MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS Vol. 28, Nos. 7 & 8, pp. 1391–1436, 2003 A Nonhomogeneous Boundary-Value Problem for the Korteweg–de Vries Equation Posed on a Finite Domain Jerry L. Bona,1,* Shu Ming Sun,2 and Bing-Yu Zhang3 1 Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois, USA 2 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA 3 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio, USA ABSTRACT Studied here is an initial- and boundary-value problem for the Korteweg–de Vries equation posed on a bounded interval with nonhomogeneous boundary conditions. This particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed. It is shown here that this initial-boundary-value problem is globally well-posed in the L2-based Sobolev space H s(0, 1) for any s
0. In addition, the mapping that associates to appropriate initial- and boundary-data the corresponding solution is shown to be analytic as a function between appropriate Banach spaces. *Correspondence: Jerry L. Bona, Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60607, USA; E-mail: bona@math.uic.edu. 1391 DOI: 10.1081/PDE-120024373 Copyright & 2003 by Marcel Dekker, Inc. 0360-5302 (Print); 1532-4133 (Online) www.dekker.com MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1392 Bona, Sun, and Zhang 1. INTRODUCTION This article is concerned with the Korteweg–de Vries equation (KdV-equation henceforth) ut þ ux þ uux þ uxxx ¼ 0 ð1:1Þ posed as an initial- and boundary-value problem. In the conception pursued here, one asks for a solution of (1.1) for ðx, tÞ 2
 Rþ where
is an interval in R, subject to an initial condition uðx, 0Þ ¼ ðxÞ, for x 2
, ð1:2Þ and appropriate boundary conditions at the ends of the interval. In applications to physical problems, the independent variable x is often a coordinate representing position in the medium of propagation, t is proportional to elapsed time, and u(x,t) is a velocity or an amplitude at the point x at time t. Here and below, if f ¼ f ðx, tÞ is a function of x and t, then fx is shorthand for @x f and similarly ft ¼ @t f . When
¼ R, the entire real line, this is the classical problem whose study was initiated by Gardner et al. (1974) and Lax (1968) in the middle 1960’s by way of the inverse scattering theory and by Sjöberg (1970) and Temam (1969) in the late 1960’s using the then new methods for the analysis of nonlinear partial differential equations, and by many others since. As will be described presently, this pure initial-value problem continues to attract attention and its mathematical theory has proved to be subtle. Another configuration that arises naturally in making predictions of waves is to take
¼ Rþ ¼ fx j x > 0g and specify uð0, tÞ for t > 0 and uðx, 0Þ ¼ 0, say, for x > 0. This corresponds to a known wavetrain generated at one end and propagating into a quiescent region of the medium of propagation. If u(0, t) is of small amplitude (small compared to one in this scaling) and has primarily low frequency content, then the waves generated by the boundary disturbance will satisfy the assumptions underlying the derivation of the KdV-equation. The semi-infinite aspect of the domain mirrors the fact that the KdV-equation written in the form (1.1) is an approximation only for waves moving in the direction of increasing values of x. Once the incoming waves encounter a boundary, reflection will come into play, and the KdV-equation is no longer expected provide an accurate rendition of reality. The problem of imposition of boundary data at the right-hand end of the domain does not arise when the KdV-equation is posed on R+ with zero initial data, say, and input from the left-hand boundary. Indeed, the zero boundary conditions at x ¼ þ1 implicit in the formulation may be imposed by function-class restrictions (e.g., uð , tÞ 2 L2 ðRþ Þ for all relevant values of t). This initial-boundary-value problem fits well with laboratory studies wherein waves are generated by a wavemaker at the left-hand end and these are monitored as they propagate down the channel, with the experiment ceasing as soon as the waves reach the other end of the channel and reflected components intrude (see Bona et al., 1981; Hammack, 1973; Hammack and Segur, 1974; Zabusky and Galvin, 1971). Similarly, when modeling surface waves arriving from deep water into near-shore zones or large-scale internal waves propagating from the deep ocean onto the continental shelf, reflection may MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1393 sometimes be safely ignored and one encounters a variable-coefficient version of Eq. (1.1) posed on Rþ  ð0, T Þ with a time-dependent Dirichlet boundary condition at x ¼ 0 (see Boczar-Karakiewicz et al., 1991; Boczar-Karakiewicz et al., (submitted) for example). The quarter-plane problem just outlined has been considered recently by the present authors (see Bona et al., 2001) and by several others (see the references in the last-quoted article) and there is a satisfactory theory of well-posedness for this problem. However, if one is interested in implementing a numerical scheme to approximate solutions of the quarter-plane problem, there arises the issue of cutting off the spatial domain. Once this is done, two more boundary conditions are needed to specify the solution completely. Because the model cannot countenance waves moving to the left, it is usual, as suggested above, to apply the model only on a time scale T short enough that significant wave motion has not reached the righthand boundary. If the right-hand boundary is located at x ¼ r, say, then it is therefore natural in regard to the physical problem to impose uðr, tÞ ¼ ux ðr, tÞ ¼ 0 for 0 t T to obtain a complete set of boundary conditions. Of course, one might also imagine imposing ux(0, t) rather than ux(r, t), but in practical situations, one does not normally have information that warrants the imposition of a second boundary condition at the left-hand, or wavemaker end of the medium of propagation. As far as mathematical analysis is concerned, it makes relatively little difference whether or not the boundary conditions are homogeneous. In consequence, consideration is given here to (1.1)–(1.2) completed by the general nonhomogeneous boundary conditions uð0, tÞ ¼ h1 ðtÞ, uðr, tÞ ¼ h2 ðtÞ, ux ðr, tÞ ¼ h3 ðtÞ, for t
0, ð1:3Þ where the initial value  and the boundary data hj, j ¼ 1, 2, 3 are given functions. The principal concern of the present essay is the well-posedness of the initial-boundaryvalue problem (IBVP henceforth) (1.1)–(1.3). That is, we aim to establish existence, uniqueness, and persistence properties of solutions corresponding to reasonable auxiliary data, together with continuous dependence of the solution upon the auxiliary data. A brief review of the mathematical theory currently available is now presented. The pure initial-value problem (IVP) for (1.1) and its relatives where the initial datum  is specified on the entire real axis R has received a lot of attention in the last three decades, both in case  lies in an L2(R)-based Sobolev space and in case  is periodic (see Bona and Scott, 1976; Bona and Smith, 1978; Bourgain, 1993a; Bourgain, 1993b; Constantin and Saut, 1988; Hammack, 1973; Hammack and Segur, 1974; Kato, 1975; Kato, 1979; Kato, 1983; Kenig et al., 1991a; Kenig et al., 1991b; Kenig et al., 1993a; Kenig et al., 1993b; Kenig et al., 1996; Lax, 1968; Miura, 1976; Russell and Zhang, 1993; Russell and Zhang, 1995; Russell and Zhang, 1996; Saut and Temam, 1976; Sun, 1996; Temam, 1969; Zhang, 1995a, 1995b, 1995c). In particular, various smoothing properties have been discovered for solutions of the (1.1) when posed on the whole line R or on a periodic domain S (e.g., the unit circle in the plane). It is those smoothing properties that enable one to prove that the IVP (1.1)–(1.2) is wellposed in the space H s(R) for s > 3/4 when posed on R and is well-posed in the space H s(S) for s > 1/2 when posed on the periodic domain S (Bourgain, 1993a; MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1394 Bona, Sun, and Zhang Bourgain, 1993b; Kenig et al., 1993b; Kenig et al., 1996). By contrast, the study of the KdV-equation posed on the half line R+ or on a finite interval has received much less attention and the results available thus far appear to be not as sharp as those for the IVP on R. For the initial-boundary-value problem (IBVP henceforth) for the KdV-equation posed on the half line R+, ut þ ux þ uux þ uxxx ¼ 0, uð0, tÞ ¼ hðtÞ, ) uðx, 0Þ ¼ ðxÞ, ð1:4Þ for x, t 2 Rþ , we have provided a review in our recent article Bona et al., (2001) (see the earlier work of Bona and Dougalis, 1980; Bona and Scott, 1974; Bona and Winther, 1989; the related article of Benjamin et al., 1972, on the BBM equation, and the recent article by Colliander and Kenig, 2002). In Bona et al. (2001), we pointed out that for the linear problem obtained from (1.4) by omitting the quadratic term, there are smoothing properties similar to those established by Kenig et al. (1991b) for (1.1)–(1.2) posed on all of R. Consequently, we were able to show the IBVP (1.4) to be well-posed in the space Cð ½0, T; H s ðRþ ÞÞ for any s > 3/4 provided the data ð, hÞ is drawn from H s ðRþ Þ  H ðsþ1Þ=3 ð0, T Þ, by applying the contractionmapping principle. The corresponding solution map was shown to be analytic. In their recent work, Colliander and Kenig (2002) showed that (1.4) is well-posed for s
0. For the KdV-equation posed on a finite interval, Bubnov (1979, 1980) studied the general two-point boundary-value problem 8 ut þ uux þ uxxx ¼ f ðx, tÞ, uðx, 0Þ ¼ 0, > > > > > < 1 uxx ð0, tÞ þ 2 ux ð0, tÞ þ 3 uð0, tÞ ¼ 0, ð1:5Þ > 1 uxx ð1, tÞ þ 2 ux ð1, tÞ þ 3 uð1, tÞ ¼ 0, > > > > : 1 ux ð1, tÞ þ 2 uð1, tÞ ¼ 0, posed on the interval ð0, 1Þ (see also the related work Bona and Dougalis (1980) on the BBM-equation). Here, i , i , j 2 R, i ¼ 1, 2, 3, j ¼ 1, 2 are real constants and assumptions are imposed so that the L2 norm of the solutions of the linear version of (1.5) (obtained by dropping the nonlinear term uux ) is decreasing. It was shown in Bubnov (1979) that for given T > 0 and f 2 H 1 ð ½0, T; L2 ð0, 1ÞÞ, there exists a T  > 0 depending on k f kH 1 ð ½0, T;L2 ð0, 1ÞÞ such that (1.5) admits a unique solution u 2 L2 ð ½0, T  ; H 3 ð0, 1ÞÞ, ut 2 L1 ð ½0, T  ; L2 ð0, 1ÞÞ \ L2 ð ½0, T  ; H 1 ð0, 1ÞÞ: In Zhang (1994), Zhang considered boundary control of the KdV-equation posed on a finite interval ð0, 1Þ with Dirichlet boundary conditions. A feedback control law was introduced to stabilize the system, leading to the initial-boundary-value problem ut þ uux þ uxxx ¼ 0, uð0, tÞ ¼ 0, uð1, tÞ ¼ 0, uðx, 0Þ ¼ ðxÞ, ) x 2 ð0, 1Þ, t
