The Planar Coleman--Gurtin model with Beltrami conductivity

TL;DR

This paper studies the planar Coleman--Gurtin heat equation with anisotropic diffusion modeled by Beltrami coefficients, proving regularization effects and constructing finite-dimensional attractors under minimal smoothness assumptions.

Contribution

It establishes regularization and attractor existence for the heat equation with rough anisotropic diffusion using advanced regularity techniques.

Findings

Solutions regularize into $D(A_\mu)$ instantly

Under additional smoothness, solutions gain $W^{2,p}$ regularity

Existence of finite-dimensional global and exponential attractors

Abstract

This article addresses the planar Coleman--Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_\mu$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $\mu\in L^\infty(\Omega)$ satisfying $\|\mu\|_{\infty}<1$. First, under no additional smoothness assumptions on $\mu$, solutions with $H^1_0(\Omega)$-based initial data enter a time-averaged $L^\infty(\Omega)$ regime, and instantaneously regularize into the second-order graph space $D(A_\mu)$. Assuming in addition $\mu\in W^{1,2}(\Omega)$, this regularization upgrades to $W^{2,p}(\Omega)$ for every $1<p<2$, and we construct regular global and exponential attractors of finite fractal dimension, for both the $L^2(\Omega)$ and $H^1_0(\Omega)$-based dynamics. The proof combines the instantaneous smoothing method of Chekroun, Di Plinio, Glatt-Holtz and Pata with maximal parabolic regularity for divergence-form operators with measurable coefficients, and with planar quasiconformal Beltrami estimates recently obtained in work by Green, Wick and the author.