SU\G(𝔸)/K·U
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All published works (122)

Coriolis-driven fluid motion on spherical surfaces

2024-12-28
Yuanzhen Shao , Gieri Simonett , Mathias Wilke
We consider the motion of an incompressible viscous fluid on a sphere, incorporating the effects of the Coriolis force. We demonstrate that global solutions exist for any divergence-free initial condition … We consider the motion of an incompressible viscous fluid on a sphere, incorporating the effects of the Coriolis force. We demonstrate that global solutions exist for any divergence-free initial condition with finite kinetic energy. Furthermore, we show that each solution converges at an exponential rate to a state that is aligned with the rotation of the sphere.
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The Navier-Stokes equations on manifolds with boundary

2024-11-05
Yuanzhen Shao , Gieri Simonett , Mathias Wilke

On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D

2024-02-10
Hengrong Du , Yuanzhen Shao , Gieri Simonett

On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D

2023-01-01
Hengrong Du , Yuanzhen Shao , Gieri Simonett
We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with … We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with prescribed energy correspond exactly with the equilibria of the system. The system is investigated in the framework of quasilinear parabolic systems and shown to be locally well-posed in an $L_p$-setting. Furthermore, we prove that constant equilibria are normally stable. In particular, we show that solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant equilibrium. Finally, we establish that the negative entropy serves as a strict Lyapunov functional and we then show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.
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The Navier-Stokes equations on manifolds with boundary

2023-01-01
Yuanzhen Shao , Gieri Simonett , Mathias Wilke
We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold $\sM$ with boundary. The motion on $\sM$ is modeled by the incompressible Navier-Stokes equations, and the … We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold $\sM$ with boundary. The motion on $\sM$ is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on $\partial\sM$. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on $\sM$ that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case $\sM$ is two-dimensional we show that solutions with divergence free initial condition in $L_2(\sM; T\sM)$ exist globally and converge to an equilibrium exponentially fast.
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$$H^\infty $$-Calculus for the Surface Stokes Operator and Applications

2022-10-25
Gieri Simonett , Mathias Wilke
Abstract We consider a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded $$H^\infty $$ <mml:math … Abstract We consider a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded $$H^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math> -calculus with angle smaller than $$\pi /2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> . As an application, we consider critical spaces for the Navier–Stokes equations on the surface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . In case $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> is two-dimensional, we show that any solution with a divergence-free initial value in $$L_2(\Sigma , \textsf{T}\Sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>Σ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mi>Σ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
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Well-posedness for magnetoviscoelastic fluids in 3D

2022-09-28
Hengrong Du , Yuanzhen Shao , Gieri Simonett
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Well-posedness for magnetoviscoelastic fluids in 3D

2022-01-01
Hengrong Du , Yuanzhen Shao , Gieri Simonett
We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal $L_p$-regularity, we establish … We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal $L_p$-regularity, we establish existence and uniqueness of local strong solutions and we show that each solution is smooth (in fact analytic) in space and time. Moreover, we give a complete characterization of the set of equilibria and show that solutions that start out close to a constant equilibrium exist globally and converge to a (possibly different) constant equilibrium. Finally, we show that every solution that is eventually bounded in the topology of the state space exists globally and converges to the set of equilibria.
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$H^\infty$-calculus for the surface Stokes operator and applications

2021-11-24
Gieri Simonett , Mathias Wilke
We consider a smooth, compact and embedded hypersurface $\Sigma$ without boundary and show that the corresponding (shifted) surface Stokes operator $\omega+A_{S,\Sigma}$ admits a bounded $H^\infty$-calculus with angle smaller than $\pi/2$, … We consider a smooth, compact and embedded hypersurface $\Sigma$ without boundary and show that the corresponding (shifted) surface Stokes operator $\omega+A_{S,\Sigma}$ admits a bounded $H^\infty$-calculus with angle smaller than $\pi/2$, provided $\omega>0$. As an application, we consider critical spaces for the Navier-Stokes equations on the surface $\Sigma$. In case $\Sigma$ is two-dimensional, we show that any solution with a divergence-free initial value in $L_2(\Sigma,\mathsf{T}\Sigma)$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
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Preface

2021-09-01
Robert Denk , Yoshikazu Giga , Hideo Kozono +3 more
s Ph.D. thesis was on integrated semigroups and differential operators on L p .On the one side, he set sights on Banach space-valued Laplace transforms and spectral theory [1,4,5] and … s Ph.D. thesis was on integrated semigroups and differential operators on L p .On the one side, he set sights on Banach space-valued Laplace transforms and spectral theory [1,4,5] and on the other side on functional analytic tools for partial differential equations, like the Schrödinger equation [2].A highlight is this direction is the monograph "Vector Valued Laplace Transforms and Cauchy Problems" [B1], published jointly with Wolfgang Arendt, Charles Batty and Frank Neubrander in the Birkhäuser Monographs Series.From 1990 to 1995, he held a position as Oberassistent at the University of Zurich.There, his research interests shifted more and more toward properties of elliptic operators arising in partial differential equations of evolution type.His publications during this period demonstrate the significance of heat kernel and Gaussian estimates as well of the H ∞ -calculus in the treatment of evolution systems [3,6,8,9].In 1995, he com-
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$H^\infty$-calculus for the surface Stokes operator and applications