0, ux ð1, tÞ ¼ ux ð0, tÞ, t
0, ð1:6Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1395 with 0 j j < 1. Note that when ¼ 0, the system (1.6) is (1.1)–(1.3) with homogeneous boundary conditions. It was shown in Zhang (1994) that Eq. (1.6) is globally well-posed in the space H 3kþ1 ð0, 1Þ for k ¼ 0, 1, . . .. In a recent article Colin and Ghidaglia (2001), the authors considered the following initial-boundary value problem ) ut þ uux þ uxxx ¼ 0, uðx, 0Þ ¼ ðxÞ, x 2 ð0, 1Þ, t
0, uð0, tÞ ¼ h1 ðtÞ, ux ð1, tÞ ¼ h2 ðtÞ, uxx ð1, tÞ ¼ h3 ðtÞ, t
0, and showed it to be locally well-posed in the space H1(0,1) with the initial data  drawn from H 1 ð0, 1Þ and the boundary data ðh1 , h2 , h3 Þ taken from the product space C 1 ½0, T Þ  C 1 ½0, T Þ  C 1 ½0, T Þ. In addition, Rosier (1997) studied the control problem for the system ) ut þ uux þ uxxx ¼ 0, uðx, 0Þ ¼ ðxÞ, x 2 ð0, 1Þ, t
0, uð0, tÞ ¼ 0, uð1, tÞ ¼ 0 ux ð1, tÞ ¼ hðtÞ, t
0, where the boundary function h is considered as a control input. Rosier showed the system is (locally) exactly controllable in the space L2(0,1). A similar problem was also considered by Zhang (1999) for the system (1.1)–(1.3) where the boundary value functions hj ðtÞ, j ¼ 1, 2, 3 are all taken to be control inputs. This system is shown to be exactly controllable in the space H s(0, 1) for any s
0 in a neighborhood of any smooth solution of the KdV-equation. (Exact controllability means, roughly, that for a given time T > 0 and a given pair of functions  and in the space H s(0, 1), there exist appropriate controls such that the corresponding system possesses a solution u which exactly equals  at t ¼ 0 and equals at t ¼ T. Put colloquially, given two states  and , there is a control h that will drive the system from  to in time T. Of course, there are obvious approximate controllability analogs of this concept. Readers who are interested in control issues are referred to the excellent review article of Russell (1978) for commentary on controllability and stabilizability of linear partial differential equations and to Russell and Zhang (1993, 1995, 1996) for theory of controllability and stabilizability of the KdV-equation. In this article, the nonhomogeneous boundary-value problem (1.1)–(1.3) is considered. The aim is to establish the well-posedness of (1.1)–(1.3) in the space H s(0, r) when the initial data is drawn from H s(0, r) and the boundary data (h1, h2, h3, ) lies in the product space H s1 (0,T )  H s2(0,T )  H s3(0,T ) for some appropriate indices s1, s2, and s3 that depend on s. As we will see later, the natural choices of s1, s2 and s3 are s1 ¼ s2 ¼ (s+1)/3 and s3 ¼ s/3. For convenience of writing, we take the underlying spatial domain (0, r) to be (0, 1) throughout. This is a restriction of no consequence as far as the theory is concerned. The well-posedness result for the IBVP (1.1)–(1.3) we establish in this article appears to require some compatibility conditions relating the initial datum (x) and the boundary data hj (t), j ¼ 1, 2, 3. A simple computation shows that if u is a C1-smooth solution of the IBVP (1.1)–(1.3), then its initial data uðx, 0Þ ¼ ðxÞ and its boundary values hj (t), j ¼ 1, 2, 3 must satisfy the following compatibility conditions: k ð0Þ ¼ h1ðkÞ ð0Þ, k ð1Þ ¼ hðkÞ 2 ð0Þ, 0k ð1Þ ¼ hðkÞ 3 ð0Þ ð1:7Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1396 Bona, Sun, and Zhang for k ¼ 0, 1, . . . , where hðkÞ j ðtÞ is the kth order derivative of hj and 0 ðxÞ ¼ ðxÞ 0   Pk1  0 k ðxÞ ¼  000 k1 ðxÞ þ k1 ðxÞ þ j¼0 j ðxÞkj1 ðxÞ ð1:8Þ for k ¼ 1, 2, . . . : When the well-posedness of (1.1)–(1.3) is considered in the space H s(0, 1) for some finite value s
0, the following s-compatibility conditions thus arise naturally. Definition 1.1. (s-compatibility) Let T > 0 and s
0 be given. A four-tuple ð, h~Þ ¼ ð, h1 , h2 , h3 Þ 2 H s ð0, 1Þ  H ðsþ1Þ=3 ð0, T Þ  H ðsþ1Þ=3 ð0, T Þ  H s=3 ð0, T Þ is said to be s-compatible if k ð0Þ ¼ hðkÞ 1 ð0Þ, k ð1Þ ¼ h2ðkÞ ð0Þ ð1:9Þ holds for k ¼ 0, 1, . . . , ½s=3  1 when s  3½s=3 k ¼ 0, 1, . . . , ½s=3 when 1=2 < s  3½s=3 3=2 and k ð0Þ ¼ hðkÞ 1 ð0Þ, 1=2, or (1.9) holds for 0k ð1Þ ¼ hðkÞ 3 ð0Þ k ð1Þ ¼ h2ðkÞ ð0Þ, holds for k ¼ 0, 1, . . . , ½s=3 when s  3½s=3 > 3=2. We adopt the convention that Eq. (1.9) is vacuous if ½s=3  1 < 0. With this compatibility notation, we may state the following two theorems, which comprise the main results of this article. Theorem 1.2. (Local well-posedness) Let T > 0 and s
0 be given. Suppose that ð, h~Þ 2 H s ð0, 1Þ  H ðsþ1Þ=3 ð0, T Þ  H ðsþ1Þ=3 ð0, T Þ  H s=3 ð0, T Þ is s-compatible. Then there exists a T  2 ð0, T depending only on the norm of ð, h~Þ in the space H s ð0, 1Þ  H ðsþ1Þ=3 ð0, T Þ  H ðsþ1Þ=3 ð0, T Þ  H s=3 ð0, T Þ such that Eqs. (1.1)–(1.3) admits a unique solution u 2 Cð ½0, T  ; H s ð0, 1ÞÞ \ L2 ð ½0, T  ; H sþ1 ð0, 1ÞÞ: Moreover, the solution depends continuously in this latter space on variations of the auxiliary data in their respective function classes. Theorem 1.3. (Global well-posedness) Let T > 0 be arbitrary and s
0. For any s-compatible ð, h~Þ 2 H s ð0, 1Þ  H where 1 ðsÞ ¼ 2 ðsÞ ¼ 1 ðsÞ ð0, T Þ  H  þ ð5s þ 9Þ=18 ðs þ 1Þ=3  þ ð5s þ 3Þ=18 s=3 if 0 1 ðsÞ ð0, T Þ  H s < 3, if s
3; if 0 s < 3, if s
3 2 ðsÞ ð0, T Þ, MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1397 and  is any positive constant, the IBVP (1.1)–(1.3) admits a unique solution u 2 Cð ½0, T; H s ð0, 1ÞÞ \ L2 ð ½0, T; H sþ1 ð0, 1ÞÞ: Moreover, the solution depends continuously on variations of the auxiliary data in their respective function classes. Remark 1.4. The global well-posedness result presented in Theorem 1.3 requires slightly stronger regularity of the boundary values hj , j ¼ 1, 2, 3 in case 0 s < 3 when compared with the local well-posedness result in Theorem 1.2. The same situation appears in the global well-posedness theory for the KdV-equation posed in a quarter plane in Bona et al. (2001). The proof of our well-posedness result for (1.1)–(1.3) relies on the smoothing properties of the associated linear problem ) ut þ ux þ uxxx ¼ f , uðx, 0Þ ¼ ðxÞ, ð1:10Þ uð0, tÞ ¼ h1 ðtÞ, uð1, tÞ ¼ h2 ðtÞ, ux ð1, tÞ ¼ h3 ðtÞ : There are three types of smoothing associated with solving (1.10); these are the smoothing effects of the solution u with respect to the forcing f, the initial value  and the boundary data hj ¼ 0, j ¼ 1, 2, 3, respectively. It will be demonstrated that (i) (ii) (iii) For  2 L2 ð0, 1Þ with f ¼ 0, hj ¼ 0, j ¼ 1, 2, 3, the solution u of (1.10) belongs to the space CðRþ ; L2 ð0, 1ÞÞ \ L2 ðRþ ; H 1 ð0, 1ÞÞ and ux 2 Cð ½0, 1, L2, t ðRþ ÞÞ; For f 2 L1 ðRþ ; L2 ð0, 1ÞÞ with  ¼ 0, hj ¼ 0, j ¼ 1, 2, 3, the solution u of Eq. (1.10) belongs to the space CðRþ ; L2 ð0, 1ÞÞ \ L2 ðRþ ; H 1 ð0, 1ÞÞ and ux 2 Cð ½0, 1, L2, t ðRþ ÞÞ; 1=3 ðRþ Þ, h3 2 L2, loc ðRþ Þ with f ¼ 0 and  ¼ 0, the solution u For h1 , h2 2 Hloc of (1.10) belongs to the space CðRþ ; L2 ð0, 1ÞÞ \ L2, loc ðRþ ; H 1 ð0, 1ÞÞ and ux 2 Cð ½0, 1, L2, t ðRþ ÞÞ. Various other related linear estimates will also be derived. Once these linear estimates are in hand, a local well-posedness result for (1.1)–(1.3) may be established using the contraction-mapping principle. The long-time results are obtained by finding global a priori estimates for smooth solutions of (1.1)–(1.3). It is interesting to note that while an L2-estimate of solutions is relatively straightforward to establish, the global H1- and H2-bounds on solutions seem difficult to obtain by the usual energytype methods. The approach used here is to obtain an L2-estimate of the time derivative ut of solutions, which, in turn, provides a global H3-estimate. Nonlinear interpolation theory (Bona and Scott, 1976; Tartar, 1972) is then used to obtain the global H s-estimates for 0 < s < 3. Global a priori H s-estimates for s > 3 are established by obtaining a priori bounds on @ kt u for k ¼ 1, 2, . . . , ½s=3. Because of its well-posedness, the IBVP (1.1)–(1.3) defines a continuous nonlinear map Ks, T from the space Xs, T ¼ H s ð0, 1Þ  H ðsþ1Þ=3 ð0, T Þ  H ðsþ1Þ=3 ð0, T Þ  H s=3 ð0, T Þ ð1:11Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1398 Bona, Sun, and Zhang to the space Cð ½0, T; H s ð0, 1ÞÞ \ L2 ð ½0, T; H sþ1 ð0, 1ÞÞ for given T > 0 and s
0. It follows readily from the proof presented here that Ks, T is locally Lipschitz continuous. In fact, the map Ks, T is much smoother than just Lipschitz. According to the local existence theory, for a given ð, h~Þ 2 Xs, T which is s-compatible, h~ ¼ ðh1 , h2 , h3 Þ, there is a unique local solution u of (1.1)–(1.3). Of course, the existence time T  for this solution need not be T. Let DðKs, T Þ connote those elements of Xs, T for which the solution exists on ½0, T Þ. As will appear from our detailed theory, DðKs, T Þ is an open neighborhood of the zero element in Xs, T if 0 s 7=2. In this case, the mapping Ks, T is analytic from DðKs, T Þ to the space Cð ½0, T; H s ð0, 1ÞÞ \ L2 ð ½0, T; H sþ1 ð0, 1ÞÞ. That is to say, for given ð, h~Þ in DðKs, T Þ, there exists a  > 0 such that for any ð1 , h~1 Þ 2 Xs, T with ð, h~Þ þ ð1 , h~1 Þ 2 DðKs, T Þ and kð1 , h~1 ÞkXs, T , then Ks, T ð þ 1 , h~ þ h~1 Þ has a Taylor series expansion which is uniformly convergent in the space Cð ½0, T; H s ð0, 1ÞÞ \ L2 ð ½0, T; H sþ1 ð0, 1ÞÞ. Each term in the Taylor series is determined by the solution of a forced linear KdV-equation. Thus one obtains the attractive result that solutions of the nonlinear problem (1.1)–(1.3) can be obtained by solving an infinite sequence of linear problems. When s > 7=2, because of the compatibility conditions, DðKs, T Þ is no longer a neighborhood of zero in Xs, T . However, we can view solutions of the IBVP (1.1)–(1.3) as a special class of solutions of IBVP’s for a system of nonlinear equations. Viewed this way, it may be shown that the IBVP for this nonlinear system is well-posed and the corresponding nonlinear map is again analytic. The approach developed in this article can also be used to obtain similar results for the following general nonhomogeneous boundary-value problem for the KdVequation: 8 ut þ ux þ uux þ uxxx ¼ f ðx, tÞ, uðx, 0Þ ¼ ðxÞ, the > > > <  u ð0, tÞ þ  u ð0, tÞ þ  uð0, tÞ ¼ h ðtÞ, 1 xx 2 x 3 1 > 1 uxx ð1, tÞ þ 2 ux ð1, tÞ þ 3 uð1, tÞ ¼ h2 ðtÞ, > > : 1 ux ð1, tÞ þ 2 uð1, tÞ ¼ h3 ðtÞ, ð1:12Þ with x 2 ð0, 1Þ and t
