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We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator ω + AS,Σ admits a bounded H ∞-calculus with angle smaller than π/2, provided ω > 0. As an... more
We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator ω + AS,Σ admits a bounded H ∞-calculus with angle smaller than π/2, provided ω > 0. As an application, we consider critical spaces for the Navier-Stokes equations on the surface Σ. In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L2(Σ,TΣ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
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It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there... more
It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. 0. Introduction. It is the main purpose of this paper to prove under mild regularity assumptionsthat Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers. In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8&0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0 , n), we ...
1. Introduction The classical Stefan problem is a model for phase transitions in solid-liquid systems and accounts for heat diiusion and exchange of latent heat in a homogeneous medium. The strong formulation of this model corresponds to... more
1. Introduction The classical Stefan problem is a model for phase transitions in solid-liquid systems and accounts for heat diiusion and exchange of latent heat in a homogeneous medium. The strong formulation of this model corresponds to a moving boundary problem involving a parabolic diiusion equation for each phase and a transmission condition prescribed at the interface separating the phases. Molecular considerations attempting to explain supercooling and dendritic growth of crystals suggest to also include surface tension on the interface separating the solid from the liquid region. In order to formulate the Stefan problem we introduce the following notations. Let be a smooth bounded domain in R
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In this chapter we introduce some basic tools from operator and semigroup theory. The class of sectorial operators is studied in detail, its functional calculus is introduced, leading to analytic semigroups and complex powers.
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In this chapter we introduce the necessary background in differential geometry of closed compact hypersurfaces in ℝ n . We investigate the differential geometric properties of embedded hypersurfaces in n-dimensional Euclidean space,... more
In this chapter we introduce the necessary background in differential geometry of closed compact hypersurfaces in ℝ n . We investigate the differential geometric properties of embedded hypersurfaces in n-dimensional Euclidean space, introducing the notions of Weingarten tensor, principal curvatures, mean curvature, tubular neighbourhood, surface gradient, surface divergence, and Laplace-Beltrami operator.
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In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and... more
In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient is that we are able to show that quasilinear parabolic evolution equations generate a smooth semiflow on the trace spaces associated with maximal regularity, which are the natural phase spaces in this framework. 1.
We consider strongly coupled quasilinear reaction-diffusion systems subject to nonlinear boundary conditions. Our aim is to develop a geometric theory for this type of equations. Such a theory is necessary in order to describe the... more
We consider strongly coupled quasilinear reaction-diffusion systems subject to nonlinear boundary conditions. Our aim is to develop a geometric theory for this type of equations. Such a theory is necessary in order to describe the dynamical behavior of solutions. In our main result we establish the existence and attractivity of center manifolds under suitable technical assumptions. The technical ingredients we need consist of the theory of strongly continuous analytic semigroups, maximal regularity, interpolation theory and evolution equations in extrapolation spaces.
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The Willmore ∞ow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of so- lutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere,... more
The Willmore ∞ow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of so- lutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere.
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We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, L_p-maximal regularity theory, and the... more
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, L_p-maximal regularity theory, and the implicit function theorem.
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Several results from differential geometry of hypersurfaces in R^n are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the... more
Several results from differential geometry of hypersurfaces in R^n are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
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Research Interests: Mathematics and Physics
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We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ Σ without boundary and flows along $$\Sigma $$ Σ . Local-in-time well-posedness is established in the... more
We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ Σ without boundary and flows along $$\Sigma $$ Σ . Local-in-time well-posedness is established in the framework of $$L_p$$ L p -$$L_q$$ L q -maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $$\Sigma $$ Σ , and we show that each equilibrium on $$\Sigma $$ Σ is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.
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Abstract: In this note we describe a new approach to establish regularityproperties for solutions of parabolic equations. It is based on maximalregularity and the implicit function theorem.
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ABSTRACT We prove that the Willmore flow can drive embedded surfaces to selfintersections in finite time.
Page 1. Classical solutions for Hele-Shaw models with surface tension Joachim ESCHER and Gieri SIMONETT Abstract It is shown that surface tension effects on the free boundary are reg-ularizing for Hele-Shaw models. ...
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We prove that the Willmore ow can drive embedded surfaces to self-inter- sections in nite time.
ABSTRACT We show that general systems of elliptic diierential operators have a bounded H1-functional calculus in Lp spaces, provided the coeecients satisfy only minimal regularity assumptions.
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ABSTRACT A survey of the results obtained in (22) is presented. In (22) the authors prove the existence of a local-in-time solution for the classical two-phase Stefan problem that is analytic in space and time. The result is based on Lp... more
ABSTRACT A survey of the results obtained in (22) is presented. In (22) the authors prove the existence of a local-in-time solution for the classical two-phase Stefan problem that is analytic in space and time. The result is based on Lp maximal regularity, which is proved first, and the implicit function theorem.
ABSTRACT It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying Lie structure.... more
ABSTRACT It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying Lie structure. Applications to Bellman equations and to a class of mean curvature flows are also discussed.
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ABSTRACT We prove that the surface-di#usion flow and the volumepreserving mean curvature flow can drive embedded hypersurfaces to self-intersections.
Abstract: This paper will address existence, uniqueness, and regularity of classical solutionsfor the quasi-stationary problem. A major difficulty in solving the set ofequations in (1.2) comes from the fact that the problem has a nonlocal... more
Abstract: This paper will address existence, uniqueness, and regularity of classical solutionsfor the quasi-stationary problem. A major difficulty in solving the set ofequations in (1.2) comes from the fact that the problem has a nonlocal character,since the solution of an elliptic boundary value problem is needed inorder to determine the normal velocity V of the moving interface \Gamma(t): On theother
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The study of the basic model for incompressible two-phase flows with phase transitions in the case of equal densities, initiated in the paper Prüss, Shibata, Shimizu, and Simonett [16], is continued here with a stability analysis of... more
The study of the basic model for incompressible two-phase flows with phase transitions in the case of equal densities, initiated in the paper Prüss, Shibata, Shimizu, and Simonett [16], is continued here with a stability analysis of equilibria and results on the asymptotic behavior of global solutions. The results parallel those for the thermodynamically consistent Stefan problem with surface tension obtained in Prüss, Simonett, and Zacher [19].
We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally... more
We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.