ϕ, v). Hence (ϕ, v) ∈ C 1 ([0, T ], T(ZDm,p N |d−2 )) and satisfies the dynamical system (2.25). m−1,p 1 −1 ∈ C([0, T ], ZNm,p Conversely if (ϕ, v) ∈ C 1 ([0, T ], T(ZDm,p |d−2 )∩C ([0, T ], ZN |d−2 ) N |d−2 )) then u = v◦ϕ by Theorem 2.1.16. Differentiating v = u ◦ ϕ we see that u satisfies equation (2.18). Combining Lemma 2.5.1 and Lemma 2.3.2 we easily get following. 80 Lemma 2.5.2. Assume that m > 3 + d/p. Then the map (ϕ, v) 7→ v ◦ ϕ−1 , m−1,p m,p 1 C 1 ([0, T ], T(ZDm,p N |d−2 )) → C([0, T ], Z̊N |d−2 ) ∩ C ([0, T ], Z̊N |d−2 ) gives a bijection between the solutions of the dynamical system (2.25) with u0 ∈ Z̊Nm,p |d−2 and m−1,p 1 the solutions of the Euler equation (2.1) in C([0, T ], Z̊Nm,p |d−2 ) ∩ C ([0, T ], Z̊N |d−2 ). Proof of Theorem 2.0.1. By Theorem 2.4.3 the Euler vector field (2.25) is real-analytic so the existence and uniqueness of solutions of ODEs on Banach spaces (see [9]) there exists (ρ0 ) there is a unique solution (ϕ, v) ∈ ρ0 > 0 and T0 > 0 such that for any u0 ∈ BZNm,p |d−2 m,p 0 C 1 ([0, T0 ], T(ZDm,p N |d−2 )). By Lemma 2.5.2 this gives a solution u ∈ C ([0, T0 ], ZN |d−2 ) ∩ C 1 ([0, T0 ], ZNm−1,p |d−2 ). Now for any ρ > 0 we use symmetry of Euler equation u 7→ uc where uc (t) = cu(ct) and take c = ρ0 ρ and T = cT0 so that for any u0 ∈ BZNm,p (ρ) we have a |d−2 m−1,p 1 solution u ∈ C([0, T ], ZNm,p |d−2 ) ∩ C ([0, T ], ZN |d−2 ). 2.6 Volume preserving asymptotic diffeomorphism The Euler vector field is defined on the tangent bundle T(ZDm,p N |d−2 ) thus the exponential map ExpE of the spray E can be defined as follows. As E is a real-analytic vector field we see that by existence and uniqueness of solutions of an analytic ODE in a Banach space (see [6]), that there exists an open neighborhood U of zero in ZNm,p |d−2 such that for any initial data (ϕ, v)|t=0 = (id, u0 ), with u0 ∈ U, the ODE on T(ZDm,p N |d−2 ) (ϕ̇, v̇) = E(ϕ, v) = (v, E2 (ϕ, v)) (2.31) m,p has a unique real-analytic solution (−2, 2) → ZDm,p N |d−2 × ZN |d−2 , t 7→ (ϕ(t; u0 ), v(t; u0 )), so that the map m,p (−2, 2) × U → ZDm,p N |d−2 × ZN |d−2 (2.32) is real-analytic. Therefore the exponential map of E, ExpE : U → ZDm,p N |d−2 , 81 u0 7→ ϕ(1; u0 ) (2.33) is also real-analytic. Furthermore the differential at (id, 0) is d0 ExpE = idZNm,p , the identity |d−2 map on ZNm,p |d−2 . Therefore by inverse function theorem on Banach spaces we get the following. Proposition 2.6.1. Assume that m > 3 + d/p. Then there exists an open neighborhood U m,p of zero in ZNm,p |d−2 and an open neighborhood V of id in ZD N |d−2 so that ExpE : U → V is a real-analytic diffeomorphism. Next we prove the following. Here U , V are the open sets as in Proposition 2.6.1 above. Proposition 2.6.2. Assume that m > 3 + d/p, then for any u0 ∈ U ∩ Z̊Nm,p |d−2 , we have that m,p ExpE (u0 ) ∈ Z̊DN |d−2 . Furthermore the exponential map m,p ExpE : U ∩ Z̊Nm,p |d−2 → V ∩ Z̊DN |d−2 (2.34) is a real-analytic diffeomorphism. m,p 1 Proof. Assume that u0 ∈ U ∩ Z̊Nm,p |d−2 and let (ϕ, v) ∈ C ([0, 1], T(ZD N |d−2 )) be a solu- tion of the dynamical system (2.25). By Lemma 2.5.2 u := v ◦ ϕ−1 ∈ C([0, 1], Z̊Nm,p |d−2 ) ∩ C 1 ([0, 1], Z̊Nm−1,p |d−2 ) is solution of Euler equation (2.1) hence in particular div u ≡ 0. By (2.25) we also see that ϕ̇ = v = u ◦ ϕ and ϕ|t=0 = id, so by Lemma 2.1.18 uniqueness of solutions m,p m,p to (2.14) give ϕ(t) ∈ Z̊DN |d−2 . Hence ExpE (u0 ) ∈ V ∩ Z̊DN |d−2 . m,p Conversely we will show that if u0 ∈ U r Z̊Nm,p |d−2 then ExpE (u0 ) ∈ V r Z̊DN |d−2 . This to- gether with the fact that ExpE : U → V is a diffeomorphism will show that the map (2.34) is bijective and therefore real-analytic diffeomorphism as U ∩ Z̊Nm,p |d−2 ⊂ U is a real-analytic m,p 1 submanifold. We let u0 ∈ U r Z̊Nm,p |d−2 and let (ϕ, v) ∈ C ((−2, 2), T(ZD N |d−2 )) be a solution of (2.25). By Lemma 2.5.1 u = v ◦ ϕ−1 is solution of (2.18) so by equation (2.24) we get that Z t ω̃(s)ds = 0 Z t 0 (div u) ◦ ϕ ds = Z t 0 div u0 ds = ln (1 + t div u0 ). 1 + s div u0 Since ϕ̇ = v = u ◦ ϕ, and ϕ|t=0 = id therefore by equation (2.16) det[ϕ](t, x) = 1 + t div u0 (x). 82 (2.35) If the initial divergence is non-zero at some point x0 , then (det[dϕ](t, x0 ) − 1) will remain m,p non-zero. And therefore ExpE (u0 ) = ϕ(1; u0 ) ∈ / Z̊DN |d−2 . Theorem 2.6.3. m,p m,p (a) Z̊DN |d−2 is a real-analytic submanifold of ZD N |d−2 . m,p m,p m,p (b) For any ψ ∈ Z̊DN |d−2 the tangent space Tψ (Z̊DN |d−2 ) to Z̊DN |d−2 at the point ψ m,p is the image of the right-translation map Z̊ , where R is ψ, Rψ Z̊Nm,p ψ N |d−2 |d−2 m,p Rψ : Z̊Nm,p |d−2 → ZN |d−2 . m,p m,p (c) The Euler vector field E is tangent to the submanifold T (Z̊DN |d−2 ) in T (ZD N |d−2 ). Proof. (a) Since U ∩ Z̊Nm,p |d−2 ⊂ U is a real-analytic submanifold by Proposition 2.6.1, its image m,p m,p is real-analytic submanifold of V. By Proposition 2.6.2 the image is V ∩ Z̊DN |d−2 . So Z̊DN |d−2 is a submanifold in open neighborhood of id in ZDm,p N |d−2 . In general, for any fixed ψ ∈ m,p m,p m,p Z̊DN |d−2 the right translation Rψ : ZD N |d−2 → ZD N |d−2 is a real analytic diffeomorphism m,p m,p so that Uψ = Rψ (U ) is a neighborhood of ψ. Furthermore Uψ ∩ Z̊DN |d−2 = Rψ (U ∩ Z̊DN |d−2 ). m,p m,p (b) At the identity id we have Tid (Z̊DN |d−2 ) = Z̊N |d−2 . Again general case follows since right m,p translation Rψ is a real analytic diffeomorphism on Z̊DN |d−2 by part (a). m,p 1 (c) Let u0 ∈ Z̊Nm,p |d−2 and let (ϕ, v) ∈ C ([0, T ], T(ZD N |d−2 )) be a solution of (2.25). Then by m,p Lemma 2.5.2 and Lemma 2.1.18 we see that in fact