arXiv subjects

Danielle Hilhorst

Publications and source records attributed to Danielle Hilhorst.

22 records · Page 2Linked to original sources

arXiv2009

Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

We study the singular limit of a spatially inhomogeneous and anisotropic reaction-diffusion equation. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. The limit problem involves motion by anisotropic mean curvature. We also prove an optimal estimate of the thickness of the transition layer.

math.AP↗

arXiv2009

Mass conserved Allen-Cahn equation and volume preserving mean curvature flow

We consider a mass conserved Allen-Cahn equation $u_t=Δu+ \e^{-2} (f(u)-\eλ(t))$ in a bounded domain with no flux boundary condition, where $\eλ(t)$ is the average of $f(u(\cdot,t))$ and $-f$ is the derivative of a double equal well potential. Given a smooth hypersurface $γ_0$ contained in the domain, we show that the solution $u^\e$ with appropriate initial data approaches, as $\e\searrow0$, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from $γ_0$.

math.AP↗

arXiv2008

Long time convergence for a class of variational phase field models

In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife models. Existence and uniqueness of the solution to the related initial boundary value problem are shown. Further regularity of the solution is deduced by exploiting the so-called regularizing effect. Then, the large time behavior of such a solution is studied and several convergence properties of the trajectory as time tends to infinity are discussed.

math.AP↗

arXiv2007

Travelling-wave analysis of a model describing tissue degradation by bacteria

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the `degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

math.AP↗