0 (cf. Bubnov, 1979, 1980). Roughly speaking, if the parameters j, j, and j, j ¼ 1, 2, 3, are chosen such that the solution u of the associated homogeneous linear problem (obtained by dropping uux and setting f ¼ 0 and hj ¼ 0 for j ¼ 1, 2, 3 in (1.12)) satisfies Z d 1 juðx, tÞj2 dx 0 ð1:13Þ dt 0 for any t > 0, the detailed techniques developed in the remainder of the article apply and one may establish that the IBVP (1.12) is well-posed in the space H s(0, 1) for any s
0. In case (1.13) is not valid, the issue of local well-posedness may be more challenging. As far as global existence is concerned, we can give conditions on (1.12) for this to hold. Indeed, because of the strong smoothing resulting from the boundedness of the domain, all that is required is to keep the L2(0, 1)— norm bounded on bounded time intervals. We will not enter into the details of this development here. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1399 With global well-posedness results in hand, a natural further question arises about the solutions of the IBVP (1.1)–(1.3), namely their long time, asymptotic behavior. Because the imposition of boundary conditions may exert a weak dissipative mechanism, it is expected that the solutions of the nonlinear system (1.1)–(1.3) will decay as t ! þ1, at least in case the initial value  is small and the boundary data hj ðtÞ, j ¼ 1, 2, 3 decay to zero as t ! 1. A special situation occurs when the boundary data are all periodic of some period , say. Experiments Bona et al., (1981) suggest that in this case the solution will eventually become time periodic of period . This has been rigorously established in Bona et al., (2003) for the quarter-plane problem (1.4) with a damping term included. It would be interesting and useful to have similar results for the finite domain problem considered here. A related question is whether the (1.1) possesses a strictly time-periodic solution if its boundary forcing h1, h2, and h3 are time-periodic functions defined on all of R (cf. Bona et al., 2003; Wayne, 1990, 1997). Those issues will be addressed in our subsequent articles. The article is organized as follows. In Sec. 2, several estimates pertaining to solutions of the linear problem (1.10) are established which display the smoothing properties mentioned earlier. In Sec. 3, the linear estimates are used to prove that (1.1)–(1.3) is locally well-posed. The global well-posedness of (1.1)–(1.3) is established in Sec. 4. Analyticity of the nonlinear map Ks, T defined by the IBVP (1.1)–(1.3) is discussed in Sec. 5. 2. LINEAR ESTIMATES AND SMOOTHING PROPERTIES In this section, various smoothing properties that accrue to the linear system Eq. (1.10) will be discussed. As (1.10) is linear, it is convenient to break up the analysis. Considered first is the problem ) ut þ ux þ uxxx ¼ 0, uðx, 0Þ ¼ ðxÞ, ð2:1Þ uð0, tÞ ¼ 0, uð1, tÞ ¼ 0, ux ð1, tÞ ¼ 0 with homogeneous boundary conditions and no forcing. Then we will consider problem (1.10) with non-trivial forcing f but with all three boundary conditions set to zero. The outcome of the analysis of these problems are recorded in Propositions 2.1 and 2.4. Next, problem (1.10) with zero forcing, but non-trivial boundary conditions is taken up. We use the Laplace transform in t to obtain a solution formula. Whilst a little complicated, the representation formulas (2.14) and following are completely explicit. Consequently, their analysis may be carried out in detail. The outcome is recorded in a sequence of propositions that conclude the section. Let A be the linear operator defined by Af ¼ f 000  f 0 : Consider A as an unbounded operator on L2(0, 1) with the domain DðAÞ ¼ f f 2 H 3 ð0, 1Þ, f ð0Þ ¼ f ð1Þ ¼ f 0 ð1Þ ¼ 0g: MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1400 Bona, Sun, and Zhang The IBVP (2.1) can be written as the initial-value problem of an abstract evolution equation in the space L2(0,1), viz du ¼ Au, uð0Þ ¼ , ð2:2Þ dt where the spatial variable is suppressed. It is easily verified that both A and its adjoint A* are dissipative, which is to say hAf , f iL2 ð0, 1Þ hA g, giL2 ð0, 1Þ 0, 0 for any f 2 DðAÞ and g 2 DðA Þ, where A g ¼ g000 þ g0 and DðA Þ ¼ f f 2 H 3 ð0, 1Þ; f ð0Þ ¼ f 0 ð0Þ ¼ f ð1Þ ¼ 0g: Thus the operator A is the infinitesimal generator of a C0-semigroup W0(t) in the space L2(0, 1) (see Pazy, 1983). By standard semigroup theory applied in the overlying space L2(0, 1), for any  2 L2 ð0, 1Þ, uðtÞ ¼ W0 ðtÞ belongs to the space Cb ðRþ ; L2 ð0, 1ÞÞ. The function u thus defined is called a mild solution of (2.1). Such solutions certainly solve (2.1) in the sense of distributions (cf. Bona and Winther (1983), Sec. 2). If  2 DðAÞ, then uðtÞ ¼ W0 ðtÞ belongs to the much smaller space Cð0, 1; H 3 ð0, 1ÞÞ \ C1 ð0, 1; L2 ð0, 1ÞÞ and uðtÞ 2 DðAÞ for all t
0. Moreover, the equation is satisfied in the sense of Cð0, 1; L2 ð0, 1ÞÞ, and in particular, pointwise almost everywhere. Such solutions are called strong solutions. For strong solutions, the boundary values are taken on pointwise. In what follows, a solution of (2.1) is either a mild solution or strong solution in the semigroup context. Proposition 2.1. For any  2 L2 ð0, 1Þ, uðtÞ ¼ W0 ðtÞ satisfies Zt u2x ð0, Þ d ¼ kk2L2 ð0, 1Þ kuð , tÞk2L2 ð0, 1Þ þ 0 ð2:3Þ and Z 1 0 xu2 ðx, tÞ dx þ 3 Z tZ 0 1 0 u2x ðx, Þ dx d ð1 þ tÞ Z 0 1 2 ðxÞ dx ð2:4Þ for any t
0. Remark 2.2. The relation (2.3) provides a trace result at x ¼ 0 which reveals a boundary smoothing effect of the system (2.1). Remark 2.3. Combining inequalities (2.3) and (2.4) gives kukL2 ð0, t;H 1 ð0, 1ÞÞ Cð1 þ tÞ1=2 kkL2 ð0, 1Þ , ð2:5Þ which is a Kato-type smoothing effect. Note that the original Kato smoothing effect for solutions of (1.1)–(1.2) posed on the whole real line R is local, which is to say, MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1401 1 ðRÞÞ. In contrast, the smoothing effect (2.5) is  2 L2 ðRÞ implies that u 2 L2 ð0, T; Hloc global. As will be seen later, this global Kato-smoothing effect alone is enough to establish the well-posedness of (1.1)–(1.3) in the space H s(0, 1) for s
0. This is in sharp contrast to the problem (1.1)–(1.2) posed on the unbounded domain R or the IBVP (1.4) posed on the unbounded domain R+, where both the Kato smoothing and the Strichartz smoothing or the Bourgain smoothing are used to establish their well-posedness in weak spaces. Proof. Assume first that  2 DðAÞ. Then uðtÞ 2 DðAÞ for any t
0 and u 2 C 1 ð0, 1; L2 ð0, 1ÞÞ. To obtain (2.3), multiply both sides of the differential equation in (2.1) by 2u and integrate over ð0, 1Þ with respect to x and over (0, t) with respect to t. Integration by parts then leads to (2.3). For inequality (2.4), multiply both sides of the equation in (2.1) by 2xu, integrate the result over ½0, 1  ½0, 1, and integrate by parts to reach the relation Z 1 0 xu2 ðx, tÞ dx þ 3 Z tZ 0 1 0 u2x ðx, Þ dx d ¼ Z 1 0 x2 ðxÞ dx þ Z tZ 0 1 0 u2 ðx, Þ dx d from which (2.4) follows on account of (2.3). If, instead,  2 L2 ð0, 1Þ, choose a sequence fn g from DðAÞ such that n converges to  in L2(0, 1) as n ! 1. Define un to be un ¼ W0 ðtÞn , n ¼ 1, 2, . . . : As we have just shown, both (2.3) and (2.1) hold with u replaced by un and  replaced by n. Let T > 0 be fixed. Then the sequence fun g is bounded in the spaces Cð ½0, T; L2 ð0, 1ÞÞ, L2 ð0, T; H 1 ð0, 1ÞÞ and Cð ½0, T; L2 ð0, 1; xdxÞÞ. Here L2 ð0, 1; xdxÞ is the weighted L2-space with the weight x. Moreover un, x ð0, tÞ  @x un ð0, tÞ is a bounded sequence in the space L2 ð0, T Þ. Thus there exists a subsequence funk g of fun g and a u 2 L2 ð0, T; H 1 ð0, 1ÞÞ \ L1 ð0, T; L2 ð0, 1ÞÞ \ L1 ð0, T; L2 ð0, 1; xdxÞÞ with ux ð0, tÞ 2 L2 ð0, T Þ such that funk g is convergent to u weakly in L2 ð0, T; H 1 ð0, 1ÞÞ and weak-star in both spaces L1 ð0, T; L2 ð0, 1ÞÞ and L1 ð0, T; L2 ð0, 1; xdxÞÞ. Furthermore, f@x unk ð0, tÞg is weakly convergent to ux ð0, tÞ in the space L2 ð0, T Þ. On account of the lower semi-continuity of the various norms with regard to weak convergence, it is adduced that (2.3) and (2.4) hold for u . On the other hand un converges strongly to u ¼ W0 ðtÞ in L1 ð0, T; L2 ð0, 1ÞÞ. We conclude that u 2 L2 ð0, T; H 1 ð0, 1ÞÞ \ L1 ð0, T; L2 ð0, 1ÞÞ \ L1 ð0, T; L2 ð0, 1; xdxÞÞ, ux ð0, tÞ 2 L2 ð0, T Þ and (2.3) and (2.4) hold for u. Next, attention is turned to the inhomogeneous linear problem ) ut þ ux þ uxxx ¼ f ðx, tÞ, uðx, 0Þ ¼ 0, uð0, tÞ ¼ 0, uð1, tÞ ¼ 0, ux ð1, tÞ ¼ 0: œ ð2:6Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1402 Bona, Sun, and Zhang In terms of the operator A defined above, one may write (2.6) as an initial-value problem for an abstract nonhomogeneous evolution equation, viz. du ¼ Au þ f , dt uð0Þ ¼ 0: ð2:7Þ By standard semigroup theory (see again Pazy, 1983), for any f 2 L1, loc ðRþ ; L2 ð0, 1ÞÞ, Z uðtÞ ¼ t 0 W0 ðt  Þ f ðÞ d ð2:8Þ belongs to the space CðRþ ; L2 ð0, 1ÞÞ and is called a mild solution of (2.7). It is a weak solution of (2.6) in the sense of distribution. In addition, if f ðtÞ 2 DðAÞ for t > 0 and Af 2 L1, loc ðRþ ; L2 ð0, 1ÞÞ, then u(t) given by (2.8) solves (2.7) a.e. on ½0, T Þ and is called a strong solution of (2.7). Proposition 2.4. There exists a constant C such that for any f 2 L1, loc ðRþ ; L2 ð0, 1ÞÞ, the solution u of Eq. (2.6) satisfies kuð , tÞkL2 ð0, 1ÞÞ þ kux ð0, ÞkL2 ð0, tÞ Ck f kL1 ð0, t;L2 ð0, 1ÞÞ ð2:9Þ and Z 1 2 xu ðx, tÞ dx þ 0 Z tZ 1 0 0 2ð1 þ tÞk f k2L1 ð0, t;L2 ð0, 1ÞÞ u2x ðx, Þ dx d ð2:10Þ for any t
