Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A regularity result for quasilinear elliptic systems with an equation of energy-type

Pierre-Etienne Druet

TL;DR

This work develops a regularity theory for quasilinear elliptic systems with energy-type equations featuring critical growth. By modeling the energy-production term as a diffusive measure and employing capacitary estimates, it proves Hölder continuity for vector-valued weak solutions on Lipschitz domains in all dimensions and, in the linear-principal-part scenario, enhanced gradient integrability along with a small-data uniqueness principle. The framework applies to electrothermal coupling problems, with concrete results for the thermistor model and non-isothermal Nernst-Planck equations, illustrating existence and regularity under rough data and mixed boundary conditions. Overall, the paper extends regularity theory to energy-transport PDE systems with minimal smoothness assumptions, providing robust a priori bounds and constructive existence results for key physical models.

Abstract

We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of physics for internal energy or entropy. Relying on the properties of diffusive measures, we prove a Hölder continuity result for these PDE systems in the case of vectorial p-growth and of low regularity features such as mixed boundary conditions on Lipschitz domains. This result is valid in all space dimensions. In the special case where the principal part is linear in the second order, we also study the higher integrability of the gradients and the uniqueness principle in the small. Finally two illustrations in the context of electrothermal coupling are considered, the stationary thermistor model and the Nernst-Planck equations of electrochemistry with variable temperature.

A regularity result for quasilinear elliptic systems with an equation of energy-type

TL;DR

This work develops a regularity theory for quasilinear elliptic systems with energy-type equations featuring critical growth. By modeling the energy-production term as a diffusive measure and employing capacitary estimates, it proves Hölder continuity for vector-valued weak solutions on Lipschitz domains in all dimensions and, in the linear-principal-part scenario, enhanced gradient integrability along with a small-data uniqueness principle. The framework applies to electrothermal coupling problems, with concrete results for the thermistor model and non-isothermal Nernst-Planck equations, illustrating existence and regularity under rough data and mixed boundary conditions. Overall, the paper extends regularity theory to energy-transport PDE systems with minimal smoothness assumptions, providing robust a priori bounds and constructive existence results for key physical models.

Abstract

We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of physics for internal energy or entropy. Relying on the properties of diffusive measures, we prove a Hölder continuity result for these PDE systems in the case of vectorial p-growth and of low regularity features such as mixed boundary conditions on Lipschitz domains. This result is valid in all space dimensions. In the special case where the principal part is linear in the second order, we also study the higher integrability of the gradients and the uniqueness principle in the small. Finally two illustrations in the context of electrothermal coupling are considered, the stationary thermistor model and the Nernst-Planck equations of electrochemistry with variable temperature.
Paper Structure (13 sections, 19 theorems, 186 equations)

This paper contains 13 sections, 19 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Quasilinear elliptic system with an equation of energy type. Notation and results
  3. Notations and basic definitions
  4. Assumptions and statement of the results
  5. Weak estimates for distributional solutions
  6. Hoelder continuity
  7. Hölder continuity for the $N$ first variables
  8. Hölder regularity in the energy equation
  9. The quasilinear case with $p_1 = \ldots = p_{N+1} =2$
  10. Uniqueness in the small
  11. Illustration I: The thermistor case with a heat source
  12. Stationary Nernst-Planck equations with variable temperature: Existence of continuous weak solutions
  13. Auxiliary results

Key Result

Theorem 2.5

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $S \subseteq \partial \Omega$ be measurable. Consider a weak solution $w = (\rho,u) \in W^{1,{\bf p}}(\Omega; \, \mathcal{O})$ with the energy dissipation property to systemcomp, BC, BC2. We assume all conditions of (A), (B$(\bf q)$) and (C) Then, there are constants $0< \lambda_1, \ldots,\lambda_{N+1}< 1$ and $M_0> 0$ such that every weak

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 37 more