ϕ ∈ C 1 ([0, T ], Z̊DN |d−2 ) and therefore v = m,p m,p m,p 1 [0, T ], T (ZD ) ). Since we already know (ϕ, v) ∈ C ) = R ( Z̊ ϕ̇ ∈ Tϕ (Z̊DN ϕ N |d−2 |d−2 N |d−2 m,p we have shown that (ϕ, v) ∈ C 1 [0, T ], T (Z̊DN |d−2 ) . In particular we see that E(id, u0 ) = d (ϕ, v) dt t=0 m,p m,p is tangent submanifold T (Z̊DN |d−2 ). Now note that for any (ϕ, v) ∈ T (Z̊DN |d−2 ) m,p and any fixed ψ ∈ Z̊DN |d−2 , E(Rψ ϕ, Rψ v) = Rψ (E(ϕ, v)) where Rψ (v, w) = (Rψ v, Rψ w). This is because Rψ maps solutions of (2.25) to solutions of (2.25) with right translated m,p initial conditions. Now take an arbitrary (ϕ0 , v0 ) ∈ T (Z̊DN |d−2 ). By part (b) there we have v0 = u0 ◦ϕ0 for some u0 ∈ Z̊Nm,p |d−2 . We have E(ϕ0 , v0 ) = Rϕ0 (E(id, u0 )). Finally Rϕ0 (E(id, u0 )) m,p m,p is tangent to T (Z̊DN |d−2 ) at (ϕ0 , v0 ) as E(id, u0 ) is tangent to T (Z̊DN |d−2 ) at (id, u0 ) and Rϕ0 : (ϕ, v) 7→ (ϕ ◦ ϕ0 , v ◦ ϕ0 ) is a C ∞ diffeomorphism. 83 2.7 Examples Example 1. We construct a divergence-free initial vector field u that is asymptotically in Am,p N and such that u̇ contains log terms. This can be done using so called stream functions in 3 dimensions. Such divergence-free vector fields are given by u = ∇ψ × ∇φ = ∇ × (ψ∇φ) . (2.36) (ψ and φ are the stream functions.) We make specific choices of stream functions to get a counter example. One such choice is by first fixing ψ(x, y, z) = φ(x, y, z) = −1 1 3 r3 − 1 1 . 2 r2 x2 2 + z2 . 2 Now we take Then we get    x   1 1   ~r + ∇ψ =  0  , ∇φ = r4 r5   z   −yz   1 1   0 . u= +   r4 r5   xy (2.37) (2.38) This is the vector field away from the origin. Near the origin we use a cut-off function that adds a compactly supported vector field to u. Note that in u the terms such as −yz r4 are asymptotics with highest harmonic in numerator. This we know is necessary to get a counter example. Now we compute the differential as follows.  2 2 2 4xyz + 5xyz , − rz4 − rz5 + 4yr6 z + 5yr7 z , − ry4 − ry5 + 4yz + r6 r7 r6   du =  0, 0, 0   2 2 2 2 y + ry5 − 4xr6 y − 5xr7 y , rx4 + rx5 − 4xy − 5xy , − 4xyz − 5xyz r4 r6 r7 r6 r7 In tr ([du]2 ) we look at the numerator, T , of and 1 r4 × 1 r 11  .   that comes from products of form 1 . r7 T = 2(5x2 y 2 + 5y 2 z 2 + 4x2 y 2 + 4y 2 z 2 ) = 18y 2 x2 + z 2 . 84  5yz 2 r7  (2.39) 1 r5 × 1 r6 It is easly to check that T can be written as a non-zero harmonic polynomial H of degree 4 modulo r2 as follows. 18 T = H + r2 7 3r2 2 +y . 5 Therefore we can write 1 T T = 7 4. 