0. Proof. Without loss of generality, we assume that u is a strong solution. The general case follows using a limiting procedure similar to that appearing in the proof of Proposition 2.1. Multiply the equation in (2.6) by 2u and integrate over ð0, 1Þ with respect to x. Integration by parts leads to d dt Z 1 0 u2 ðx, tÞ dx þ u2x ð0, tÞ 2k f ð , tÞkL2 ð0, 1Þ kukL2 ð0, 1Þ from which (2.9) follows. To prove (2.10), multiply both sides of the equation in (2.6) by 2xu and integrate over the rectangle ð0, 1Þ  ð0, tÞ in space-time. After integrations by parts, it is seen that Z 0 1 xu2 ðx, tÞ dx þ 3 Z tZ ¼2 0 Z 0 t 1 0 Z tZ 0 1 0 u2x ðx, Þ dx d xf ðx, Þuðx, Þ dx d þ Z tZ 0 1 0 u2 ðx, Þ dx d kx1=2 f ð , ÞkL2 ð0, 1Þ kx1=2 uð , ÞkL2 ð0, 1Þ d þ Z tZ 0 0 1 u2 ðx, Þ dx d MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain Z t kx 1=2 1 1 sup kx1=2 uð , Þk2L2 ð0, 1Þ þ 20  t 2 Z tZ 1 u2 ðx, Þ dx d, þ Z 1=2 sup kx 0  t uð , ÞkL2 ð0, 1Þ 0 1403 f ð , ÞkL2 ð0, 1Þ d þ t 0 Z tZ 0 0 1 u2 ðx, Þ dx d 2 kx1=2 f ð , ÞkL2 ð0, 1Þ d 0 0 œ which yields the inequality (2.10). Next, consider the non-homogeneous boundary-value problem ) ut þ ux þ uxxx ¼ 0, uðx, 0Þ ¼ 0, uð0, tÞ ¼ h1 ðtÞ, uð1, tÞ ¼ h2 ðtÞ, ux ð1, tÞ ¼ h3 ðtÞ: ð2:11Þ A common approach to (2.11) is to render its boundary conditions homogeneous as follows. The solution u of (2.11) can be written as uðx, tÞ ¼ wðx, tÞ þ vðx, tÞ with vðx, tÞ ¼ ð1  xÞh1 ðtÞ þ xh2 ðtÞ þ xð1  xÞðh3 ðtÞ  h2 ðtÞ þ h1 ðtÞÞ and w satisfying wt þ wx þ wxxx ¼ vt  vx , wð0, tÞ ¼ 0, wð1, tÞ ¼ 0, ) wðx, 0Þ ¼ 0, wx ð1, tÞ ¼ 0: ð2:12Þ Thus to solve (2.11), one only need solve (2.12), which can be done by applying Proposition 2.2. Here we assume hj ð0Þ ¼ 0 for j ¼ 1, 2, 3. However there is a serious drawback to this approach; it is required that h0j 2 L1 ðRÞ for j ¼ 1, 2, 3 to obtain even a mild solution u of (2.12) in the space Cb ð0, T; L2 ð0, 1ÞÞ \ L2 ð0, T; H 1 ð0, 1ÞÞ. Furthermore, for such a mild solution u, although both uð0, tÞ and uð1, tÞ are defined thanks to the Kato smoothing, it seems that the trace of ux ðx, tÞ at x ¼ 1 does not make sense since ux is only known to be in the space L2 ð0, T; L2 ð0, 1ÞÞ. This suggests that a stronger boundary smoothing property of (2.11) is needed if one wants to solve (2.11) in the space Cb ð0, T; L2 ð0, 1ÞÞ \ L2 ð0, T; H 1 ð0, 1ÞÞ. Our approach to solve (2.11) is to seek an explicit solution formula in terms of its boundary values via the Laplace transform as we did in Bona et al., (2001) for the KdV-equation in a quarter plane. Applying the Laplace transform with respect to t, (2.11) is converted to ) su^ ðx, sÞ þ u^ x ðx, sÞ þ u^ xxx ðx, sÞ ¼ 0, ð2:13Þ u^ ð0, sÞ ¼ h^1 ðsÞ, u^ ð1, sÞ ¼ h^2 ðsÞ, u^ x ð1, sÞ ¼ h^3 ðsÞ, where u^ ðx, sÞ ¼ and Z þ1 0 est uðx, tÞ dt MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1404 Bona, Sun, and Zhang h^j ðsÞ ¼ Z þ1 0 est hj ðtÞ dt, j ¼ 1, 2, 3: The solution u^ ðx, sÞ of (2.13) can be written in the form u^ ðx, sÞ ¼ 3 X j¼1 cj ðsÞej ðsÞx , where j ðsÞ, j ¼ 1, 2, 3, are the three solutions of the characteristic equation s þ  þ 3 ¼ 0 and cj ¼ cj ðsÞ, j ¼ 1, 2, 3, solve the linear system 8 ^ > > < c1 þ c2 þ c3 ¼ h1 ðsÞ, c1 e1 ðsÞ þ c2 e2 ðsÞ þ c3 e3 ðsÞ ¼ h^2 ðsÞ, > > : c1 1 ðsÞe1 ðsÞ þ c2 2 ðsÞe2 ðsÞ þ c3 3 ðsÞe3 ðsÞ ¼ h^3 ðsÞ : Let (s) be the determinant of the coefficient matrix and i (s) the determinants of the matrices that are obtained by replacing the ith-column of (s) by the column vector ðh^1 ðsÞ, h^2 ðsÞ, h^3 ðsÞÞT , i ¼ 1, 2, 3. Cramer’s rule implies that cj ¼ j ðsÞ , ðsÞ j ¼ 1, 2, 3: Taking the inverse Laplace transform of u^ yields uðx, tÞ ¼ 1 2i Z rþi1 ri1 est u^ ðx, sÞ ds ¼ Z 3 X 1 rþi1 st j ðsÞ j ðsÞx e e ds 2i ri1 ðsÞ j¼1 for any r > 0. The solution u of (2.11) may also be written in the form uðx, tÞ ¼ u1 ðx, tÞ þ u2 ðx, tÞ þ u3 ðx, tÞ where um ðx, tÞ solves (2.11) with hj  0 when j 6¼ m, m, j ¼ 1, 2, 3; thus um has the representation Z 3 X 1 rþi1 st j, m ðsÞ j ðsÞx ^ e e hm ðsÞ ds  Wm ðtÞhm um ðx, tÞ ¼ ðsÞ 2i ri1 j¼1 ð2:14Þ for m ¼ 1, 2, 3. Here j, m ðsÞ is obtained from j ðsÞ by letting h^m ðtÞ ¼ 1 and hk ðtÞ  0 for k 6¼ m, k, m ¼ 1, 2, 3. It is straightforward to determine that in the last two formulas, the right-hand sides are continuous with respect to r for r
0. As the left-hand sides do not depend on r, it follows that we may take r ¼ 0 in these MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1405 formulas and in those appearing below. Write um in the form um ðx, tÞ ¼ Z 3 X 1 þi1 st j, m ðsÞ j ðsÞx ^ e hm ðsÞ ds e 2i 0 ðsÞ j¼1 Z 3 X 1 0 st j, m ðsÞ j ðsÞx ^ þ hm ðsÞ ds e e 2i i1 ðsÞ j¼1  Im ðx, tÞ þ IIm ðx, tÞ, for m ¼ 1, 2, 3. Letting s ¼ ið 3  Þ with 1  < þ1 in the characteristic equation s þ  þ 3 ¼ 0, the three roots are given in terms of  by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32  4  i þ þ , 1 ð Þ ¼ i, 2 ð Þ ¼ 2 ð2:15Þ þ 3 ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  32  4  i ¼ 2 and thus Im ðx, tÞ and IIm ðx, tÞ may be written in the form Im ¼ Z þ 3 X 1 þ1 ið 3 Þt þj ð Þx j, m ð Þ 2 ^þ e e þ ð Þ ð3  1Þhm ð Þ d  2 1 j¼1 and IIm ¼ Z 3 X 1 þ1 ið 3 Þt j ð Þx  j, m ð Þ 2 ^ e e  ð Þ ð3  1Þhm ð Þ d 2  1 j¼1 3 þ þ ^ where h^þ m ð Þ ¼ hm ðið   ÞÞ,  ð Þ and j, m ð Þ are obtained from ðsÞ and j, m ðsÞ, 3 respectively, by replacing s with ið   Þ and j ðsÞ with þ j ð Þ, for j ¼ 1, 2, 3. Notice that, with an obvious notation,  ð Þ ¼ þ ð Þ and  ð Þ ¼ þ j, m j, m ð Þ for j ¼ 1, 2, 3,  þ ^ ^ and hm ð Þ ¼ hm ð Þ. The next result is a technical lemma that will find frequent use in this section. It plays the same role in dealing with the finite-interval problem as does Plancherel’s theorem for the pure initial-value problem posed on the line. Lemma 2.5. For any f 2 L2 ð0 þ 1Þ, let Kf be the function defined by Z þ1 e ð Þx f ð Þ d Kf ðxÞ ¼ 0 where ð Þ is a continuous complex-valued function defined on ð0, 1Þ satisfying the following two conditions: (i) There exist  > 0 and b > 0 such that   Re ð Þ sup
b; 0< < MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1406 (ii) Bona, Sun, and Zhang There exists a complex number  þ i such that ð Þ lim !þ1 ¼  þ i: Then there exists a constant C such that for all f 2 L2 ð0, 1Þ, kKf kL2 ð0, 1Þ CðkeRe ðÞ f ð ÞkL2 ðRþ Þ þ k f ð ÞkL2 ðRþ Þ Þ: Proof. Observe that kKf k2L2 ð0, 1Þ Notice also that ð yÞ þ1 0 Z þ1 0 0 Z þ1  Re ð ðsÞþ ðyÞÞ 0 ¼ keRe Z 1Z Z e Re ð ðsÞxÞ 0 þ1 Z þ1 Z 1 j f ðsÞj ds eRe ð þ1 Z eRe ð ð yÞxÞ 0 ðsÞþ ð yÞÞx j f ð yÞj dy dx dx j f ðsÞjj f ð yÞj ds dy  j f ðsÞ f ð yÞj ds dy þ1 jReð ðsÞ þ ð yÞÞj 0  0Z  þ1 eRe ðsÞ j f ðsÞj ds    keRe ð Þ f ð ÞkL2 ðRþ Þ    0 jReð ðsÞ þ ð yÞÞj L2 ðRþ Þ  Z þ1   j f ðsÞj ds   k f kL2 ðRþ Þ : þ jReð ðsÞ þ ð yÞÞjL2 ðRþ Þ 0 e 1=2 f ð yÞkL2 ðRþ Þ keRe ðÞ f ð ÞkL2 ðRþ Þ and for any y 2 ð0, þ 1Þ, y jReð ð yÞ þ ð yÞÞj C : þ1 Using the integral version of Minkowski’s inequality yields  Z  þ1 eRe ðsÞ j f ðsÞj ds       0 jReð ðsÞ þ ð yÞÞj L2 ðRþ Þ  Z  þ1 eRe ð yÞ j f ð yÞjy d    ¼   0 jReð ð yÞ þ ð yÞÞj  L2 ðRþ Þ   Z þ1  Re ð yÞ f ð yÞy    e d   Reð ð yÞ þ ð yÞÞ 0 þ L2 ðR Þ C Z þ1 0 CkeRe 1 d keRe pffiffiffiffi ð1 þ Þ ðÞ f kL2 ðRþ Þ ðÞ f ð ÞkL2 ðRþ Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1407 for some absolute constant C. The same argument also gives  Z þ1   j f ðsÞj ds   Ck f kL2 ðRþ Þ :  jReð ðsÞ þ ð yÞÞjL2 ðRþ Þ 0 The proof is complete. œ Lemma 2.6. Let a > 0 be given. For any f 2 L2 ð0, aÞ, let Gf be the function defined by Za Gf ðxÞ ¼ eið Þx f ð Þ d 0 where  is a continuous real-valued function defined on the interval ½0, a which is C1 on the open interval ð0, aÞ and such that there is a constant C1 for which ð1=j0 ð ÞjÞ C1 for 0 < < a. Then there exists a constant C such that for all f 2 L2 ð0, aÞ, Ck f kL2 ð0, aÞ : kGf kL2 ð0, 1Þ Proof. Let ! ¼ ð Þ. Since 0 ð Þ 6¼ 0 for 2 ð0, aÞ,  is strictly monotone and so is invertible. Let ¼ 1 ð!Þ denote its inverse. Note that d! ¼ 0 ð Þd and so by a change of variables, we may write Gf in terms of ! thusly: Z ðaÞ 1 Gf ðxÞ ¼ ei!x f ð1 ð!ÞÞ 0 1 d!:  ð ð!ÞÞ ð0Þ It follows from Plancherel’s theorem that Z 1 ðaÞ  1 2 1 f ð ð!Þ 0 1 2 ð0Þ  ð ð!ÞÞ Z 1 a 1 ¼ d j f ð Þj2 0 2 0 j ð Þj Z C1 a j f ð Þj2 d 2 0 kGf k2L2 ð0, 1Þ ¼ 2 d! œ which is the advertised inequality. The following three propositions provide estimates for u1 , u2 , and u3 , respectively. They show clearly various smoothing properties that accrue through implementation of the boundary conditions for the linear system (2.7) (cf. Remark 2.3). Proposition 2.7. There exists a constant C such that ku1 kL2 ðRþ ;H 1 ð0, 1ÞÞ þ sup ku1 ð , tÞkL2 ð0, 1Þ 0 t<þ1 and @x u1 2 Cb ð ½0, 1; L2 ðRþ ÞÞ with sup k@x u1 ðx, ÞkL2 ðRþ Þ Ckh1 kH 1=3 ðRþ Þ x2ð0, 1Þ for all h1 2 H 1=3 ðRþ Þ: Ckh1 kH 1=3 ðRþ Þ ð2:16Þ ð2:17Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1408 Bona, Sun, and Zhang Proof. Since 1 ðsÞ þ 2 ðsÞ þ 3 ðsÞ  0, it is readily seen that 1, 1 ðsÞ ¼ ð3 ðsÞ  2 ðsÞÞe1 ðsÞ , 2, 1 ðsÞ ¼ ð1 ðsÞ  3 ðsÞÞe2 ðsÞ , 3, 1 ðsÞ ¼ ð2 ðsÞ  1 ðsÞÞe3 ðsÞ and thus ðsÞ ¼ ð3 ðsÞ  2 ðsÞÞe1 ðsÞ þ ð1 ðsÞ  3 ðsÞÞe2 ðsÞ þ ð2 ðsÞ  1 ðsÞÞe3 ðsÞ : In consequence, it follows readily that as a function of the variable  introduced above and defined by the relation s ¼ ið 3  Þ, pffi þ  3 1, 1 ð Þ 2 ,  e þ ð Þ pffiffi þ 2, 1 ð Þ  3  e , þ ð Þ þ 3, 1 ð Þ 1 þ ð Þ as  ! þ1. An application of Lemma 2.5 produces a constant C such that 2  3 Z þ1  þ   2  X 2 j, 1 ð Þ Re þj ð Þ 2 2  kI1 ð , tÞkL2 ð0, 1Þ þ 1 h^þ  e  þ 1 ð Þð3  1Þ d    ð Þ 1 j¼1 Z þ1 2 2 2 C jh^þ 1 ð Þj ð3  1Þ d 1 C Z þ1 0 Z  ð1 þ Þ2=3  0 þ1 2  ei h1 ðÞ d  d Ckh1 k2H 1=3 ðRþ Þ : The same argument applied to II1 ðx, tÞ gives kII1 ð , tÞkL2 ð0, 1Þ Ckh1 kH 1=3 ðRþ Þ : Thus (2.16) holds. To prove (2.17), observe that Z þ 3 X 1 þ1 ið 3 Þt þ þ ð Þx 1, j ð Þ 2 ^þ j @x I1 ðx, tÞ ¼ e j ð Þe þ ð Þ ð3  1Þh1 ð Þ d 2  1 j¼1 Z 3 X 1 þ1 i t þ þ þ j, 1 ðð ÞÞ ^ h ði Þ d ¼ e j ðð ÞÞej ðð ÞÞx þ  ðð ÞÞ 1 2 0 j¼1 where ð Þ is the real solution of ¼ 3   for 
1. Using the Plancherel Theorem (with respect to t) yields that for any x 2 ð0, 1Þ, 2  Z 3 X þ ðð ÞÞ ^ 1 þ1  þ j, 1 ðð ÞÞx þ 2 2 k@x I1 ðx, ÞkL2 ðRþ Þ j ðð ÞÞe j þ ðð ÞÞ  jh1 ði Þj d :  2  0 j¼1 MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1409 Thus, one finds there is a constant C such that Z 0 1 k@x I1 ðx, Þk2L2 ðRþ Þ dx sup k@x I1 ðx, Þk2L2 ðRþ Þ x2ð0, 1Þ C 3 Z X j¼1 C 3 Z X j¼1 0 þ1  þ1 0  þ j ðð   þ ÞÞx 2 j, 1 ðð j  þ   ðð 2 þ ÞÞ sup jej ðð x2ð0, 1Þ ð1 þ Þ2=3 jh^1 ði Þj2 d 2 ÞÞ ^  jh ði Þj2 d ÞÞ  1 Ckh1 k2H 1=3 ðRþ Þ : The following estimates were used in obtaining the last inequality:  pffiffi  sup je1 ð Þx j2 C, sup je2 ð Þx j2 C e 3 þ 1 , x2ð0, 1Þ x2ð0, 1Þ sup je3 ð Þx j2 x2ð0, 1Þ  pffiffi  C1 e 3 þ 1 : To see @x I1 is continuous from ½0, 1 to the space L2 ðRþ Þ, choose any x0 2 ½0, 1 and x 2 ð0, 1Þ and observe that @x I1 ðx, tÞ  @x I1 ðx0 , tÞ Z 3 X  þ 1 þ1 i t þ e j ðð ÞÞ ej ðð ¼ 2 0 j¼1 ÞÞx þ  ej ðð ÞÞx0  þ j, 1 ðð ÞÞ ^ h ði Þ d : þ ðð ÞÞ 1 Using the Plancherel theorem with respect to t as above yields k@x I1 ðx, Þ  @x I1 ðx0 , Þk2L2 ðRþ Þ  Z 3 X þ 1 þ1  þ j ðð ÞÞðej ðð  2 0 j¼1 C 3 Z X j¼1 þ1 0 ÞÞx e þ j ðð ÞÞx0 ð1 þ Þ2=3 jh^1 ði Þj2 d : 2  þ j, 1 ðð ÞÞ ^ Þ þ  jh ði Þj2 d  ðð ÞÞ  1 An application of Fatou’s lemma gives lim k@x I1 ðx, Þ  @x I1 ðx0 , Þk2L2 ðRþ Þ x!x0  Z 3 X  þ 1 þ1  þ j ðð ÞÞ lim ej ðð x!x0 2 0  j¼1 ¼0 ÞÞx e þ j ðð ÞÞx0 2   þ j, 1 ðð ÞÞ ^  jh ði Þj2 d þ  ðð ÞÞ  1 MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1410 Bona, Sun, and Zhang Similar considerations establish that Z1 k@x I2 ðx, Þk2L2 ðRþ Þ dx sup k@x I2 ðx, Þk2L2 ðRþ Þ x2ð0, 1Þ 0 Ckh1 k2H 1=3 ðRþ Þ and I2 ðx, Þ 2 Cb ð ½0, 1; L2 ðRþ ÞÞ. The proof is complete. œ Proposition 2.8. There exists a constant C such that ku2 kL2 ðRþ ;H 1 ð0, 1ÞÞ þ sup ku2 ð , tÞkL2 ð0, 1Þ 0 t<þ1 Ckh2 kH 1=3 ðRþ Þ ð2:18Þ and @x u2 2 Cb ð ½0, 1; L2, t ðRþ ÞÞ with sup k@x u2 ðx, ÞkL2 ðRþ Þ x2ð0, 1Þ Ckh2 kH 1=3 ðRþ Þ ð2:19Þ for all h2 2 H 1=3 ðRþ Þ. Proof. Let u2 ðx, tÞ ¼ I2 ðx, tÞ þ II2 ðx, tÞ with Z 3 X 1 þi1 st j, 2 