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Matthias Köhne

Publications and source records attributed to Matthias Köhne.

13 recordsLinked to original sources
arXiv2026

Co-moving volumes and the Reynolds transport theorem for two-phase flows

We consider the local kinematics at fluid interfaces in sharp-interface two-phase flows with phase change and interfacial slip. In this setting the governing velocity field is discontinuous at the phase boundary, with possible jumps of both normal and tangential components, and the associated kinematic initial value problems may fail to be uniquely solvable. A physically consistent example exhibits this non-uniqueness and, in addition, rapid loss of boundary regularity: smooth initial control volumes can instantaneously develop edges, while their phasewise parts may form cusps. Motivated by these phenomena, we use concepts from differential inclusions to define co-moving volumes as attainable sets. For such attainable-set co-moving volumes in three-dimensional two-phase flows, we prove the Reynolds transport theorem first in boundary-integral form and then in divergence form. A key ingredient is a boundary-integral form of the single-phase Reynolds transport theorem for families of compact regular closed sets whose space-time tubes are Lipschitz domains. We also provide a short proof of this single-phase result by applying the divergence theorem in space-time; this proof does not require the motion to be generated by an ambient velocity field.

math.AP↗
arXiv2024

Optimal Regularity for the Stokes Equations on a 2D Wedge Domain Subject to Navier Boundary Conditions

We consider the Stokes equations subject to Navier boundary conditions on a two-dimensional wedge domain with opening angle $θ_0 \in (0,\,π)$. We prove existence and uniqueness of solutions with optimal regularity in an $L^p$-setting. The results are based on optimal regularity results for the Stokes equations subject to perfect slip boundary conditions on a two-dimensional wedge domain that have been obtained by the authors in [15]. Based on a detailed study of the corresponding trace operator on anisotropic Sobolev-Slobodeckij type function spaces on a two-dimensional wedge domain we are able to generalize the results proved in [15] to the case of inhomogeneous boundary conditions. Existence and uniqueness of solutions to the Stokes equations subject to (inhomogeneous) Navier boundary conditions are then obtained using a perturbation argument.

math.AP↗
arXiv2023

Stability Analysis for a Class of Heterogeneous Catalysis Models

We prove stability for a class of heterogeneous catalysis models in the $L_p$-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

math.AP↗
arXiv2021

Multiplication in Vector-Valued Anisotropic Function Spaces and Applications to Non-Linear Partial Differential Equations

We study multiplication as well as Nemytskij operators in anisotropic vector-valued Besov spaces $B^{s, ω}_p$, Bessel potential spaces $H^{s, ω}_p$, and Sobolev-Slobodeckij spaces $W^{s, ω}_p$. Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar-valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector-valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs.

math.FA↗
arXiv2019

Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the $L^p$-setting, i.p. it is $W^{2,p}$ in space. This improves known results in the literature to a large extend. For instance, in [21, Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting $W^{2,p}$ regularity, however, is restricted to the interval $1 0$, depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range $1<p<\infty$. We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every $p\in(1,\infty)$ on the entire $L^p$-space. On the other hand, for the Stokes operator in the space of solenoidal fields $L^p_σ$ we obtain optimal Sobolev regularity for the full range $1<p<\infty$ and for all opening angles less that $π$. Roughly speaking, this relies on the fact that an existing $ bad$ part of $L^p$ for the Laplacian is complementary to the space of solenoidal vector fields.

math.AP↗
arXiv2019

A Kinematic Evolution Equation for the Dynamic Contact Angle and some Consequences

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid-fluid interface and mass transfer across the fluid-fluid interface.

physics.flu-dyn↗
arXiv2016

Global Strong Solutions for a Class of Heterogeneous Catalysis Models

We consider a mathematical model for heterogeneous catalysis in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. The system under consideration consists of a diffusion-advection system inside the bulk phase and a reaction-diffusion-sorption system modeling the processes on the catalytic wall and the exchange between bulk and surface. We assume Fickian diffusion with constant coefficients, sorption kinetics with linear growth bound and a network of chemical reactions which possesses a certain triangular structure. Our main result gives sufficient conditions for the existence of a unique global strong $L^2$-solution to this model, thereby extending by now classical results on reaction-diffusion systems to the more complicated case of heterogeneous catalysis.

math.AP↗
arXiv2016

Strong Well-Posedness for a Class of Dynamic Outflow Boundary Conditions for Incompressible Newtonian Flows

Based on energy considerations, we derive a class of dynamic outflow boundary conditions for the incompressible Navier-Stokes equations, containing the well-known convective boundary condition but incorporating also the stress at the outlet. As a key building block for the analysis of such problems, we consider the Stokes equations with such dynamic outflow boundary conditions in a halfspace and prove the existence of a strong solution in the appropriate Sobolev-Slobodeckij-setting with $L_p$ (in time and space) as the base space for the momentum balance. For non-vanishing stress contribution in the boundary condition, the problem is actually shown to have $L_p$-maximal regularity under the natural compatibility conditions. Aiming at an existence theory for problems in weakly singular domains, where different boundary conditions apply on different parts of the boundary such that these surfaces meet orthogonally, we also consider the prototype domain of a wedge with opening angle $\fracπ{2}$ and different combinations of boundary conditions: Navier-Slip with Dirichlet and Navier-Slip with the dynamic outflow boundary condition. Again, maximal regularity of the problem is obtained in the appropriate functional analytic setting and with the natural compatibility conditions.

math.AP↗
arXiv2012

On Two-Phase Flows with Soluble Surfactant

The presence of surfactants has a pronounced effect on the surface tension and, hence, on the stress balance at the phase separating interface of two-phase flows. The transport of momentum induced by the local variations of the capillary forces are known as Marangoni effects. Here we study a model, which assumes the surfactant to be soluble in one of the adjacent bulk phases and which represents a generalization of the two-phase Navier-Stokes equations. Based on maximal Lp-regularity results for suitable linearizations we obtain local well-posedness of this model. We employ recent results from the Lp-theory of two-phase flows without surfactant.

math.AP↗
arXiv2012

On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows

We derive a class of energy preserving boundary conditions for incompressible Newtonian flows and prove local-in-time well-posedness of the resulting initial boundary value problems, i.e. the Navier-Stokes equations complemented by one of the derived boundary conditions, in an Lp-setting in domains, which are either bounded or unbounded with almost flat, sufficiently smooth boundary. The results are based on maximal regularity properties of the underlying linearisations, which are also established in the above setting.

math.AP↗
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.

math.AP↗
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.

math.AP↗