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The Planar Coleman--Gurtin model with Beltrami conductivity

Francesco Di Plinio

Abstract

This article addresses the planar Coleman--Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_μ$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $μ\in L^\infty(Ω)$ satisfying $\|μ\|_{\infty}<1$. First, under no additional smoothness assumptions on $μ$, solutions with $H^1_0(Ω)$-based initial data enter a time-averaged $L^\infty(Ω)$ regime, and instantaneously regularize into the second-order graph space $D(A_μ)$. Assuming in addition $μ\in W^{1,2}(Ω)$, this regularization upgrades to $W^{2,p}(Ω)$ for every $1<p<2$, and we construct regular global and exponential attractors of finite fractal dimension, for both the $L^2(Ω)$ and $H^1_0(Ω)$-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.

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 , typical of heterogeneous or composite media and encoded by a Beltrami coefficient satisfying . First, under no additional smoothness assumptions on , solutions with -based initial data enter a time-averaged regime, and instantaneously regularize into the second-order graph space . Assuming in addition , this regularization upgrades to for every , and we construct regular global and exponential attractors of finite fractal dimension, for both the and -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.
Paper Structure (30 sections, 20 theorems, 192 equations)

This paper contains 30 sections, 20 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Hereditary heat conduction in anisotropic media: the model
  3. Assumptions on the nonlinearities and the memory kernel
  4. Remark on Trudinger--Moser alternatives
  5. Physical motivation
  6. Main results and technical novelties
  7. Structure of the paper
  8. Functional setting and elliptic toolbox
  9. Operator-theoretic setting for the Beltrami diffusion
  10. Memory spaces and past history formulation
  11. Beltrami elliptic toolbox
  12. Parabolic maximal regularity and first-order smoothing
  13. Main results
  14. Standing assumptions and basic dissipative structure
  15. Time-averaged and $\mathcal{V}$-instantaneous smoothing
  16. ...and 15 more sections

Key Result

Lemma 2.2

Assume (K1). For every $r\in\mathbb R$ and every $\eta\in D(T_r)$, Consequently, $T$ generates a strongly continuous semigroup of contractions on ${\mathcal{M}^{r}}$, given explicitly by the right-translation

Theorems & Definitions (35)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: CGDP2010
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 25 more