ðsÞ j ðsÞx ^ I2 ðx, tÞ ¼ h2 ðsÞ ds e e 2i 0 ðsÞ j¼1 and II2 ðx, tÞ ¼ Z 3 X 1 0 st j, 2 ðsÞ j ðsÞx ^ e h2 ðsÞ ds: e ðsÞ 2i i1 j¼1 As in the proof of Proposition 2.7, one has I2 ðx, tÞ ¼ Z þ 3 X 1 þ1 ið 3 Þt þj ð Þx j, 2 ð Þ ð32  1Þh^þ e e 2 ð Þ d þ 2  ð Þ 1 j¼1 3 þ 3 ^ where h^þ 2 ð Þ ¼ h2 ðið   ÞÞ and j, 2 ð Þ ¼ j, 2 ðið   ÞÞ for j ¼ 1, 2, 3. Note that 1, 2 ðsÞ ¼ 2 e2  3 e3 , 3, 2 ðsÞ ¼ 1 e1  3 e2 : 2, 2 ðsÞ ¼ 3 e3  1 e1 , One readily obtains that, as  ! þ1, þ 1, 2 ð Þ  1, þ ð Þ pffi þ  3 2, 2 ð Þ  e 2 , þ  ð Þ þ 3, 2 ð Þ  1: þ ð Þ Using Lemma 2.5, it is adduced that there is a constant C for which 2  3 Z þ1  þ  X  2 j, 2 ð Þ  Re þj ð Þ 2 2  kI2 ð , tÞkL2 ð0, 1Þ þ 1bigÞ2 h^þ  e  þ 2 ð Þð3  1Þ d   ð Þ  j¼1 1 Z þ1  þ  h^2 ð Þð32  1Þ2 d Ckh2 k2 1=3 þ : C H ðR Þ 1 MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1411 The same estimate holds for II2 ðx, tÞ. To prove (2.19), note that Z 3 X þ þ 1 þ1 ið 3 Þt þ j, 2 ð Þ ð32  1Þh^þ e j ð Þej ð Þx þ 2 ð Þ d 2  ð Þ 1 j¼1 Z 3 X þ þ 1 þ1 i t þ j, 2 ðð ÞÞ ^ h ði Þ d : ¼ e j ðð ÞÞej ðð ÞÞx þ 2 0  ðð ÞÞ 2 j¼1 @x I2 ðx, tÞ ¼ Using the Plancherel theorem (with respect to t), 2 Z þ1  þ 3  X  ðð ÞÞ þ 1  ^þ  j, 2 j ðð ÞÞx 2 k@x I2 ðx, Þk2L2 ðRþ Þ þ j ðð ÞÞe þ ðð ÞÞ  jh2 ði Þj d ,  2  0 j¼1 from which follows k@x I1 ðx, tÞk2L2 ð0, 1;L2 ðRþ ÞÞ sup k@x I1 ðx, Þk2L2 ðRþ Þ x2ð0, 1Þ C 3 Z X j¼1 C 3 Z X j¼1 0 þ1  þ1 0  þ 2  þ  sup ej ðð j ðð ÞÞ x2ð0, 1Þ 2 ð1 þ Þ2=3 jh^þ 2 ði Þj d    þ ÞÞx 2 j, 2 ðð  þ   ðð 2 ÞÞ ^  jh ði Þj2 d ÞÞ  2 Ckh2 k2H 1=3 ðRþ Þ : The same estimate holds for II2 ðx, tÞ. Moreover, a similar argument as that used in the proof of Proposition 2.7 shows that both I2 and II2 are continuous from ½0, 1 to the space L2 ðRþ Þ. The proof is complete. œ Proposition 2.9. There exists a constant C such that ku3 kL2 ðRþ ;H 1 ð0, 1ÞÞ þ sup ku3 ð , tÞkL2 ð0, 1Þ Ckh3 kL2 ðRþ Þ 0 t<þ1 ð2:20Þ and @x u3 2 Cb ð ½0, 1; L2 ðRþ ÞÞ with sup k@x u3 ðx, ÞkL2 ðRþ Þ x2ð0, 1Þ Ckh3 kL2 ðRþ Þ ð2:21Þ for all h3 2 L2 ðRþ Þ. Proof. The function u3 ðx, tÞ can be written in the form u3 ðx, tÞ ¼ I3 ðx, tÞ þ II3 ðx, tÞ with Z 3 X 1 þi1 st j, 3 ðsÞ j ðsÞx ^ e e h3 ðsÞ ds I3 ðx, tÞ ¼ 2i 0 ðsÞ j¼1 and II3 ðx, tÞ ¼ Z 3 X 1 0 st j, 3 ðsÞ j ðsÞx ^ e h3 ðsÞ ds: e 2i i1 ðsÞ j¼1 MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1412 Bona, Sun, and Zhang As in the proof of Proposition 2.7, one has Z þ 3 X 1 þ1 ið 3 Þt þj ð Þx j, 3 ð Þ ð32  1Þh^þ e e I3 ðx, tÞ ¼ 2 ð Þ d þ 2  ð Þ 1 j¼1 3 þ 3 ^ where h^þ 3 ð Þ ¼ h3 ðið   ÞÞ and j, 3 ð Þ ¼ j, 3 ðið   ÞÞ, j ¼ 1, 2, 3. Since 1, 3 ðsÞ ¼ e3 ðsÞ  e2 ðsÞ , 2, 3 ðsÞ ¼ e1 ðsÞ  e3 ðsÞ and 3, 3 ðsÞ ¼ e2 ðsÞ  e1 ðsÞ , it follows that þ 1, 3 ð Þ  1 , þ ð Þ pffiffi þ 2, 3 ð Þ  1 eð 3=2Þ , þ  ð Þ þ 3, 2 ð Þ  1 þ ð Þ as  ! þ1. The remainder of the proof follows the lines developed above. œ Let h~ðtÞ ¼ ðh1 ðtÞ, h2 ðtÞ, h2 ðtÞÞ and write the solution u of (2.11) as uðtÞ ¼ Wb ðtÞh~ ¼ 3 X j¼1 Wj ðtÞhj ð2:22Þ where the spatial variable x is suppressed and the Wj are as defined in (2.14). For s
0 and T > 0, let Hs, T ¼ H ðsþ1Þ=3 ð0, T Þ  H ðsþ1Þ=3 ð0, T Þ  H s=3 ð0, T Þ: For any h~ 2 Hs, T ,  1=2 kh~kHs, T  kh1 k2H ðsþ1Þ=3 ð0, T Þ þ kh2 k2H ðsþ1Þ=3 ð0, T Þ þ kh3 k2H s=3 ð0, T Þ : If T ¼ 1, denote Hs, T by Hs . Combining Propositions 2.7–2.9 yields the following theorem about the linear IBVP (2.11). Theorem 2.10. For any h~ 2 H0 , the IBVP Eq. (2.11) admits a unique solution uðx, tÞ ¼ ½Wb ðtÞh~ðtÞðxÞ which belongs to the space Cb ðRþ ; L2 ð0, 1ÞÞ \ L2 ðRþ ; H 1 ð0, 1ÞÞ with ux 2 Cb ð ½0, 1; L2 ðRþ ÞÞ. Moreover there exists a constant C such that kukL2 ðRþ ;H 1 ð0, 1ÞÞ þ sup kuð , tÞkL2 ð0, 1Þ 0 t<þ1 Ckh~kH0 ð2:23Þ and sup kux ðx, ÞkL2 ðRþ Þ x2ð0, 1Þ for all h~ 2 H0 . Ckh~kH0 ð2:24Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1413 Remark 2.11. The estimate (2.23) reveals a Kato-type smoothing effect of the system (2.11) while (2.24) shows that the system (2.11) possesses a stronger smoothing effect, the so-called sharp Kato smoothing (see Kato (1983), Kenig et al. (1991a); Vega (1988)). It is this smoothing property that provides the rationale for being able to impose in a strong sense the boundary condition ux ð1, tÞ ¼ h3 ðtÞ in (2.11). Remark 2.12. The condition imposed on h~ in Theorem 2.10 appears to be sharp in the context of our approach to the analysis. From the explicit solution formula for (2.11), using arguments similar to those appearing in the proof of Propositions 2.7– 2.9, one shows that u 2 Cb ð ½0, 1; Ht1=3 ðRþ ÞÞ and sup kuðx, ÞkH 1=3 ðRþ Þ x2ð0, 1Þ Ckh~kH0 : Of course, this does not mean that results with s < 0 are not possible, mearly that the present argument would not be adequate to the task. Finally we return to the homogeneous IBVP (2.1) to show that it possesses the sharp Kato-smoothing property. Let a function  be defined on the interval ð0, 1Þ and let  be its extension by zero to the whole line R. Assume that v ¼ vðx, tÞ is the solution of vt þ vx þ vxxx ¼ 0, vðx, 0Þ ¼  ðxÞ for x 2 R, t
0. If g1 ðtÞ ¼ vð0, tÞ, g2 ðtÞ ¼ vð1, tÞ, g3 ðtÞ ¼ vx ð1, tÞ, then in terms of Wb ðtÞ defined in (2.22), vg~ ¼ vg~ðx, tÞ  Wb ðtÞ~ g is the corresponding solution of the nonhomogeneous boundary-value problem Eq. (2.1) with boundary conditions hj ðtÞ ¼ gj ðtÞ, j ¼ 1, 2, 3, for t
0. It is clear that for x 2 ð0, 1Þ, the function vðx, tÞ  vg~ðx, tÞ solves the IBVP (2.1), and this in turn leads to a representation of the semigroup W0 ðtÞ in terms of Wb ðtÞ and WR ðtÞ, where WR ðtÞ is the C0 -semigroup in the space L2 ðRÞ generated by the operator AR defined by AR f ¼ f 0  f 000 with domain DðAR Þ ¼ H 3 ðRÞ and vðx, tÞ ¼ WR ðtÞ ðxÞ. Proposition 2.13. For any  2 L2 ð0, 1Þ, if  is its zero-extension to R, then W0 ðtÞ may be written in the form W0 ðtÞ ¼ WR ðtÞ  Wb ðtÞ~ g for any x 2 ð0, 1Þ, t > 0, where g~ ¼ ðg1 , g2 , g3 Þ, g1 ðtÞ ¼ vð0, tÞ,  with v ¼ WR ðtÞ . g2 ðtÞ ¼ vð1, tÞ, g3 ðtÞ ¼ vx ð1, tÞ, MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1414 Bona, Sun, and Zhang Remark 2.14. Of course, g~ depends upon the particular extension  of  chosen here. However, the intrusion of  is simply as an intermediary for obtaining the trace estimate in Proposition 2.16 below. It plays no other role in the theory. To have appropriate estimates of W0 ðtÞ, the following trace result related to the semi-group WR ðtÞ is needed. Lemma 2.15. There exists a constant C such that for any WR ðtÞ ðxÞ satisfies sup kvðx, ÞkH 1=3 ðRÞ x2R 2 L2 ðRÞ, vðx, tÞ ¼ Ck kL2 ðRÞ : Moreover, vx 2 Cb ðR; L2 ðRÞÞ and sup kvx ðx, ÞkL2 ðRÞ x2R Ck kL2 ðRÞ : Proof. This lemma follows as a special case of Lemma 2.1 in Kenig et al., (1991b) except the continuity of vx ðx, Þ from R to the space L2 ðRÞ, which can be verified using Fatou’s lemma and the argument that appears in the proof of Proposition 2.7. œ The following estimate for W0 ðtÞ follows from Proposition 2.13, Lemma 2.15, and the estimates of Wb ðtÞ established earlier in Theorem 2.10. Proposition 2.16. For any  2 L2 ð0, 1Þ, u ¼ W0 ðtÞ has the property ux 2 Cb ð ½0, 1; L2, t ðRþ ÞÞ and there exists a constant C such that sup kux ðx, ÞkL2 ðRþ Þ CkkL2 ð0, 1Þ : x2ð0, 1Þ We conclude this section with the following proposition which, like the foregoing results, will be needed later (see Proposition 3.2). Proposition 2.17. Let T > 0 be given and Zt uðx, tÞ ¼ W0 ðt  Þ f ð , Þ d: 0 Then sup kux ðx, ÞkL2 ð0, T Þ x2ð0, 1Þ C Z 0 T k f ð , ÞkL2 ð0, 1Þ d: Proof. Observe that Z Zt ux ðx, tÞ ¼ @x ðW0 ðt  Þ f ð , ÞÞ d ¼ 0 T 0 ð0, tÞ ðÞ@x ðW0 ðt  Þ f ð , ÞÞ d MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1415 where 1 if  2 ð0, tÞ; 0 if  > t: Using the Minkowski’s integral inequality gives us ð0, tÞ ðÞ ¼ n Z kux ðx, ÞkL2 ð0, T Þ ¼ T 0 Z T 0 Z 0 Z 1=2 T  ð0, tÞ ðÞ@x ðW0 ðt  Þ f ð , ÞÞ2 dt 1=2 T  @x ðW0 ðt  Þ f ð , ÞÞ2 dt  d d: Thus, invoking Proposition 2.16 (with the initial time Þ gives us sup kux ðx, ÞkL2 ð0, T Þ x2ð0, 1Þ Z T sup x2ð0, 1Þ 0 C Z 0 T Z  T 1=2  @x ðW0 ðt  Þ f ð , ÞÞ2 dt d k f ð , ÞkL2 ð0, 1Þ d: œ The proof is complete. Remark 2.18. It is worth highlighting the crucial role played by the formulas (2.14) and following, which provided an explicit representation of solutions directly in terms of the boundary data. Our theory devolves in large part on the efficacy of these formulas. 3. LOCAL WELL-POSEDNESS In this section, attention will be given to the full nonlinear IBVP  ut þ ux þ uux þ uxxx ¼ 0, uðx, 0Þ ¼ ðxÞ, uð0, tÞ ¼ h1 ðtÞ, uð1, tÞ ¼ h2 ðtÞ, ux ð1, tÞ ¼ h3 ðtÞ ð3:1Þ introduced at the outset of our discussion. For any T > 0 and s
0, let Xs, T be as defined in (1.11) with its usual product topology and let Ys, T be the collection of v 2 Cð ½0, T; H s ð0, 1ÞÞ \ L2 ð ½0, T; H sþ1 ð0, 1ÞÞ with vx 2 Cð ½0, 1; L2 ð0, T ÞÞ. A norm k kYs, T on the space Ys, T is defined by  1=2 kvkYs, T :¼ kvk2Cð ½0, T;H s ð0, 1ÞÞ þ kvk2L2 ð ½0, T;H sþ1 ð0, 1ÞÞ þ kvx k2Cð ½0, 1;L2 ð0, T ÞÞ for v 2 Ys, T .a The space Ys, T possesses the following helpful property. a The reader may notice that the space Ys;T need not include the finiteness of kvx kCð½0;1;L2 ð0;TÞÞ for the arguments that follow to be valid, and hence for proving well-posedness of the IBVP Eq. (3.1) in the space H s ð0; 1Þ. However, by keeping this term, we are able to determine at a stroke that solutions of Eq. (3.1) possess the sharp Kato smoothing effect. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1416 Bona, Sun, and Zhang Lemma 3.1. Let s