2021-01-01
Gieri Simonett , Mathias Wilke
We consider a smooth, compact and embedded hypersurface $Σ$ without boundary and show that the corresponding (shifted) surface Stokes operator $ω+A_{S,Σ}$ admits a bounded $H^\infty$-calculus with angle smaller than $π/2$, … We consider a smooth, compact and embedded hypersurface $Σ$ without boundary and show that the corresponding (shifted) surface Stokes operator $ω+A_{S,Σ}$ admits a bounded $H^\infty$-calculus with angle smaller than $π/2$, provided $ω&gt;0$. 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 $L_2(Σ,\mathsf{T}Σ)$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
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On the Navier–Stokes equations on surfaces

2020-11-11
Jan Prüß , Gieri Simonett , Mathias Wilke
Abstract We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and flows along $$\Sigma $$ … Abstract We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and flows along $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . Local-in-time well-posedness is established in the framework of $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> - $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math> -maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> , and we show that each equilibrium on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> 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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On the Navier-Stokes equations on surfaces

2020-05-02
Jan Pruess , Gieri Simonett , Mathias Wilke
We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $\Sigma$ without boundary and flows along $\Sigma$. Local-in-time well-posedness is established in … We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $\Sigma$ without boundary and flows along $\Sigma$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $\Sigma$ and we show that each equilibrium on $\Sigma$ 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.

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

2020-01-01
Jeremy LeCrone , Yuanzhen Shao , Gieri Simonett
We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform … We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are $ C^{1+\alpha} $–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are $ C^{1+\alpha} $–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.
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On the Navier-Stokes equations on surfaces

2020-01-01
Jan Pruess , Gieri Simonett , Mathias Wilke
We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $Σ$ without boundary and flows along $Σ$. Local-in-time well-posedness is established in … We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $Σ$ without boundary and flows along $Σ$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $Σ$ and we show that each equilibrium on $Σ$ 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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On quasilinear parabolic equations and continuous maximal regularity

2019-10-17
Jeremy LeCrone , Gieri Simonett
We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global … We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
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A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

2019-07-26
Giusy Mazzone , Jan Prüß , Gieri Simonett
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The Rayleigh–Taylor instability for the Verigin problem with and without phase transition

2019-05-06
Jan Prüß , Gieri Simonett , Mathias Wilke
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A new approach to the regularity of solutions for parabolic equations

2019-04-24
Joachim Escher , Jan Prüß , Gieri Simonett

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

2019-01-01
Jeremy LeCrone , Yuanzhen Shao , Gieri Simonett
We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform … We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are $C^{1+\alpha}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are $C^{1+\alpha}$-close to a sphere, and we prove that these solutions become spherical as time goes to infinity.
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On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

2019-01-01
Giusy Mazzone , Jan Prüß , Gieri Simonett
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on … We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray--Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.
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The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

2019-01-01
Jeremy LeCrone , Yuanzhen Shao , Gieri Simonett
We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform … We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are $C^{1+\alpha}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are $C^{1+\alpha}$-close to a sphere, and we prove that these solutions become spherical as time goes to infinity.
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On the microscopic bidomain problem with FitzHugh–Nagumo ionic transport

2018-12-03
Gieri Simonett , Jan Prüß
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On the motion of a fluid-filled rigid body with Navier Boundary conditions

2018-09-07
Giusy Mazzone , Jan Pruess , Gieri Simonett
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on … We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution a la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.
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On the motion of a fluid-filled rigid body with Navier Boundary conditions

2018-09-07
Giusy Mazzone , Jan Pruess , Gieri Simonett
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on … We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution \`a la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.
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On quasilinear parabolic equations and continuous maximal regularity

2018-08-02
Jeremy LeCrone , Gieri Simonett
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global … We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
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The Verigin problem with and without phase transition

2018-05-03
Jan Prüß , Gieri Simonett
Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent … Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.
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On the microscopic bidomain problem with FitzHugh-Nagumo ionic transport

2018-02-10
Jan Pruess , Gieri Simonett
The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong … The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension $d\leq 3$, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria parallel those of the classical FitzHugh-Nagumo system in ODE's. These properties of the bidomain equations are obtained combining recent results on Dirichlet-to-Neumann operators, on critical spaces for parabolic evolution equations, and qualitative theory of evolution equations.
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The Rayleigh-Taylor instability for the Verigin problem with and without phase transition