11 r r r We see that T r4 contains a (k − d)th non-zero spherical harmonic (Here k = 7 and d = 3). T r11 Therefore when computing ∆−1 tr ([du]2 ), the term will give rise to log terms. Example 2. Here we will provide another example of a initial velocity vector field that develops log-term. In this example we show that u0 ∈ ZNm,p |1 is not enough to avoid log-term and that in fact the divergence-free conditions are needed to avoid it. Such an example is easy to construct since there is no divergence-free condition here. Consider the velocity field given asymptotically as  1 r  u=   +  1 x r3 r  0 0  .   The differential is easily computed as  y −4x2 +r 2 x − r 3 + r 6 , − r 3 −  du =  0, 0,   0, 0, So that tr([du]2 ) = − rx3 + −4x2 +r 2 r6 2 (2.40) 4xy , r6 contains the term − rz3 − 0 0  4xz r6  −2x(−4x2 +r2 ) r9  .   = 1 8x3 −2xr 2 . r6 r3 (2.41) Now if the numerator (8x3 −2xr2 ) contains a harmonic polynomial modulo r2 of degree k−d = 6−3 = 3 then inverting the Laplacian we will get asymptotics with log-terms. A direct computation shows that we can write x3 = h + 35 xr2 , such that h = x3 − 35 xr2 is harmonic polynomial. However when computing Q̃(u) we see that no log-terms appear. 85 2.8 Remarks on solutions with constant asymptotic term From the Euler equation we can see apriori that for any spatially constant vector field ~c(t), the transformation p 7→ p + ~c · ~x transforms the Euler equation to ut + u · ∇u = −∇p − ~c(t). (2.42) Now comparing the asymptotics we see that ~a0 (t) satisfies ~a˙ 0 (t) = −~c(t). Note that ~c(t) was arbitrary and this gives non-uniqueness. However uniqueness can be obtained by restricting to solutions such that p(x) = o(|x|). However it is not obvious that solutions to (2.42) exist. We can follow same approach and construct a corresponding dynamical system that gives solutions to (2.42). L ZNm,p |d−2 . This space is still a Ba- (ϕ̇, v̇) = (v, Ẽ2 (ϕ, v)) ≡ Ẽ(ϕ, v) (2.43) m,p We can append constants to ZNm,p |d−2 to obtain Z̃N |d−2 := R m,p nach algebra in Am,p 0;N and therefore we can similarly define Z̃DN |d−2 and obtain a dynamical m,p m,p system analogous to (2.25) on Z̃DN |d−2 × Z̃N |d−2 . (ϕ, v)|t=0 = (id, u0 ) u0 ∈ Z̃Nm,p |d−2 where Ẽ2 (ϕ, v) = E2 (ϕ, v) − ~c(t). This vector is field is analytic and therefore has a unique solution, depending on ~c(t), which gives a solution uc of the Euler equation. 2.9 Asymptotics preserved by Euler equation Combined with the divergence-free condition on the asymptotics we will see that certain asymptotic terms are constant in Euler equation. This is true for dimensions greater than 3. Let us write u = χ(r) X 1 J~k X rd−2 ~ J˜ + f + f = χ(r) k rk 2k k r r 0≤k≤N 0≤k≤N 86 then divergence is given by div u = χ(r) X rd−2 ˜ (−λk J~k · ~r + r2 div J~k ) + f. 2k+2 r 0≤k≤N And so on the asymptotic terms the condition div u ≡ 0 gives from (2.19) λk J~k · ~r = r2 div J~k here λk = 2k − d + 2. Using this we first write Q(u) = X k,l d−2 d−2 r r J~˜k · d J~˜l tr d 2k 2l r r (2.44) X r2(d−2) (−λk xγ J˜k,α + r2 ∂γ J˜k,α )(−λl xα