0 be given. There exists a constant C such that for any T > 0 and u, v 2 Ys, T , ZT kðuð , tÞvð , tÞÞx kH s ð0, 1Þ dt CðT 1=2 þ T 1=3 ÞkukYs, T kvkYs, T : ð3:2Þ 0 Proof. The proof is given for 0 s 1. The proof for other values of s is similar. Notice first that ZT ZT ZT kuð , tÞvx ð , tÞkL2 ð0,1Þ dt: kux ð ,tÞvð ,tÞkL2 ð0,1Þ dt þ kðuð , tÞvð , tÞÞx kL2 ð0, 1Þ dt 0 0 0 Using the Poincare inequality, there obtains kuð , tÞvx ð ,tÞkL2 ð0,1Þ kuð ,tÞkL1 ð0, 1Þ kvx ð ,tÞkL2 ð0,1Þ 1=2 Cðkuð , tÞkL2 ð0,1Þ þ kuð , tÞk1=2 L2 ð0,1Þ kux ð , tÞkL2 ð0,1Þ Þkvx ð ,tÞkL2 ð0,1Þ : These two terms, when integrated with respect to t, are bounded thusly: ZT ZT kvx ð ,tÞkL2 ð0,1Þ dt kuð ,tÞkL2 ð0,1Þ kvx ð ,tÞkL2 ð0,1Þ dt sup kuð ,tÞkL2 ð0,1Þ 0 t T 0 0 T 1=2 sup kuð ,tÞkL2 ð0,1Þ 0 t T CT 1=2 Z T 0 1=2 kvx ð ,tÞk2L2 ð0,1Þ dt kukY0,T kvkY0,T and Z T 0 1=2 kuð , tÞk1=2 L2 ð0, 1Þ kux ð , tÞkL2 ð0, 1Þ kvx ð , tÞkL2 ð0, 1Þ dt Z sup kuð , tÞk1=2 L2 ð0, 1Þ 0 t T CT 1=3 kukY0, T kvkY0, T : T 0 1=4 kux ð , tÞk2L2 ð0, 1Þ dt Z 0 T 3=4 kvx ð , tÞkL4=3 dt 2 ð0, 1Þ The last three inequalities combine to establish that ZT kuð , tÞvx ð , tÞkL2 ð0, 1Þ dt CðT 1=2 þ T 1=3 ÞkukY0, T kvkY0, T : 0 Similarly, one sees that ZT kux ð , tÞvð , tÞkL2 ð0, 1Þ dt 0 CðT 1=2 þ T 1=3 ÞkukY0, T kvkY0, T : In consequence, estimate (3.2) holds with s ¼ 0. To see that (3.2) is true for s ¼ 1, argue as follows. Observe that kðuð , tÞvð , tÞÞx ÞkH 1 ð0, 1Þ kðuð , tÞvð , tÞÞx kL2 ð0, 1Þ þ kðuð , tÞvð , tÞÞxx kL2 ð0, 1Þ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1417 and that kðuð , tÞvð , tÞÞxx kL2 ð0, 1Þ kðux ð , tÞvð , tÞÞx kL2 ð0, 1Þ þ kðuð , tÞvx ð , tÞÞx kL2 ð0, 1Þ : The inequality (3.2) with s ¼ 0, just established, gives Z T CðT 1=2 þ T 1=3 Þðkux kY0, T kvkY0, T þ kukY0, T kvx kY0, T Þ kðuð , tÞvð , tÞÞxx kL2 ð0, 1Þ dt 0 CðT 1=2 þ T 1=3 ÞkukY1, T kvkY1, T which together with (3.2) (again with s ¼ 0) yields Z T 0 kuð , tÞvð , tÞkH 1 ð0, 1Þ dt CðT 1=2 þ T 1=3 ÞkukY1, T kvkY1, T : The estimate (3.2) with 0 < s < 1 now follows from the nonlinear interpolation theory developed in Bona and Scott (1976). The proof is complete. œ The next step is to show that the IBVP (3.1) is locally well-posed in the space X0, T . Proposition 3.2. Let T > 0 be given. For any ð, h~Þ 2 X0, T with h~ ¼ ðh1 , h2 , h3 Þ, there is a T  2 ð0, T depending on kð, h~ÞkX0, T such that the IBVP Eq. (3.1) admits a unique solution u 2 Y0, T  . Moreover, for any T 0 < T  , there is a neighborhood U of ð, h~Þ such that the IBVP Eqs. (1.1)–(1.3) admits a unique solution in the space Y0, T 0 for any ð , h~1 Þ 2 U and the corresponding solution map from U to Y0, T 0 is Lipschitz continuous. Proof. Write the IBVP (3.1) in its integral equation form Zt uðtÞ ¼ W0 ðtÞ þ Wb ðtÞh~  W0 ðt  Þðuux ÞðÞ d 0 ð3:3Þ where the operator Wb ðtÞ is as defined in formulas (2.14) and (2.22) in Sec. 2 and the spatial variable is suppressed throughout. For given ð, h~Þ 2 X0, T , let r > 0 and  > 0 be constants to be determined. Let S, r ¼ fv 2 Y0,  , kvkY0,  rg: The set S, r is a closed, convex, and bounded subset of the space Y0,  and therefore is a complete metric space in the topology induced from Y0,  . Define a map  on S, r by Zt ðvÞ ¼ W0 ðtÞ þ Wb ðtÞh~  W0 ðt  Þðvvx ÞðÞ d 0 for v 2 S, r . The crux of the matter is the following inequality. For any v 2 S, r , kðvÞkY0,  C0 kð, h~ÞkX0, T þ C1 Z  kvvx ð , ÞkL2 ð0, 1Þ d 0 1=2 C0 kð, h~ÞkX0, T þ C1 ð þ 1=3 Þkvk2Y0,  MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1418 Bona, Sun, and Zhang where C0 and C1 are constants. As the norm on Y0,  has three parts, this amounts to three inequalities, all of which follow immmediately from the linear estimates in Sec. 2 and in Lemma 3.1. Choosing r > 0 and  > 0 so that ( r ¼ 2C0 kð, h~ÞkX0, T , C1 ð1=2 þ 1=3 Þ r ð3:4Þ 1 2, then kðvÞkY0,  r for any v 2 S, r . Thus, with such a choice of r and ,  maps S, r into S, r . The same inequalities allow one to deduce that for r and  chosen as in (3.4), kðv1 Þ  ðv2 ÞkY0,  1 kv  v2 kY0,  2 1 for any v1 , v2 2 S, r . In other words, the map  is a contraction mapping of Sr,  . Its fixed point u ¼ ðuÞ is the unique solution of the IBVP (1.1)–(1.3) in S, r . œ Consider the forced linear problem ut þ ux þ uxxx ¼ f , uð0, tÞ ¼ h1 ðtÞ, uðx, 0Þ ¼ ðxÞ, uð1, tÞ ¼ h2 ðtÞ, ux ð1, tÞ ¼ h3 ðtÞ: ) ð3:5Þ Applying the linear estimates derived in Sec. 2, for ð, h~Þ 2 X0, T and f 2 L1 ð0, T; L2 ð0, 1ÞÞ, the corresponding solution u of (3.5) belongs to the space Y0, T and satisfies kukY0, T Cðkð, h~ÞkX0, T þ k f kL1 ð0, T;L2 ð0, 1ÞÞ Þ ð3:6Þ for some constant C independent of , hj , j ¼ 1, 2, 3 and f. The next lemma gives an estimate for solutions of (3.5) in the space Ys, T with s in the range of 0 s 3. Lemma 3.3. For given T > 0 and s in the range ½0, 3, let there be given f 2 W s=3, 1 ð ½0, T; L2 ð0, 1ÞÞ and ð, h~Þ 2 Xs, T satisfying the compatibility conditions ð0Þ ¼ h1 ð0Þ, ð0Þ ¼ h1 ð0Þ, ð1Þ ¼ h2 ð0Þ ð1Þ ¼ h2 ð0Þ, 0 ð1Þ ¼ h3 ð0Þ if 1/2 < s if 3/2 < s 3/2 , or 3. ð3:7Þ Then Eq. (3.5) admits a unique solution u 2 Ys, T and kukYs, T Cðkð, h~ÞkXs, T þ k f kW s=3, 1 ð0, T;L2 ð0, 1ÞÞ Þ ð3:8Þ for some constant C > 0 independent of , h~, and f. Moreover, if s ¼ 3, ut 2 Y0, T and kut kY0, T Cðkð, h~ÞkX3, T þ k f kW s=3, 1 ð0, T;L2 ð0, 1ÞÞ Þ: MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1419 Proof. The proof is provided for s ¼ 3 since the result for other values of s can be established by interpolation and (3.6). For the solution u of (3.5), let v ¼ ut . Then the function v is a solution of ) vt þ vx þ vxxx ¼ ft , vðx, 0Þ ¼ f ðx, 0Þ  000 ðxÞ  0 ðxÞ, ð3:9Þ vð0, tÞ ¼ h01 ðtÞ, vð1, tÞ ¼ h02 ðtÞ, vx ð1, tÞ ¼ h03 ðtÞ: Applying (3.6) to v in (3.9) yields that Cðk ft kL1 ð0, T;ð0, 1ÞÞ þ kð f ð , 0Þ  000 ð Þ  0 ð Þ, h~0 ÞkX0, T Þ: kvkY0, T Define uðx, tÞ ¼ Z t 0 vðx, Þ d þ ðxÞ: Then uðx, 0Þ ¼ ðxÞ and Zt uð0, tÞ ¼ vð0, Þ d þ ð0Þ 0 Zt h01 ðÞ d þ ð0Þ ¼ 0 ¼ h1 ðtÞ  h1 ð0Þ þ ð0Þ ¼ h1 ðtÞ: Similarly, uð1, tÞ ¼ h2 ðtÞ and ux ð1, tÞ ¼ h3 ðtÞ. Furthermore, it is easily verified that ut ðx, tÞ þ ux ðx, tÞ þ uxxx ðx, tÞ Zt ¼ vðx, tÞ þ ðvx ðx, Þ þ vxxx ðx, ÞÞ d þ 0 ðxÞ þ 000 ðxÞ 0 Zt ¼ vðx, 0Þ þ ð ft ðx, Þ  vx ðx, Þ  vxxx ðx, ÞÞ d 0 Zt þ ðvx ðx, Þ þ vxxx ðx, ÞÞ d þ 0 ðxÞ þ 000 ðxÞ ¼ 0: 0 Thus u solves the IBVP (3.5). Since uxxx ¼ f  ut  ux ¼ f  v  ux , it follows that u 2 Y3, T and satisfies (3.8) with s ¼ 3. The proof is complete. œ Here is the promised local well-posedness result for the IBVP (3.1) in Xs, T . Theorem 3.4. Let T > 0 and s
0 be given. Suppose that ð, h~Þ 2 Xs, T satisfies the s-compatibility conditions. Then there exists a T  2 ð0, T depending only on kð, h~ÞkXs, T such that Eq. (3.1) admits a unique solution u 2 Ys, T  with @ jt u 2 Ys3j, T  MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1420 Bona, Sun, and Zhang for j ¼ 0, 1, 2, . . . , ½s=3  1, ½s=3. Moreover, for any T 0 < T  , there is a neighborhood U of ð, h~Þ such that the IBVP Eq. (3.1) admits a unique solution in the space Ys, T 0 for any ð , h~1 Þ 2 U and the corresponding solution map is Lipschitz continuous. Proof. For given s-compatible ð, h~Þ 2 Xs, T , let r > 0 and  > 0 be given and S, r be the collection of functions v in the space Cð ½0, ; L2 ð0, 1ÞÞ \ L2 ð0, ; H 1 ð0, 1ÞÞ satisfying @ tj v 2 Y3,  , for j ¼ 0, 1, 2, . . . , ½s=3  1 and @ t½s=3 v 2 Ys3½s=3,  , and k@½s=3 vkYs3½s=3 þ t ½s=31 X j¼0 k@ tj vkY3,  r: Let Y s,  ¼ Ys3½s=3,   ½s=31 Y Y3,  j¼0 with the usual product topology. Then the set S, r may be viewed as a closed subset of Y s,  via the mapping v ! ðv, @t v, . . . , @ x½s=3 vÞ  v~, and therefore is a complete metric space. For any v 2 S, r , consider the system of equations 9 P  k ðkÞ ðkÞ ðkÞ ð j Þ ðkjÞ 1 k! = uðkÞ ¼  þ u þ u , u ðx, 0Þ ¼  ðxÞ, @ v v k xxx x t j¼0 j!ðkjÞ! 2 x ð3:10Þ ; ðkÞ ðkÞ ðtÞ, uð1, tÞ ¼ h ðtÞ, u ð1, tÞ ¼ h ðtÞ, uð0, tÞ ¼ hðkÞ x 1 2 3 ðkÞ for k ¼ 0, 1, 2, . . . , ½s=3, where uðkÞ  @ kt u, vðkÞ  @ kt v and k , h1ðkÞ , hðkÞ 2 , h3 are defined in (1.7) and (1.8). By Lemma 3.3, the IBVP (3.10) defines a map  from S, r to the space Y s,  . Moreover,   kð~ vÞkY s,  Ckð, h~ÞkXs, T þ C 1=2 þ 1=3 k~ vkY s,  for some constant C independent of h~, , and . Thus, the argument presented in the proof of Proposition 3.2 shows that  is a contraction map from S, r to S, r if r and  are appropriately chosen. As a result, its fixed point u~ 2 S, r is the unique solution of (3.5). Thus the proof is complete when s 3. In case s > 3, the result just established shows that uð j Þ 2 Cð ½0, ; H 3 ð0, 1ÞÞ \ L2 ð ½0, ; H 4 ð0, 1ÞÞ for j ¼ 0, 1, . . . , ½s=3  1 and u½s=3 2 Ys3½s=3,  ¼ Cð ½0, ; H s3½s=3 ð0, 1ÞÞ \ L2 ð ½0, ; H sþ13½s=3 ð0, 1ÞÞ: In case k ¼ ½s=3  1, (3.10) implies that ð ½s=31Þ uxxx ¼ utð ½s=31Þ  uxð ½s=31Þ 1  @x 2 ð ½s=31Þ X j¼0 C j½s=31 uð j Þ uð ½s=31jÞ ! : MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1421 We thereby arrive at the conclusion uð ½s=31Þ 2 Cð ½0, ; H s3½s=3þ2 ð0, 1ÞÞ: It is further implied that the left-hand side of the last equation belongs to Cð ½0, ; H s3ð ½s=3 ð0, 1ÞÞ \ L2 ð ½0, ; H sþ13½s=3 ð0, 1ÞÞ: Consequently, it must be the case that uð ½s=31Þ 2 Cð ½0, ; H sþ33ð ½s=3 ð0, 1ÞÞ \ L2 ð ½0, ; H sþ43½s=3 ð0, 1ÞÞ: Repeating this argument L2 ð ½0, ; H sþ1 ð0, 1ÞÞ with if necessary yields that u 2 Cð ½0, ; H s ð0, 1ÞÞ \ @ jt u 2 Cð ½0, ; H s3j ð0, 1ÞÞ \ L2 ð ½0, ; H sþ13j ð0, 1ÞÞ for j ¼ 1, 2, . . . , ½s=3  1 and @ t½s=3 u 2 Cð ½0, ; H s3½s=3 ð0, 1ÞÞ \ L2 ð ½0, ; H sþ13½s=3 ð0, 1ÞÞ: œ The proof is complete. 4. GLOBAL WELL-POSEDNESS The results presented in Theorem 3.4 are local in the sense that the time interval ð0, T  Þ on which the solution exists depends on kð, h~ÞkXs, T . In general, the larger kð, h~ÞkXs, T , the smaller will be T  . However, if T  ¼ T no matter what the size of kð, h~ÞkXs, T , the IBVP (3.1) is said to be globally well-posed. In this section we study global well-posedness of the problem (3.1). First we introduce a helpful Banach space. For given s
0 and T > 0, let Zs, T  H s ð0, 1Þ  H þð5sþ9Þ=18 ð0, T Þ  H þð5sþ9Þ=18 ð0, T Þ  H þð5sþ3Þ=18 ð0, T Þ if 0 s Zs, T 3 and  Xs, T if s > 3, where  is any positive constant. Of course, for s 3, Zs, T depends on , but this dependence is suppressed. The Sobolev indices when s lies in ½0, 3 may look a little odd. We feel it likely that they are an artifact of our proof. The strange indices derive from slightly inadequate smoothing results and are the best we can do with what is in hand. Note this inelegance ceases as soon as s
3, and hence for the case of classical solutions when s > 7=2. The same issue arose in Bona et al., (2001) for the quarter-plane problem, so the issue does not necessarily devolve upon the third boundary condition ux ð1, tÞ ¼ h3 ðtÞ. Theorem 4.1. Let T > 0 and s
0. For any s-compatible ð, h~Þ 2 Zs, T , the IBVP Eq. (3.1) admits a unique solution u 2 Ys, T with @ jt u 2 Ys3j, T for j ¼ 0, 1, 2, . . . , ½s=3. Moreover, the corresponding solution map of the IBVP Eq. (3.1) is Lipschitz continuous. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1422 Bona, Sun, and Zhang Proof of Theorem 4.1. In the context of an established local well-posedness result, it suffices to prove the following global a priori H s -estimate for smooth solutions of the IBVP (3.1). Proposition 4.2. For given T > 0 and s