2018-01-18
Jan Pruess , Gieri Simonett , Mathias Wilke
Isothermal incompressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that … Isothermal incompressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh-Taylor instability.
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Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids

2018-01-01
Gieri Simonett , Mathias Wilke

On quasilinear parabolic equations and continuous maximal regularity

2018-01-01
Jeremy LeCrone , Gieri Simonett
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global … We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
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The Rayleigh-Taylor instability for the Verigin problem with and without phase transition

2018-01-01
Jan Pruess , Gieri Simonett , Mathias Wilke
Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that … Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh-Taylor instability.
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On the microscopic bidomain problem with FitzHugh-Nagumo ionic transport

2018-01-01
Jan Pruess , Gieri Simonett
The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong … The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension $d\leq 3$, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria parallel those of the classical FitzHugh-Nagumo system in ODE's. These properties of the bidomain equations are obtained combining recent results on Dirichlet-to-Neumann operators, on critical spaces for parabolic evolution equations, and qualitative theory of evolution equations.
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On the motion of a fluid-filled rigid body with Navier Boundary conditions

2018-01-01
Giusy Mazzone , Jan Pruess , Gieri Simonett
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on … We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution \`a la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.
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Critical spaces for quasilinear parabolic evolution equations and applications

2017-10-17
Jan Prüß , Gieri Simonett , Mathias Wilke
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On the Muskat problem

2016-10-01
Jan Prüß , Gieri Simonett
Of concern is the motion of two fluids separated by a free interface in a porous medium,where the velocities are given by Darcy's law.We consider the case with and without … Of concern is the motion of two fluids separated by a free interface in a porous medium,where the velocities are given by Darcy's law.We consider the case with and without phase transition.It is shown that the resulting models can be understood as purely geometric evolution laws,where the motion of the separating interface depends in a non-local way on the mean curvature.It turns out that the models are volume preserving and surface area reducing,the latter property giving rise to a Lyapunov function.We show well-posedness of the models, characterize all equilibria,and study the dynamic stability of the equilibria.Lastly, we show that solutions which do not develop singularities exist globallyand converge exponentially fast to an equilibrium.
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Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero

2016-09-06
Gieri Simonett , Mathias Wilke
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Well-posedness and long-time behavior for the Westervelt equation with absorbing boundary conditions of order zero

2016-03-07
Gieri Simonett , Mathias Wilke
We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed by Kaltenbacher & Shevchenko. We apply the concept of … We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed by Kaltenbacher & Shevchenko. We apply the concept of maximal regularity of type $L_p$ to prove global well-posedness for small initial data. Moreover, we show that the solutions regularize instantaneously which means that they are $C^\infty$ with respect to time $t$ as soon as $t>0$. Finally, we show that each equilibrium is stable and each solution which starts sufficiently close to an equilibrium converges at an exponential rate to a possibly different equilibrium.

On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension

2016-01-13
Jan Prüß , Yuanzhen Shao , Gieri Simonett
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, … 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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On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension

2016-01-01
Jan Prüß , Senjo Shimizu , Gieri Simonett +1 more
Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics (Prüss et al., Evol Equ Control Theory 1:171–194, 2012; Prüss and Shimizu, J Evol … Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics (Prüss et al., Evol Equ Control Theory 1:171–194, 2012; Prüss and Shimizu, J Evol Equ 12:917–941, 2012; Prüss et al., Commun Part Differ Equ 39:1236–1283, 2014; see also Prüss et al., Interfaces Free Bound 15:405–428, 2013) is extended to the case of temperature-dependent surface tension. We prove well-posedness in an L p -setting, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally, and if its limit set contains a stable equilibrium it converges to this equilibrium in the natural state manifold for the problem as time goes to infinity.
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Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids

2016-01-01
Gieri Simonett , Mathias Wilke

Linear Stability of Equilibria

2016-01-01
Jan Prüß , Gieri Simonett
In this chapter we investigate the spectral properties of the linearizations Lj of the six problems at a given equilibrium. We show that the dimension of the kernel N(Lj) equals … In this chapter we investigate the spectral properties of the linearizations Lj of the six problems at a given equilibrium. We show that the dimension of the kernel N(Lj) equals the dimension of the tangent space of the manifold of equilibria ε, the eigenvalue 0 is semi-simple for Lj, and the intersection of the spectrum of Lj with the imaginary axis is {0}.