J˜l,γ + r2 ∂α J˜l,γ ) 2k+2l+4 r k,l;γ,α X r2(d−2) 2 4 ˜ ˜ ˜ ˜ ˜ ˜ ˜ = λ λ x x J J − 2r (kx J ∂ J + lx J ∂ J ) + r (∂ J ∂ J ) k l γ α k,α l,γ γ k,α β l,γ α l,γ γ k,α γ k,α β l,γ r2k+2l+4 k,l;γ,α X r2(d−2) ~˜ · ~r)(J~J˜ · ~r) + r2 B (2.45) λ λ ( J = k l k l k+l+4−2d 2(k+l+2) r k,l X r2(d−2) ~˜ div J~˜ + r2 B 4 = . (2.46) r div J k l k+l+4−2d r2(k+l+2) k,l = So that Q(u) has numerators in Zk+l+2|2d−2 . So △−1 (Q(u)) has numerators in Zk+l|2d−4 expect top harmonic terms are added that come from the non-local part of the operator. Now writing u · ∇u we get for the asymptotic terms [u · ∇u]α = = = X rd−2 Jl,β k,l r2l ∂β rd−2 Jk,α X r2(d−2) = (−λk Jl,β xβ Jk,α + r2 Jl,β ∂β Jk,α ) 2(k+l+1) r2k r k,l X r2(d−2) (−λk (J~l · ~x)Jk,α + r2 J~l · ∇Jk,α ) 2(k+l+1) r k,l X r2(d−2) 2~ ~ (r2 λk λ−1 l div Jl Jk,α + r Jl · ∇Jk,α ) 2(k+l+1) r k,l And so the numerator above lies in Zk+l+1|2d−2 . Therefore in ut = −u · ∇u + ∇ ◦ △−1 ◦ Q̃(u) the modes between d − 2 and 2d − 4. So there are d − 3 modes preserved. 87 88 Appendices 89 Eulers equation in complex coordinates Using the complex structure of C = R2 we rewrite the Euler equation in terms of the holomorphic projection of the vector field v. The vector field v = A(x, y)∂x + B(x, y)∂y in R2 can be written as v = a(z, z̄)∂z + b(z, z̄)∂z̄ , where a = A + iB, b = ā. Here we use following transformations ∂x = ∂z + ∂z̄ , ∂y = i(∂z − ∂z̄ ). The real vector field v is now completely determined by one complex function a(z, z̄) and we call it the complex function associated to the field v. We will use the above expression to write divergence and Lapalce operators in terms of the Cauchy operator ∂z and the complex function a. The divergence and the gradient operators. Direct computation for divergence gives following expression: div v = ∂x A + ∂y B = (∂z + ∂z̄ )A + i(∂z − ∂z̄ )B = ∂z (A + iB) + ∂z̄ (A − iB) = ∂z a + ∂z̄ b = 2 Re(∂z a) 89 Similar computation gives v · ∇v = (aaz + āaz̄ )∂z + (aāz + āāz̄ )∂z̄ and applying divergence div a = 2 Re az to this, we get div (u · ∇u) = ∂z (aaz + āaz̄ ) + ∂z̄ (āāz̄ + aāz ) (47) = (az )2 + (āz̄ )2 + 2āz az̄ = 2(az )2 + 2|az̄ |2 Now if p : R2 → R is a scalar function such as the pressure term in Euler equation then we can write its gradient field as ∇p = (∂x p)∂x + (∂y p)∂y = px (∂z + ∂z̄ ) + py (i(∂z − ∂z̄ )) = (px + ipy )∂z + (px − ipy )∂z̄ = 2pz̄ ∂z + 2pz ∂z̄ Finally let us compute the curl of v as curl v = 1 az − āz̄ . 2i (48) Note that for a = a(z, z̄) divergence free we obtain that curl v = −i ∂z a. Lemma .0.1. The Euler equation in R2 can be written in terms of complex function a : C → C and real pressure function p : R2 → R as at + (aaz + āaz̄ ) = −2∂z̄ p, div a = az + āz̄ = 0. (49) Proof. 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