0, there exists a continuous and nondecreasing function s : Rþ ! Rþ such that for any smooth solution u of Eq. (3.1), sup kuð , tÞkH s ð0, 1Þ 0 t T ~ s ðkð, hÞkZs, T Þ: ð4:1Þ The proof of Proposition 4.3 consists of four parts. In part (i), estimate (4.1) is shown to be true for s ¼ 0. In part (ii), estimate (4.1) is shown to be true for s ¼ 3. Then, in part (iii), Tartar’s nonlinear interpolation theory is used to show that (4.1) holds for 0 < s < 3. The validity of (4.1) for other values of s is established in part (iv). Part (i). For a smooth solution u of the IBVP (3.1), write u ¼ w þ v, where v solves ) vt þ vx þ vxxx ¼ 0, vðx, 0Þ ¼ ðxÞ, ð4:2Þ vð0, tÞ ¼ h1 ðtÞ, vð1, tÞ ¼ h2 ðtÞ, vx ð1, tÞ ¼ h3 ðtÞ, with ðxÞ ¼ ð1  xÞh1 ð0Þ þ xh2 ð0Þ þ xð1  xÞðh3 ð0Þ  h2 ð0Þ þ h1 ð0ÞÞ and w solves wt þ wx þ wwx þ wxxx ¼ ðwvÞx  vvx , wð0, tÞ ¼ 0, wð1, tÞ ¼ 0, wx ð1, tÞ ¼ 0: ) wðx, 0Þ ¼ ðxÞ  ðxÞ, ð4:3Þ By Lemma 3.3 Ckð , h~ÞkXs, T kvkYs, T for 0 s 3. In particular, for 3=2 < s kvkYs, T ð4:4Þ 3, Cðkh1 kH s=3 ð0, T Þ þ kh2 kH s=3 ð0, T Þ þ kh3 kH ðs1Þ=3 ð0, T Þ Þ ð4:5Þ since k kH s ð0, 1Þ Cðkh1 kH s=3 ð0, T Þ þ kh2 kH s=3 ð0, T Þ þ kh3 kH ðs1Þ=3 ð0, T Þ Þ: Multiply both sides of the equation in (4.3) by w and integrate over ð0, 1Þ with respect to x. Integration by parts leads to Z1 Z1 d 2 2 kwð , tÞkL2 ð0, 1Þ C jvx ð , tÞw ð , tÞj dx þ C jvx ð , tÞvð , tÞwð , tÞj dx: dt 0 0 Observe that MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain Z 1 0 jvx ð , tÞw2 ð , tÞj dx 1423   sup vx ðx, tÞkwð , tÞk2L2 ð0, 1Þ x2ð0, 1Þ C kvð , tÞkH 3=2þ ð0, 1Þ kwð , tÞk2L2 ð0, 1Þ and Z 1 0 jvx ð , tÞvð , tÞwð , tÞj dx   sup vx ðx, tÞkvð , tÞkL2 ð0, 1Þ kwð , tÞkL2 ð0, 1Þ x2ð0, 1Þ C kvð , tÞk2H 3=2þ ð0, 1Þ kwð , tÞkL2 ð0, 1Þ where  is any fixed positive constant. In consequence, one has that d kwð , tÞkL2 ð0, 1Þ dt C kvð , tÞkH 3=2þ ð0, 1Þ kwð , tÞkL2 ð0, 1Þ þ C kvð , tÞk2H 3=2þ ð0, 1Þ for any t
0 The estimate (4.1) with s ¼ 0 then follows by using Gronwall’s inequality and (4.5). Part (ii). For a smooth solution u, v ¼ ut solves vt þ vx þ ðuvÞx þ vxxx ¼ 0, vð0, tÞ ¼ h01 ðtÞ, vðx, 0Þ ¼  ðxÞ vð1, tÞ ¼ h02 ðtÞ, vx ð1, tÞ ¼ h03 ðtÞ ) where  ðxÞ ¼ 0 ðxÞ  0 ðxÞðxÞ  000 ðxÞ. By Lemma 3.3, there exists a constant C > 0 such that for any T 0 T, kvkY0, T 0 Ckð , h~0 ÞkX0, T þ CðT 01=2 þ T 01=3 ÞkukY0, T kvkY0, T : Choose T 0 T such that CðT 01=2 þ T 01=3 ÞkukY0, T ¼ 1=2; with such a choice, kvkY0, T 0 2Ckð , h~0 ÞkX0, T : Note that T 0 only depends on kukY0, T , and therefore depends only on kð, h~ÞkZ0, T by the estimate proved in Part (i). By a standard density argument, kvkY0, T C1 kð, h~ÞkZ3, T where C1 depends only on T and kð, h~ÞkZ0, T . The estimate (4.1) with s ¼ 3 then follows from v ¼ ðuxxx þ ux þ uux Þ by a now familiar argument. Part (iii). Here is a précis of the (real) interpolation theory as it will be used below. Let B0 and B1 be two Banach spaces such that B1 B0 with the inclusion map continuous. Let f 2 B0 and, for t
0, define Kð f , tÞ ¼ inf fk f  gkB0 þ tkgkB1 g: g2B1 For 0 <  < 1 and 1 p þ1, define MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1424 Bona, Sun, and Zhang ½B0 , B1 , p ¼ B, p ¼ ( f 2 B0 : k f k, p ¼ Z þ1 1=p p p1 Kð f , tÞ t 0 dt < þ1 ) with the usual modification for the case p ¼ þ1. Then B, p is a Banach space with norm kk, p . Given two pairs of indices ð1 , p1 Þ and ð2 , p2 Þ as above, then ð1 , p1 Þ ! ð2 , p2 Þ means 1 < 2 , 1 ¼ 2 or and p1 > p2 : If ð1 , p1 Þ ! ð2 , p2 Þ then B2 , p2 B1 , p1 with the inclusion map continuous. Theorem 4.3. (Bona and Scott, 1976) Let B0j and B1j be Banach spaces such that B j1 B0j with continuous inclusion mappings, j ¼ 1, 2. Let  and q lie in the ranges 0 <  < 1 and 1 q þ1. Suppose A is a mapping such that (i ) A : B1, q ! B20 and for f , g 2 B1, q , kAf  AgkB2 0 C0 ðk f kB1, q þ kgkB1, q Þk f  gkB1 0 and (ii) A : B11 ! B21 and for h 2 B11 kAhkB2 1 C1 ðkhkB1, q ÞkhkB1 , 1 where Cj : Rþ ! Rþ are continuous nondecreasing functions, j ¼ 0, 1. Then if ð, pÞ
ð, qÞ, A maps B1, p into B2, p and for f 2 B1, p kAf kB2, p Cðk f kB1, q Þk f kB1, p , where for r > 0, CðrÞ ¼ 4C0 ð4rÞ1 C1 ð3rÞ : Remark 4.4. This theorem is identical with Theorem 2 of Tartar (1972) except that Tartar makes the more restrictive assumption that the constants C0 and C1 depend only on the B10 norms of the functions in question. Theorem 4.3 was used by Bona and Scott to give the first proof of global well-posedness of the pure initialvalue problem for the KdV-equation on the whole line in fractional order Sobolev spaces H s(R). To prove that estimate (4.1) holds for T > 0 and 0 s 3, let Z s, T ¼ fð, h~Þ 2 Zs, T satisfying s-compatibility conditiong with the inherited norm from the space Zs, T . Choose B10 ¼ Z 0, T , B11 ¼ Z 3, T , B20 ¼ Cð ½0, T; L2 ð0, 1ÞÞ, B21 ¼ Cð ½0, T; H 3 ð0, 1ÞÞ: Let A be the solution map of the IBVP (3.1): u ¼ Að, h~Þ. For given s with 0 < s < 3, choose p ¼ 2 and  ¼ s=3. Then MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain B2, p ¼ Cð ½0, T; H s ð0, 1ÞÞ, 1425 B1, p ¼ Z s, T : In this case, assumption (ii) of Theorem 4.3 is (4.1) with s ¼ 3, which we have already proved. It remains to verify assumption (i) of Theorem 4.3. To this end, let u1 ¼ Að1 , h~1 Þ, u2 ¼ Að2 , h~2 Þ, and w ¼ u1  u2 . It is seen that w solves the variable coefficient problem ) wt þ wx þ ðzwÞx þ wxxx ¼ 0, vðx, 0Þ ¼ 1 ðxÞ  2 ðxÞ wð0, tÞ ¼ h1, 1 ðtÞ  h2, 1 ðtÞ, wx ð1, tÞ ¼ h1, 2 ðtÞ  h2, 2 ðtÞ, wð1, tÞ ¼ h1, 3 ðtÞ  h2, 3 ðtÞ with z ¼ ð1=2Þðu1  u2 Þ. Applying Lemma 3.3 with s ¼ 0 yields that, for any 0 T 0 T, kwkY0, T 0 Cðkð1 , h~1 Þ  ð2 , h~2 ÞkX0, T þ kzwkL1 ð0, T 0 ;L2 ð0, 1ÞÞ Þ   Ckð1 , h~1 Þ  ð2 , h~2 ÞkX0, T þ C T 01=2 þ T 01=3 kzkY0, T kwkY0, T 0 : Because of Part (ii), the estimate kzkY0, T ðkð1 , h~1 kZ0, T þ kð2 , h~2 ÞkZ0, T Þ is obtained for z. If T 0 is chosen such that   C T 01=2 þ T 01=3 kzkY0, T ¼ 1=2 then it follows that kwkY0, T 0 2Ckð1 , h~1 Þ  ð2 , h~2 ÞkX0, T : kwkY0, T ðkð1 , h~1 ÞkZ0, T þ kð2 , h~2 ÞkZ0, T Þkð1 , h~1 Þ  ð2 , h~2 ÞkX0, T : ð4:6Þ Since T only depends on kzkY0, T which in turn only depends on kð1 , h~1 ÞkZ0, T þ kð2 , h~2 ÞkZ0, T , by a standard extension argument, one arrives at 0 Thus assumption (i) of Theorem 4.3 is satisfied. Estimate (4.1) is established for 0 < s < 3 by invoking Theorem 4.3. Part (iv). We prove that (4.1) holds for 3 < s < 6. The same argument can be invoked for s
6. For a smooth solution u of the IBVP (3.1), v ¼ ut solves ) vt þ vx þ ðuvÞx þ vxxx ¼ 0, vðx, 0Þ ¼  ðxÞ vð0, tÞ ¼ h01 ðtÞ, vð1, tÞ ¼ h02 ðtÞ, Applying Lemma 3.3 for any 0 < T 0 kvkYs3, T 0 vx ð1, tÞ ¼ h03 ðtÞ : T gives the inequality   Ckð, h~ÞkZs, T þ C T 01=2 þ T 01=3 kvkYs3, T 0 kukYs3, T for some constant C > 0 independent of T 0 and ð, h~Þ. Thus, if one chooses T 0 such that MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1426 Bona, Sun, and Zhang   C T 01=2 þ T 01=3 kukYs3, T ¼ 1=2, then 2Ckð, h~ÞkZs, T : kvkYs3, T 0 Because T 0 only depends on kukYs3, T , which by the estimate (4.1) proved in Part (iii) only depends on kð, h~ÞkZs3, T , one obtains ~ ~ ~ ~ s3 ðkð, hÞkZs3, T Þkð, hÞkZs, T : kvkYs3, T Consequently, kukYs, T C s3 ðkð, hÞkZs3, T Þkð, hÞkZs, T œ and the proof is complete. 5. ANALYTICITY For given T > 0 and s
0, let X s, T be the collection of all s-compatible functions ð, h~Þ 2 Xs, T . By the definition of s-compatibility, X s, T is a linear subspace of the Banach space Xs, T if and only if 0 s 7=2. When 0 s 7=2, we consider X s, T as a Banach space with its norm inherited from Xs, T . By the results established in Secs. 3 and 4, the IBVP (3.1) defines a nonlinear map KI from X s, T to the space Ys, T for any s
0. From the proofs of the results given in Sec. 3, the map KI is known to be locally Lipschitz continuous from DðKI Þ, the domain of KI , to Ys, T . In this section it is shown that this nonlinear map KI is analytic. More precisely, when 0 s 7=2, for any g 2 DðKI Þ, there exists an  > 0 such that for any w 2 X s, T with kwkX s, T , we have g þ w 2 X s, T and KI ðg þ wÞ has the following Taylor series expansion: KI ðg þ wÞ ¼ KI ðgÞ þ 1 X K ðnÞ ðgÞ½wn  I n¼1 n! where KIðnÞ ðgÞ is the nth order Fréchet derivative of KI evaluated at g and the series converges in the space YTs . In case s > 7=2, the Taylor series expansion does not hold as just written since the space X s, T is no longer a linear vector space. In this case, we consider the initial-boundary value problem for a general m-nonlinear system, which includes the IBVP (3.1) as a special case, and show that the corresponding nonlinear solution map KI is analytic in this context. In pursuit of this program, we present a well-posedness result for the linearized KdV-equation with variable coefficients, viz. ) ut þ ux þ ðauÞx þ uxxx ¼ f ðx, tÞ, uðx, 0Þ ¼ ðxÞ, ð5:1Þ uð0, tÞ ¼ h1 ðtÞ, uð1, tÞ ¼ h2 ðtÞ, ux ð1, tÞ ¼ h3 ðtÞ: MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1427 Proposition 5.1. Let 0 s 3 and T > 0 be given. Assume that a 2 Ys, T . Then for any f 2 H s=3 ð ½0, T; H ð3sÞ=3 ð0, 1ÞÞ and ð, h~Þ 2 X s, T , Eq. (5.1) admits a unique solution u 2 Ys, T satisfying kukYs, T Cðkð, h~ÞkXs, T þ k f kH s=3 ð ½0, T;H s=3 ð0, 1ÞÞ Þ where C > 0 only depends on kakYs, T . Proof. The proof is similar to that of Theorem 3.4, and so we only provide a sketch. For given 0 <  T and r > 0, let S, r ¼ fw 2 Ys,  : kwkYs,  rg: For specified a 2 Ys, T , f 2 H s=3 ð ½0, T; H s=3 ð0, 1ÞÞ and ð, h~, f Þ 2 Xs, T  H s=3 ð ½0, T; H s=3 ð0, 1ÞÞ with ð, h~Þ 2 X s, T , consider a map  : S, r ! Ys,  defined by u ¼ ðvÞ where u is the unique solution of ut þ ux þ uxxx ¼ f ðx, tÞ  ðavÞx , uð0, tÞ ¼ h1 ðtÞ, uð1, tÞ ¼ h2 ðtÞ, ) uðx, 0Þ ¼ ðxÞ, ux ð1, tÞ ¼ h3 ðtÞ, for v 2 S, r . Applying Lemmas 3.3 and 3.1 yields kðvÞÞkYs,  Choose 0 <  Ckð, h~, f ÞkXs, T H s=3 ð0, T;H s=3 ð0, 1ÞÞ þ Cð1=3 þ 1=2 ÞkakYs, T kvkYs,  : T and r such that r ¼ 2Ckð, h~, f ÞkXs, T H s=3 ð0, T;H s=3 ð0, 1ÞÞ ð5:2Þ Cð1=3 þ 1=2 ÞkakYs, T r ð5:3Þ and 1=2: It follows that kðvÞkYs,  r for any v 2 S, r and that for any v1 , v2 2 S, r , kððv1 Þ  ðv2 ÞÞkYs,  1 kv  v2 kYs,  : 2 1 Thus  is a contraction from S, r to S, r . Its unique fixed point is the desired solution of (5.1) for 0 t . However, since  is chosen according to (5.2) and (5.3) which only depends on kakYs, T , this local argument can be iterated to extend the solution to the entire temporal interval 0 t T. The proof is complete. œ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1428 Bona, Sun, and Zhang Formally, if KI is an analytic mapping from X s, T to Ys, T , then, for n ¼ 0, 1, 2, . . . , its n-th order Fréchet derivative KIðnÞ ðgÞ at g 2 X s, T exists and is the symmetric, n-linear map from X s, T to Ys, T given by ( !) n X @n ðnÞ KI ðgÞ½w1 , . . . :, wn  ¼ k wk K gþ @1 @n I k¼1 0,..., 0 for any w1 , w2 , . . . , wn 2 X s, T . The homogeneous polynomial KIðnÞ ðgÞ½wn  of degree n induced by KIðnÞ ðgÞ, where wn ¼ ðw, w, . . . , wÞ (n components), is  dn K ðg þ wÞ KIðnÞ ðgÞ½wn  ¼ dn I ¼0 for w ¼ ðw , wh Þ 2 X s, T . If we define yn by yn ¼ KIðnÞ ðgÞ½wn , then it is formally ascertained that for 0 < t < T, ð y1 , y2 , . . . , yn Þ solves the system of the equations ) @t y1 þ @x y1 þ @x ðuy1 Þ þ @ 3x y1 ¼ 0, y1 ðx, 0Þ ¼ w ðxÞ, ð5:4Þ y1 ð0, tÞ ¼ wh1 ðtÞ, y1 ð1, tÞ ¼ wh2 ðtÞ, @x y1 ð1, tÞ ¼ wh3 ðtÞ and @t yk þ @x yk þ @x ðuyk Þ þ yk ðx, 0Þ ¼ 0, @ 3x yk yk ð0, tÞ ¼ 0, ! 9 k1 k 1X > @x ð yj ykj Þ, = ¼ 2 j¼0 j > ; yk ð1, tÞ ¼ 0, @yk ð1, tÞ ¼ 0 ð5:5Þ for 2 k n, where u ¼ KI ðgÞ and w ¼ ðw , wh1 , wh2 , wh3 Þ 2 X s, T . On the other hand, for any g ¼ ð, h~Þ 2 DðKI Þ, let u ¼ KI ðgÞ and consider solving the linear systems (5.4)–(5.5). It follows from Proposition 5.1 that (5.4)–(5.5) define a homogeneous polynomial of degree n from X s, T to Ys, T as described by the following proposition. Proposition 5.2. Let T > 0, 0 s 3, and g 2 X s, T be given and let u ¼ KI ðgÞ. Then Eqs. (5.4)–(5.5) define a homogeneous polynomial KIðnÞ ðgÞ½wn  of degree n from X s, T to Ys, T . Moreover, there exists a constant c3 such that kyn kYs, T cn3 n!kwknX s, T ð5:6Þ for any n