Generalized Stokes Problems

2016-01-01
Jan Prüß , Gieri Simonett
This chapter is devoted to maximal L p -regularity of one-phase linear generalized Stokes problems on domains $$ {\Omega} \subset {\mathbb{R}}^{n} $$ which are either $$ {\mathbb{R}}^{n} $$ , $$ … This chapter is devoted to maximal L p -regularity of one-phase linear generalized Stokes problems on domains $$ {\Omega} \subset {\mathbb{R}}^{n} $$ which are either $$ {\mathbb{R}}^{n} $$ , $$ {\mathbb{R}}_{ + }^{n} $$ , or domains with compact boundary $$ \partial {\Omega} $$ of class C 3, i.e., interior or exterior domains. Here we only consider the physically natural boundary conditions no-slip, pure slip, outflow, and free. As in Chap. 6, our approach is based on vector-valued Fourier multiplier theory, perturbation, and localization. It turns out that due to the divergence condition (and the pressure), the analysis for the half-space as well as the localization procedure are much more involved than in the previous chapter. Nevertheless, besides some extra compatibility condition which comes from the divergence condition, the main results will parallel those in Chap. 6.

Local Well-Posedness and Regularity

2016-01-01
Jan Prüß , Gieri Simonett
In this chapter we study local well-posedness and regularity of the solutions of Problems (P1)~(P6). Here we employ without further comments the notations introduced in Chapters 1 and 2, in … In this chapter we study local well-posedness and regularity of the solutions of Problems (P1)~(P6). Here we employ without further comments the notations introduced in Chapters 1 and 2, in particular those in connection with Conditions (H1)~(H6) from Chapter 1, the Hanzawa transform, and the transformed problems on the fixed domain Ω\Σ in Section 1.3. In the first section of this chapter we reformulate Problems (P1)~(P6) in a way which is amenable to a joint analysis, which will be based on maximal Lp-regularity as well as on the contraction mapping principle in Section 9.2, and on the implicit function theorem for dependence on the data in Section 9.3. For regularity we employ in Section 9.4 the so-called parameter trick, which is also based on maximal Lp-regularity and the implicit function theorem. This way we can show that the solutions obtained in Section 9.2 are in fact classical solutions. The proofs for the technical results on the nonlinearities are postponed to the last section of this chapter.

Vector-Valued Harmonic Analysis

2016-01-01
Jan Prüß , Gieri Simonett
In this chapter, operator-valued Fourier multiplier results for vector-valued L p -spaces are derived and discussed. These form the basic tools for the proof of various results on maximal L … In this chapter, operator-valued Fourier multiplier results for vector-valued L p -spaces are derived and discussed. These form the basic tools for the proof of various results on maximal L p -regularity which are needed for the nonlinear problems.

Operator Theory and Semigroups

2016-01-01
Jan Prüß , Gieri Simonett
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 … 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.

Moving Interfaces and Quasilinear Parabolic Evolution Equations

2016-01-01
Jan Prüß , Gieri Simonett
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili

Problems and Strategies

2016-01-01
Jan Prüß , Gieri Simonett
The purpose of this introductory chapter is to explain the problems to be considered in the main part of this book in some detail. We derive their physical origin from … The purpose of this introductory chapter is to explain the problems to be considered in the main part of this book in some detail. We derive their physical origin from first principles, discuss some of the main structural properties of the models, and describe the strategies of our analytical approach. All the notions and properties relating to differential geometry of hypersurfaces will be introduced and explained in Chapter 2.

Elliptic and Parabolic Problems

2016-01-01
Jan Prüß , Gieri Simonett
In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in … In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in Chapter 1.

Quasilinear Parabolic Evolution Equations

2016-01-01
Jan Prüß , Gieri Simonett
In this chapter we consider abstract quasilinear parabolic problems of the form. In this chapter we consider abstract quasilinear parabolic problems of the form.
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Common Coauthors

Coauthor Papers Together
Jan Prüß 39
Jan Pruess 27
Mathias Wilke 25
Joachim Escher 15
Yuanzhen Shao 13
Jeremy LeCrone 11
Rico Zacher 7
Uwe Mayer 6
Giusy Mazzone 5
Senjo Shimizu 5
Hengrong Du 4
Jürgen Saal 3
Yoshihiro Shibata 3
Juergen Saal 2
Herbert Amann 2
Philippe Clément 2
Mary Ann Horn 1
Xuan Thinh Duong 1
Yoshikazu Giga 1
Stig–Olof Londen 1
Matthias Hieber 1
Robert Denk 1
Edriss S. Titi 1
Graham Webb 1
Jan Pr�ss 1
Hideo Kozono 1
Jan Pr�� 1
Dieter Bothe 1
Marcelo M. Disconzi 1

