arXiv subjects

Mathias Wilke

Publications and source records attributed to Mathias Wilke.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of a Cahn-Hilliard-Canham-Helfrich system for the evolution of a two-phase membrane

The coupling of the evolution of a surface with evolution equations defined on that surface is of relevance in many applications and has been in the focus of interest in the analysis of parabolic PDEs in recent years. In applications the evolution of two-phase vesicles and biomembranes is governed by flows decreasing an energy which involves Canham-Helfrich-type curvature energies coupled to a Ginzburg-Landau energy. We derive a new Cahn-Hilliard-Canham-Helfrich system for the evolution of two-phase membranes. The resulting system is highly non-linear and we use the theory of quasi-linear parabolic evolution equations in weighted $L_p$-spaces to show the existence of a strong local-in-time solution and hence demonstrate that the derived system is well-posed.

math.AP↗

arXiv2025

Nematic liquid crystals: Ericksen-Leslie theory with general stress tensors

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the $L_p$-setting. More precisely, the existence and uniqueness of a local, strong $L_p$-solution to the general system is proved and it is shown that the director $d$ satisfies $|d|_2\equiv 1$ provided this holds for its initial data $d_0$. In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi's relation.

math.AP↗

arXiv2024

Coriolis-driven fluid motion on spherical surfaces

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.

math.AP↗

arXiv2023

The Navier-Stokes equations on manifolds with boundary

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.

math.AP↗

arXiv2023

Linear and quasilinear evolution equations in the context of weighted $L_p$-spaces

In 2004, the article "Maximal regularity for evolution equations in weighted $L_p$-spaces" by J. Pr\"{u}ss and G. Simonett has been published in Archiv der Mathematik. We provide a survey of the main results of that article and outline some applications to semilinear and quasilinear parabolic evolution equations which illustrate their power.

math.AP↗

arXiv2022

$L_p$-$L_q$-theory for a quasilinear non-isothermal Westervelt equation

We investigate a quasilinear system consisting of the Westervelt equation from nonlinear acoustics and Pennes bioheat equation, subject to Dirichlet or Neumann boundary conditions. The concept of maximal regularity of type $L_p$-$L_q$ is applied to prove local and global well-posedness. Moreover, we show by a parameter trick that the solutions regularize instantaneously. Finally, we compute the equilibria of the system and investigate the long-time behaviour of solutions starting close to equilibria.

math.AP↗

arXiv2021

$H^\infty$-calculus for the surface Stokes operator and applications

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.

math.AP↗

arXiv2020

On the Navier-Stokes equations on surfaces

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.

math.AP↗

arXiv2020

Well-Posedness and Qualitative Behaviour of a Two-phase Navier-Stokes/Mullins-Sekerka system with boundary contact

We consider a coupled two-phase Navier-Stokes/Mullins-Sekerka system describing the motion of two immiscible, incompressible fluids inside a bounded container. The moving interface separating the liquids meets the boundary of the container at a constant ninety degree angle. This common interface is unknown and has to be determined as a part of the problem. We show well-posedness and investigate the long-time behaviour of solutions starting close to certain equilibria. We prove that for equal densities these solutions exist globally in time, are stable, and converge to an equilibrium solution at an exponential rate.

math.AP↗

arXiv2019

Well-Posedness and qualitative behaviour of the Mullins-Sekerka problem with ninety-degree angle boundary contact

We show local well-posedness for the Mullins-Sekerka system with ninety degree angle boundary contact. We will describe the motion of the moving interface by a height function over a fixed reference surface. Using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well-posedness of the full nonlinear problem via the contraction mapping principle. Here one difficulty lies in choosing the right space for the Neumann trace of the height function and showing maximal $L_p-L_q$-regularity for the linear problem. In the second part we show that solutions starting close to certain equilibria exist globally in time, are stable, and converge to an equilibrium solution at an exponential rate.

math.AP↗

arXiv2018

The Rayleigh-Taylor instability for the Verigin problem with and without phase transition

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.

math.AP↗

arXiv2017

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.

math.AP↗

arXiv2017

On critical spaces for the Navier-Stokes equations

The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the $L_p$-$L_q$ setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $\mathcal{H}^\infty$-calculus with $\mathcal{H}^\infty$-angle 0, and the real and complex interpolation spaces of these operators are identified.

math.AP↗

arXiv2017

Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in cylindrical domains

This article is concerned with the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical domain, which are separated by a sharp interface. In case that the heavy fluid is situated on top of the light fluid, one expects that the fluid on top sags down into the lower phase. This effect is known as the Rayleigh-Taylor-Instability. We present a rather complete analysis of the corresponding free boundary problem which involves a contact angle. Our main result implies the existence of a critical surface tension with the following property. In case that the surface tension of the interface separating the two fluids is smaller than the critical surface tension, one has Rayleigh-Taylor-Instability. On the contrary, if the interface has a greater surface tension than the critical value, the instability effect does not occur and one has exponential stability of a flat interface. The last part of this article is concerned with the bifurcation of nontrivial equilibria in multiple eigenvalues. The invariance of the corresponding bifurcation equation with respect to rotations and reflections yields the existence of bifurcating subcritical equilibria. Finally it is proven that the bifurcating equilibria are unstable.

math.AP↗

arXiv2016

Addendum to the Paper "On Quasilinear Parabolic Evolution Equations in Weighted $L_p$-Spaces II"

This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper [6]. The improvement is explained in Section 1 and proved in Section 2. The importance of the extension is demonstrated in Section 3 with an application to the Navier-Stokes system in critical $L_q$-spaces.

math.AP↗

arXiv2016

Stability of equilibrium shapes in some free boundary problems involving fluids

In this paper the motion of two-phase, incompressible, viscous fluids with surface tension is investigated. Three cases are considered: (1) the case of heat-conducting fluids, (2) the case of isothermal fluids, and (3) the case of Stokes flows. In all three situations, the equilibrium states in the absence of outer forces are characterized and their stability properties are analyzed. It is shown that the equilibrium states correspond to the critical points of a natural physical or geometric functional (entropy, available energy, surface area) constrained by the pertinent conserved quantities (total energy, phase volumes). Moreover, it is shown that solutions which do not develop singularities exist globally and converge to an equilibrium state.

math.AP↗

arXiv2016

Well-posedness and long-time behavior for the Westervelt equation with absorbing boundary conditions of order zero

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.

math.AP↗

arXiv2015

On thermodynamically consistent Stefan problems with variable surface energy

A thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities, it is proved that it exists globally in time and converges towards an equilibrium of the problem. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable if surface heat capacity is small; however, if kinetic undercooling is absent, they are stable if surface heat capacity is sufficiently large.

math.AP↗