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Mathias Wilke

Publications and source records attributed to Mathias Wilke.

32 records · Page 2Linked to original sources
arXiv2015

Optimal regularity and long-time behavior of solutions for the Westervelt equation

We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal $L_p$-regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal $L_p$-regularity for abstract quasilinear parabolic equations.

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arXiv2014

On incompressible two-phase flows with phase transitions and variable surface tension

Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics [10,11,12,16] is extended to the case of temperature-dependent surface tension. We prove well-posedness in an Lp-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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arXiv2014

Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics

The mass-based Maxwell-Stefan approach to one-phase multicomponent reactive mixtures is mathematically analyzed. It is shown that the resulting quasilinear, strongly coupled reaction-diffusion system is locally well-posed in an $L_p$-setting and generates a local semiflow on its natural state space. Solutions regularize instantly and become strictly positive if their initial components are all nonnegative and nontrivial. For a class of reversible mass-action kinetics, the positive equilibria are identified: these are precisely the constant chemical equilibria of the system, which may form a manifold. Here the total free energy of the system is employed which serves as a Lyapunov function for the system. By the generalized principle of linearized stability, positive equilibria are proved to be normally stable.

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arXiv2013

On quasilinear parabolic evolution equations in weighted Lp-spaces II

Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.

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arXiv2013

Qualitative behaviour of incompressible two-phase flows with phase transitions: The case of non-equal densities

Our study of a basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics in the case of constant but non-equal densities of the phases, begun by the first two authors is continued. We extend our well-posedness result to general geometries, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally. If its limit set contains a stable equilibrium it converges to this equilibrium as time goes to infinity, in the natural state manifold for the problem in an Lp-setting.

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arXiv2012

Well-Posedness and Qualitative Behaviour of Solutions for a Two-Phase Navier-Stokes-Mullins-Sekerka System

We consider a two-phase problem for two incompressible, viscous and immiscible fluids which are separated by a sharp interface. The problem arises as a sharp interface limit of a diffuse interface model. We present results on local existence of strong solutions and on the long-time behavior of solutions which start close to an equilibrium. To be precise, we show that as time tends to infinity, the velocity field converges to zero and the interface converges to a sphere at an exponential rate.

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arXiv2012

Global well-posedness and exponential stability for Kuznetsov's equation in L_p-spaces

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal L_p-regularity for parabolic equations and the implicit function theorem.

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arXiv2012

$L_p$-theory for a Cahn-Hilliard-Gurtin system

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.

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arXiv2010

Fractional White Noise Perturbations of Parabolic Volterra Equations

Aim of this work is to extend the results of Clément, Da Prato & Prüss on the fractional white noise perturbation with Hurst parameter 0<H<1. We will obtain similar results and it will turn out that the regularity of the solution u(t) of the stochastic Volterra equation increases with Hurst parameter H.

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arXiv2010

Qualitative Behaviour of Solutions for the Two-Phase Navier-Stokes Equations with Surface Tension

The two-phase free boundary value problem for the isothermal Navier-Stokes system is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an Lp-setting, and that it generates a local semiflow on the induced phase manifold. If the phases are connected, the set of equilibria of the system forms a (n+1)-dimensional manifold, each equilibrium is stable, and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity. The latter is proved by means of the energy functional combined with the generalized principle of linearized stability.

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arXiv2010

On conserved Penrose-Fife type models

In this paper we investigate quasilinear parabolic systems of conserved Penrose-Fife type. We show maximal $L_p$ - regularity for this problem with inhomogeneous boundary data. Furthermore we prove global existence of a solution, provided that the absolute temperature is bounded from below and above. Moreover, we apply the Lojasiewicz-Simon inequality to establish the convergence of solutions to a steady state as time tends to infinity.

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arXiv2009

On quasilinear parabolic evolution equations in weighted $L_p$-spaces

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted $L_p$-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the $\om$-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) $L_p$-spaces.

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