Commonly Cited References

Linear and Quasilinear Parabolic Problems

1995-01-01
Herbert Amann
In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted … In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted to concrete realizations of linear parabolic evolution equations by general parabolic systems.There we discuss the various function spaces that are needed and useful, and the generation of analytic semigroups by general elliptic boundary value problems.The last volume contains the abstract nonlinear theory as well as various applications to concrete systems, illustrating the scope and the flexibility of the general results.Of course, each one of the three volumes contains much material of independent interest related to our main subject.In writing this book I had help from many friends, collegues, and students.It is a pleasure to thank all of them, named or unnamed.I am particularly indebted to P. Quittner and G. Simonett, who critically and very carefully read, not only the whole manuscript of this first volume but also many earlier versions that were produced over the years and will never be published, and pointed out numerous mistakes and improvements.Large parts of the first volume, and of earlier versions as well, were also read and commented on by D. Daners, J. Escher, and P
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Nonlinear analytic semiflows

1990-01-01
Sigurd Angenent
Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), … Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t )) x ( t ) + f ( x ( t ))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

A Center Manifold Analysis for the Mullins–Sekerka Model

1998-03-01
Joachim Escher , Gieri Simonett

On convergence of solutions to equilibria for quasilinear parabolic problems

2008-12-10
Jan Prüß , Gieri Simonett , Rico Zacher
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Moving Interfaces and Quasilinear Parabolic Evolution Equations

2016-01-01
Jan Prüß , Gieri Simonett
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili

Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension

2012-10-18
Jan Prüß , Gieri Simonett , Rico Zacher
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Analytic solutions for a Stefan problem with Gibbs-Thomson correction

2003-01-16
Joachim Escher , Jan Prüß , Gieri Simonett
We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the … We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lpregularity for the linearized problem, which is fully developed in this paper as well.

Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space

1988-06-01
Giuseppe Da Prato , Alessandra Lunardi

The Surface Diffusion Flow for Immersed Hypersurfaces

1998-11-01
Joachim Escher , Uwe Mayer , Gieri Simonett
We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we … We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.
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Analyticity of the interface in a free boundary problem

1996-05-01
Joachim Escher , Gieri Simonett

Existence of analytic solutions for the classical Stefan problem

2007-03-09
Jan Prüß , Jürgen Saal , Gieri Simonett
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Analytic Semigroups and Optimal Regularity in Parabolic Problems

1995-01-01
Alessandra Lunardi

Optimal L p -L q -estimates for parabolic boundary value problems with inhomogeneous data

2007-03-05
Robert Denk , Matthias Hieber , Jan Prüß

Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension

2012-10-29
Matthias Köhne , Jan Prüß , Mathias Wilke
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Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

2001-03-01
Philippe Clément , Gieri Simonett
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Two-phase free boundary problem for viscous incompressible thermo-capillary convection

1995-01-01
Naoto Tanaka
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Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

1993-01-01
Herbert Amann
It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. … It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

On quasilinear parabolic evolution equations in weighted L p -spaces II

2014-03-10
Jeremy LeCrone , Jan Prüß , Mathias Wilke
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Maximal regularity for evolution equations in weighted L p -spaces

2004-05-01
Jan Pr�ss , Gieri Simonett

Critical spaces for quasilinear parabolic evolution equations and applications

2017-10-17
Jan Prüß , Gieri Simonett , Mathias Wilke
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Classical solutions for Hele-Shaw models with surface tension

1997-01-01
Joachim Escher , Gieri Simonett
It is shown that surface tension effects on the free boundary are regularizing for Hele-Shaw models. This implies, in particular, existence and uniqueness of classical solutions for a large class … It is shown that surface tension effects on the free boundary are regularizing for Hele-Shaw models. This implies, in particular, existence and uniqueness of classical solutions for a large class of initial data. As a consequence, we give a rigorous proof of the fact that homogeneous Hele-Shaw flows with positive surface tension are volume preserving and area shrinking.
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Involutive structures on the tangent bundle of symmetric spaces

2001-02-01
N. J. Kalton , Lutz Weis

ℛ-boundedness, Fourier multipliers and problems of elliptic and parabolic type

2003-01-01
Robert Denk , Matthias Hieber , Jan Prüß

On Thermodynamically Consistent Stefan Problems with Variable Surface Energy

2015-10-05
Jan Prüß , Gieri Simonett , Mathias Wilke
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Small-time existence for the Navier-Stokes equations with a free surface

1987-07-01
Geneviève Allain

Large-time existence of surface waves in incompressible viscous fluids with or without surface tension

1995-01-01
Atusi Tani , Naoto Tanaka

Addendum to the paper “On quasilinear parabolic evolution equations in weighted $$L_p$$ L p -spaces II”

2017-02-03
Jan Prüß , Mathias Wilke

Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights

2011-11-13
Martin Meyries , Roland Schnaubelt
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Large-time regularity of viscous surface waves

1984-12-01
J. Thomas Beale

Parabolic Equations for Curves on Surfaces Part I. Curves with p-Integrable Curvature

1990-11-01
Sigurd Angenent
This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening … This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if the curve becomes singular in finite time. A detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part.