2, where c3 ¼ c3 ðT, kukYs, T Þ, and it may be that c3 ! þ1 as T ! þ1 or kukYs, T ! þ1, but in any case c3 ! 0 if T ! 0. Proof. The proof is a straightforward consequence of the linear estimates in Sec. 2 and Proposition 5.1 (cf. Zhang, 1995b), Proposition 3.3 for a detailed argument in related circumstances). œ MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain Define a Taylor polynomial Pn ðwÞ of degree n for w 2 X s, T by n n X X KIðkÞ ðgÞ½wk  yk ¼ KI ðgÞ þ Pn ðwÞ ¼ k! k! k¼1 k¼0 1429 ð5:7Þ and a Taylor series by PðwÞ ¼ 1 X K ðkÞ ðgÞ½wk  I k¼0 k! : ð5:8Þ Proposition 5.3. Let T > 0 and 0 s 3 be given. For any g ¼ ð, h~Þ 2 DðKI Þ, there exists an  > 0 depending only on kKI ðgÞkYs, T such that the formal Taylor series (5.8) is uniformly convergent in the space Ys, T with respect to w 2 X s, T with kwkX s, T . Moreover, if v ¼ PðwÞ, then v 2 Ys, T solves the problem ) vt þ vx þ vvx þ vxxx ¼ 0, vðx, 0Þ ¼ ðxÞ þ w ðxÞ ð5:9Þ vð0, tÞ ¼ h1 þ wh1 , vð1, tÞ ¼ h2 þ wh2 , vx ð1, tÞ ¼ h3 þ wh3 for 0 t T. Proof. It is readily seen that the sequence fPn ðwÞg1 n¼0 of Taylor polynomials is Cauchy in Ys, T uniformly for w in the ball of radius  in X s, T for suitable . Indeed, because of Proposition 5.2, for m
n
0,   m m m ky k X X X yk  k Ys, T   kPn ðwÞ  Pm ðwÞkYs, T ¼  ck3 khkkX s, T :   k¼n k!  k! k¼n k¼n Ys, T If  is chosen so that  1=ð2c3 Þ, then for w 2 X s, T with kwkX s, T , m X 1 kPn ðwÞ  Pm ðwÞkYs, T k 2 k¼n ð5:10Þ which goes to zero uniformly as n, m ! 1. Since fPn ðwÞg1 n¼0 is a Cauchy sequence in the space Ys, T , it makes sense to define v ¼ PðwÞ 2 Ys, T as its limit as n ! 1. It is then readily verified that v solves the IBVP (3.1). The proof is complete. œ The following theorem is now adduced. Theorem 5.4. (Analyticity) For any T > 0 and 0 s 3, the IBVP Eq. (3.1) establishes an analytic map KI from the space X s, T to the space Ys, T in the sense that for any g 2 DðKI Þ there exists an  > 0 such that for any w 2 X s, T with kwkX s, T , the Taylor series expansion 1 X KIðnÞ ðgÞ½wn  KI ðg þ wÞ ¼ n! n¼0 MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1430 Bona, Sun, and Zhang converges in the space Ys, T . Moreover, the convergence is uniform with regard to w in the aforementioned ball in X s, T . Remark 5.5. The above theorem holds also for 3 < s 7=2. Since the reasoning in this case is similar to that put forward for the system discussed below, we include analysis for this range of s in our next theorem. Now, consider the case wherein s > 3. This situation is a little more involved than the previous case because the compatibility conditions are no longer linear restrictions. One could attempt to deal directly with the geometric situation implied by the nonlinear compatibility conditions, but another approach presents itself which is more transparent. That is to link the single equation faithfully with a class of systems to be discussed presently. As in the Sec. 4, for any s > 3, write s ¼ 3m þ s0 where m > 0 is an integer and 0 < s0 3. For T > 0, define the space Y sT to be Y sT ¼ Y3, T  Y3, T   Y3, T  Ys0 , T and the space X sT as X sT ¼ X3, T  X3, T   X3, T  Xs0 , T : Consider the system u~ðx, 0Þ ¼ ~ðxÞ u~t þ u~x þ ðFð~ uÞ~ uÞx þ u~xxx ¼ 0, u~ð0, tÞ ¼ h~1 , u~ð1, tÞ ¼ h~2 , u~x ð1, tÞ ¼ h~3 ) ð5:11Þ where u~ ¼ ðu0 , u1 , . . . , um ÞT , ~ ¼ ð0 , 1 , . . . , m ÞT , h~j ¼ ðhj, 0 , hj, 1 , . . . , hj, m ÞT for j ¼ 1, 2, 3 and Fð~ uÞ ¼ ð1=2Þ u20 , m X m u u 2u0 u1 , . . . , k k mk k¼0 !T : By Theorem 4.1, for any s-compatible ð, h1 , h2 , h3 Þ 2 Xs, T , the IBVP (3.1) has a unique solution u 2 Ys, T . If one defines 0 by 0 ¼  and let k be obtained from  k by (3.4) with hj, k ¼ hðkÞ j , uk ¼ @ t u for j ¼ 1, 2, 3 and k ¼ 0, 1, . . . , m, then s ð~, h~1 , h~2 , h~3 Þ 2 X T and u~ is a solution of (5.11). In this sense, the IBVP (3.1) is a specialization of the system (5.11). Theorem 5.6. Let T > 0 and s > 3 be given with s ¼ 3m þ s0 and 0 s0 < 3. Then for any ð~, h~Þ 2 X sT , the system (5.11) admits a unique solution u~ 2 Y sT . MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1431 Proof. Observe that the nonlinear system (5.11) consists of initial-boundary-value problems for m þ 1 scalar equations. Among them, the first one is the IBVP (3.1) which only involves u0 . The second one involves only u0 and u1 . If u0 is known, then the second IBVP is a linear problem. Similar remarks apply for the rest of the equations. Thus we can solve the nonlinear system by solving for u0 from the first equation, plugging u0 into the second equation and solving the corresponding linearized problem to obtain u1 , etc. Using Theorem 4.1 and Proposition 5.1, it is deduced inductively that uk 2 Y3, T for k ¼ 0, 1, . . . , m  1. Now the equation related to um has the form ) um ðx, 0Þ ¼ m ðxÞ @t um þ @x um þ @x ðaum Þ þ @ 3x um ¼ f , ð5:12Þ um ð0, tÞ ¼ h1, m ðtÞ, um ð1, tÞ ¼ h2, m ðtÞ, @x um ð1, tÞ ¼ h3, m ðtÞ 0 where f 2 Cð ½0, T; H s ð0, 1ÞÞ and a 2 Y 3m T are known. Using Lemmas 4.1–4.6, the contraction principle and arguments similar to those appearing in the proof of Theorem 4.1, it can be shown that for any ðm , h1, m , h2, m , h3, m Þ 2 X s0 , T , (5.12) admits a unique solution um 2 Ys0 , T . The proof is complete. œ The last result implies the nonlinear system (5.11) defines a nonlinear map KI from the space X sT to Y sT for given T > 0 and s ¼ 3m þ s0 with 0 s0 < 3. We claim this map KI is analytic from X sT to Y sT . For the purpose of establishing this contention, consider the linearized system corresponding to the nonlinear system (5.11), namely ) @t w~ þ @x w~ þ @x ðJð~ aÞw~ Þ þ @ 3x w~ ¼ f~, w~ ðx, 0Þ ¼ ~ðxÞ, ð5:13Þ w~ ð0, tÞ ¼ h~1 ðtÞ, w~ ð1, tÞ ¼ h~2 ðtÞ, @x w~ ð1, tÞ ¼ h~3 ðtÞ where J is the Jacobian matrix of F at u~ ¼ a~, viz.  @Fð~ uÞ Jð~ aÞ ¼ @~ u u~¼~a ! k X  k  ¼ ði, jÞakj þ aj ði, k  jÞ j j¼0 0 k, i m and ði, jÞ ¼ 1 0 if i ¼ j, if i ¼ 6 j: Proposition 5.7. Let T > 0 and s > 3 be given. Suppose a~ 2 Y sT and f~ 2 FTs ¼ L1 ð0, T; H 3 ð0, 1ÞÞ  0  L1 ð0, T; H 3 ð0, 1ÞÞ  L1 ð0, T; H s ð0, 1ÞÞ: Then for any ð~, h~1 , h~2 , h~3 Þ 2 X sT , Eq. (5.13) admits a unique solution w~ 2 Y sT . Moreover, kw~ kY Ts where ðk~ akY sT Þðkð~, h~1 , h~2 , h~3 ÞkX sT þ k f~kFTs Þ : Rþ ! Rþ is a continuous nondecreasing function. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 1432 Bona, Sun, and Zhang Proof. The proof is similar to that of Proposition 5.1 and therefore omitted. For given u~ ¼ KI ðð~, h~ÞÞ with ð~, h~Þ 2 X sT , consider the linear systems 9 ~ ~,= y~1 ðx, 0Þ ¼ w uÞ~ y1 Þ þ @ 3x y~1 ¼ 0, @t y~1 þ @x y~1 þ @x ðJð~ y~1 ð0, tÞ ¼ w~ h~ ðtÞ, y~1 ð1, tÞ ¼ w~ h~ ðtÞ, 1 2 and @x y~1 ð1, tÞ ¼ w~ h~ ðtÞ; for 2 n ð5:14Þ 3 y1 , . . . , y~n1 Þ, uÞ~ yn Þ þ @ 3x y~n ¼ Fn ð~ @t y~n þ @x y~n þ @x ðJð~ y~n ð0, tÞ ¼ 0, œ y~n ð1, tÞ ¼ 0, @x y~n ð1, tÞ ¼ 0 n y y i i, j ni, kj ! y~n ðx, 0Þ ¼ 0, ) ð5:15Þ N, where Fn ¼ ð fn, 0 , fn, 1 , . . . , fn, m ÞT with fn, k k X n1 X 1 k ¼  @x j 2 j¼0 i¼1 for k ¼ 0, 1, . . . , m. Proposition 5.8. Given T > 0, s > 3, and g~ ¼ ð~, h~Þ 2 Y sT , let u~ ¼ KI ðð~, h~ÞÞ. Then the systems (5.14)–(5.15) defines a homogeneous polynomial KðnÞ gÞ½w~ n  of degree n from I ð~ s s X T to Y T . Moreover, there exists a constant C > 0 such that Cn n!kw~ knX sT k~ yn kY sT for any n
2, where C ¼ CðT, k~ ukY sT Þ. Here C may go to þ1 when T ! 1 or k~ ukY sT ! 1, but must go to 0 if T ! 0, Proof. This follows from Proposition 5.4 by direct computation. œ ~ Þ of degree n for w~ 2 X sT by Define a Taylor polynomial Pn ðw Pn ðw~ Þ ¼ N X KðkÞ ðgÞ½w~ k  I k! k¼0 ¼ KI ð~ gÞ þ N X y~k k¼1 k! , ð5:16Þ and a Taylor series by Pðw~ Þ ¼ 1 X ~k KðkÞ ð~ gÞ½w I k¼0 k! : ð5:17Þ A proof similar to that given for Proposition 5.3 yields the following proposition. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Korteweg–de Vries Equation on Finite Domain 1433 Proposition 5.9. Let T > 0 and s > 0 be given. For any g~ ¼ ð~, h~1 , h~2 , h~3 ÞÞ 2 X sT , there exists an  > 0 depending only on kKI ð~ gÞkY sT such that the formal Taylor series (5.17) is uniformly convergent in the space Y sT with respect to w~ 2 X sT with kw~ kX sT . Moreover, if v~ ¼ Pðw~ Þ, then v~ 2 Y sT solves the problem @t v~ þ @x v~ þ @x ðFð~vÞx v~Þ þ @ 3x v~ ¼ 0, ~ h~ ðtÞ, v~ð0, tÞ ¼ h~1 ðtÞ þ w 1 v~ðx, 0Þ ¼ ~ðxÞ þ w~ ~ðxÞ v~ð1, tÞ ¼ h~2 ðtÞ þ w~ h~ ðtÞ, 2 9 = @x v~ð1, tÞ ¼ h~3 ðtÞ þ w~ h~ ðtÞ ; 3 ð5:18Þ for 0 t T. Consequently, we have the following theorem. Theorem 5.10. (Analyticity) For any T > 0 and s > 3, the nonlinear problem (5.11) establishes a map KI from the space X sT to the space Y sT . The map KI is analytic from X sT to Y sT in the sense that for any g~ 2 X sT , there exists an  > 0 such that for any w~ 2 X sT with kw~ kX sT , the Taylor series expansion ~Þ ¼ KI ð~ gþw 1 X KðnÞ gÞ½w~ n  I ð~ n! n0 converges in the space Y sT . Moreover, the convergence is uniform with regard to h~ in the aforementioned ball in X sT . ACKNOWLEDGMENTS JLB was partially supported by the National Science Foundation and by the W. M. Keck Foundation. SMS was partially supported by the National Science Foundation. BYZ was partially supported by the Charles H. Taft Memorial Fund. The manuscript benefitted very considerably from detailed commentary by an anonymous referee. The authors are grateful for this advice. REFERENCES Benjamin, T. B., Bona, J. L., Mahony, J. J. (1972). Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Royal Soc. London Ser. A 272:47–78. Boczar-Karakiewicz, B., Bona, J. L., Pelchat, B. (1991). Interaction of internal waves with the sea bed on continental shelves. Continental Shelf Res. 11:1181–1197. Boczar-Karakiewicz, B., Bona, J. L., Romńczyk, W., Thornton, E. B. Seasonal and interseasonal variability of sand bars at Duck, NC, USA. 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