The Planar Coleman--Gurtin model with Beltrami conductivity

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 $\Vert \mu {\Vert }_{\infty }<1$. First, under no additional smoothness assumptions on $\mu$, solutions with $H_0^1(\Omega )$-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 $H_0^1(\Omega )$-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.

arXiv comment30 pages; submitted