The $H^{\infty}-$ calculus and sums of closed operators

2001-10-01
N. J. Kalton , Lutz Weis
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The heat equation shrinking convex plane curves

1986-01-01
Michael E. Gage , Richard S. Hamilton
Let M and M' be Riemannian manifolds and F: M -» M' a smooth map Let M and M' be Riemannian manifolds and F: M -» M' a smooth map
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Small-time existence for the three-dimensional navier-stokes equations for an incompressible fluid with a free surface

1996-01-01
Atusi Tani

On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations

2010-01-01
Jan Pruess , Gieri Simonett
The two-phase free boundary problem with surface tension and downforce gravity for the Navier-Stokes system is considered in a situation where the initial interface is close to equilibrium.The boundary symbol … The two-phase free boundary problem with surface tension and downforce gravity for the Navier-Stokes system is considered in a situation where the initial interface is close to equilibrium.The boundary symbol of this problem admits zeros in the unstable halfplane in case the heavy fluid is on top of the light one, which leads to the well-known Rayleigh-Taylor instability.Instability is proved rigorously in an Lp-setting by means of an abstract instability result due to Henry [12].
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On quasilinear parabolic evolution equations in weighted L p -spaces

2010-02-18
Matthias Köhne , Jan Prüß , Mathias Wilke
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On Critical Spaces for the Navier–Stokes Equations

2017-10-05
Jan Prüß , Mathias Wilke
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Classical Solutions of Multidimensional Hele--Shaw Models

1997-09-01
Joachim Escher , Gieri Simonett
Previous article Next article Classical Solutions of Multidimensional Hele--Shaw ModelsJoachim Escher and Gieri SimonettJoachim Escher and Gieri Simonetthttps://doi.org/10.1137/S0036141095291919PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractExistence and uniqueness of classical solutions for … Previous article Next article Classical Solutions of Multidimensional Hele--Shaw ModelsJoachim Escher and Gieri SimonettJoachim Escher and Gieri Simonetthttps://doi.org/10.1137/S0036141095291919PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractExistence and uniqueness of classical solutions for the multidimensional expanding Hele--Shaw problem are proved.[1] S. Agmon, , A. Douglis and , and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), pp. 623–727. cpa CPMAMV 0010-3640 Commun. Pure Appl. Math. CrossrefISIGoogle Scholar[2] Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, Vol. 89, Birkhäuser Boston Inc., 1995xxxvi+335, Abstract linear theory 96g:34088 CrossrefGoogle Scholar[3] Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, Vol. 89, Birkhäuser Boston Inc., 1995xxxvi+335, Abstract linear theory 96g:34088 CrossrefGoogle Scholar[4] Sigurd Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91–107 91c:34085 CrossrefISIGoogle Scholar[5] Google Scholar[6] Giuseppe Da Prato and , Pierre Grisvard, Equations d'évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl. (4), 120 (1979), 329–396 81d:34052 CrossrefGoogle Scholar[7] Emmanuele DiBenedetto and , Avner Friedman, The ill‐posed Hele‐Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. 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(4), 21 (1994), 235–266 95g:35045 Google Scholar[12] Joachim Escher and , Gieri Simonett, Maximal regularity for a free boundary problem, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 463–510 96k:35142 CrossrefGoogle Scholar[13] Joachim Escher and , Gieri Simonett, Analyticity of the interface in a free boundary problem, Math. Ann., 305 (1996), 439–459 98d:35242 CrossrefISIGoogle Scholar[14] A. Fasano and , M. Primicerio, New results on some classical parabolic free‐boundary problems, Quart. Appl. Math., 38 (1980/81), 439–460 82g:35061 CrossrefISIGoogle Scholar[15] A. Fasano and , M. Primicerio, Blow‐up and regularization for the Hele‐Shaw problem, IMA Vol. Math. Appl., Vol. 53, Springer, New York, 1993, 73–85 97b:76121 Google Scholar[16] Avner Friedman, Time dependent free boundary problems, SIAM Rev., 21 (1979), 213–221 81g:76097 LinkISIGoogle Scholar[17] Google Scholar[18] Ei‐ichi Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J. 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Math., Vol. 155, Dekker, New York, 1994, 523–536 94m:34163 Google Scholar[24] Google ScholarKeywordsclassical solutionsHele--Shaw modelmoving boundary problemmaximal regularity Previous article Next article FiguresRelatedReferencesCited byDetails Comoving mesh method for certain classes of moving boundary problems2 August 2022 | Japan Journal of Industrial and Applied Mathematics, Vol. 12 Cross Ref Steady-state solutions for the Muskat problem8 March 2022 | Collectanea Mathematica, Vol. 16 Cross Ref A tumor growth model with autophagy: The reaction-(cross-)diffusion system and its free boundary limitDiscrete and Continuous Dynamical Systems - B, Vol. 0, No. 0 Cross Ref Global existence and stability for the modified Mullins–Sekerka and surface diffusion flowMathematics in Engineering, Vol. 4, No. 6 Cross Ref Hyperbolic Solutions to Bernoulli's Free Boundary Problem12 February 2021 | Archive for Rational Mechanics and Analysis, Vol. 240, No. 2 Cross Ref Introduction23 March 2021 Cross Ref Global Simply Connected Weak Solutions23 March 2021 Cross Ref Asymptotic analysis of a contact Hele-Shaw problem in a thin domain13 August 2020 | Nonlinear Differential Equations and Applications NoDEA, Vol. 27, No. 5 Cross Ref Well-Posedness and Stability Results for Some Periodic Muskat Problems6 June 2020 | Journal of Mathematical Fluid Mechanics, Vol. 22, No. 3 Cross Ref On the Banach Manifold of Simple Domains in the Euclidean Space and Applications to Free Boundary Problems26 June 2019 | Acta Applicandae Mathematicae, Vol. 167, No. 1 Cross Ref A Cahn--Hilliard Model for Cell MotilityAlessandro Cucchi, Antoine Mellet, and Nicolas Meunier17 August 2020 | SIAM Journal on Mathematical Analysis, Vol. 52, No. 4AbstractPDF (662 KB)Motion of sets by curvature and derivative of capacity potentialJournal of Differential Equations, Vol. 267, No. 1 Cross Ref The String Equation for Some Rational Functions31 January 2019 Cross Ref The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results1 January 2019 | Analysis & PDE, Vol. 12, No. 2 Cross Ref Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesisJournal of Mathematical Analysis and Applications, Vol. 468, No. 1 Cross Ref Asymptotic behavior of solutions of a free-boundary tumor model with angiogenesisNonlinear Analysis: Real World Applications, Vol. 44 Cross Ref Viscous displacement in porous media: the Muskat problem in 2D26 June 2018 | Transactions of the American Mathematical Society, Vol. 370, No. 10 Cross Ref Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesisJournal of Differential Equations, Vol. 265, No. 2 Cross Ref Some geometrical properties of free boundaries in the Hele-Shaw flowsApplied Mathematics and Computation, Vol. 323 Cross Ref On a Model for a Sliding Droplet: Well-posedness and Stability of Translating Circular SolutionsPatrick Guidotti and Christoph Walker20 March 2018 | SIAM Journal on Mathematical Analysis, Vol. 50, No. 2AbstractPDF (473 KB)Classical Solvability of the Radial Viscous Fingering Problem in a Hele–Shaw Cell with Surface Tension19 December 2017 | Journal of Mathematical Sciences, Vol. 228, No. 4 Cross Ref Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relationJournal of Differential Equations, Vol. 262, No. 10 Cross Ref Rate of Convergence of General Phase Field Equations in Strongly Heterogeneous Media Toward Their Homogenized LimitM. 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Application of the Hele--Shaw Problem for Power-Law FluidsPierre Cardaliaguet and Elisabeth Rouy1 August 2006 | SIAM Journal on Mathematical Analysis, Vol. 38, No. 1AbstractPDF (243 KB)Quasistationary Approximation for the Stefan ProblemJournal of Mathematical Sciences, Vol. 132, No. 4 Cross Ref Free boundary problems with surface tension conditionsNonlinear Analysis: Theory, Methods & Applications, Vol. 63, No. 5-7 Cross Ref Approximate Solutions of the Cahn-Hilliard Equation via Corrections to the Mullins-Sekerka Motion21 April 2005 | Archive for Rational Mechanics and Analysis, Vol. 178, No. 1 Cross Ref The Hele-Shaw problem with surface tension in a half-plane: A model problemJournal of Differential Equations, Vol. 216, No. 2 Cross Ref The Hele–Shaw problem with surface tension in a half-planeJournal of Differential Equations, Vol. 216, No. 2 Cross Ref A Variational Approach to the Hele–Shaw Flow with InjectionCommunications in Partial Differential Equations, Vol. 30, No. 9 Cross Ref Global classical solution of quasi-stationary Stefan free boundary problemApplied Mathematics and Computation, Vol. 160, No. 3 Cross Ref All Time Smooth Solutions of the One-Phase Stefan Problem and the Hele-Shaw FlowCommunications in Partial Differential Equations, Vol. 29, No. 1-2 Cross Ref Smooth solutions to a class of free boundary parabolic problems6 October 2003 | Transactions of the American Mathematical Society, Vol. 356, No. 3 Cross Ref Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces Cross Ref Global classical solution of free boundary problem for a coupled systemApplied Mathematics-A Journal of Chinese Universities, Vol. 18, No. 1 Cross Ref A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor GrowthCommunications in Partial Differential Equations, Vol. 28, No. 3-4 Cross Ref A Free Boundary Problem for an Elliptic-Hyperbolic System: An Application to Tumor GrowthXinfu Chen and Avner Friedman1 August 2006 | SIAM Journal on Mathematical Analysis, Vol. 35, No. 4AbstractPDF (159 KB)The Intermediate Surface Diffusion Flow on Spheres1 January 2003 | Journal of Nonlinear Mathematical Physics, Vol. 10, No. Supplement 1 Cross Ref Chapter 5 Nonlinear parabolic equations and systems Cross Ref Об эволюции слабого возмущения окружности в задаче о течении Хил - ШоуУспехи математических наук, Vol. 57, No. 6 Cross Ref Moving Surfaces and Abstract Parabolic Evolution Equations Cross Ref Local and global existence results for anisotropic Hele–Shaw flows14 November 2011 | Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 129, No. 2 Cross Ref A Center Manifold Analysis for the Mullins–Sekerka ModelJournal of Differential Equations, Vol. 143, No. 2 Cross Ref Volume 28, Issue 5| 1997SIAM Journal on Mathematical Analysis History Published online:01 August 2006 InformationCopyright © 1997 Society for Industrial and Applied MathematicsKeywordsclassical solutionsHele--Shaw modelmoving boundary problemmaximal regularityMSC codes35R3535K5535S3076D99PDF Download Article & Publication DataArticle DOI:10.1137/S0036141095291919Article page range:pp. 1028-1047ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Elliptic Partial Differential Equations of Second Order

2001-01-01
David Gilbarg , Neil S. Trudinger
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On the two-phase Navier–Stokes equations with surface tension

2010-10-01
Jan Prüß , Gieri Simonett
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of … The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.
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Geometric Theory of Semilinear Parabolic Equations

1981-01-01
Daniel Henry
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On Hele-Shaw models with surface tension

1996-01-01
Joachim Escher , Gieri Simonett
It is shown that surface tension effects on the free boundary have a regularizing effect for Hele-Shaw models, which implies existence and uniqueness of classical solutions for general initial domains. It is shown that surface tension effects on the free boundary have a regularizing effect for Hele-Shaw models, which implies existence and uniqueness of classical solutions for general initial domains.
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Function spaces on singular manifolds

2012-05-04
Herbert Amann
Abstract It is shown that most of the well‐known basic results for Sobolev‐Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^n$\end{document} , continue to … Abstract It is shown that most of the well‐known basic results for Sobolev‐Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^n$\end{document} , continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the singularities, are introduced. These results are of importance for the study of partial differential equations on piece‐wise smooth domains.
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Analytic Solutions for the Two-phase Navier-Stokes Equations with Surface Tension and Gravity

2011-01-01
Jan Prüß , Gieri Simonett
We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing … We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously.
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Existence Results for the Quasistationary Motion of a Free Capillary Liquid Drop

1997-06-30
Matthias Günther , Georg Prokert
We consider instationary creeping flow of a viscous liquid drop with free boundary driven by surface tension. This yields a nonlocal surface motion law involving the solution of the Stokes … We consider instationary creeping flow of a viscous liquid drop with free boundary driven by surface tension. This yields a nonlocal surface motion law involving the solution of the Stokes equations with Neumann boundary conditions given by the curvature of the boundary. The surface motion law is locally reformulated as a fully nonlinear parabolic (pseudodifferential) equation on a smooth manifold. Using analytic expansions, invariance properties, and a priori estimates we give, under suitable presumptions, a short-time existence and uniqueness proof for the solution of this equation in Sobolev spaces of sufficiently high order. Moreover, it is shown that if the initial shape of the drop is near the ball, then the evolution problem has a solution for all positive times which exponentially decays to the ball.
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On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities

2012-03-01
Jan Prüß , Yoshihiro Shibata , Senjo Shimizu +1 more
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved … The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique ofmaximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.
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Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle

1998-01-01
Matthias Hieber , Jan Prüß
Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class … Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.
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On the closedness of the sum of two closed operators

1987-06-01
Giovanni Dore , Alberto Venni

Weak and variational methods for moving boundary problems

1982-01-01
Charles M. Elliott , J. R. Ockendon

The Dirichlet-Neumann operator on continuous functions

1994-01-01
Joachim Escher

Flow by mean curvature of convex surfaces into spheres

1984-01-01
Gerhard Huisken
The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2], [5], … The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2], [5], [9]. A reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal. In this paper we take a more classical point of view: Consider a compact, uniformly convex w-dimensional surface M = Mo without boundary, which is smoothly imbedded in R. Let Mo be represented locally by a diffeomorphism
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