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On regularity for nonhomogeneous parabolic systems with a skew-symmetric part in BMO

Guoming Zhang

TL;DR

The paper addresses regularity for nonhomogeneous parabolic systems with a skew-symmetric BMO part in the coefficient, allowing unbounded $A$. It adopts a global variational method that splits time via $∂_{t} = D_{t}^{1/2} H_{t} D_{t}^{1/2}$ to reveal hidden coercivity and derives an improved Caccioppoli inequality and a reverse Hölder inequality for gradients. A key contribution is removing the boundedness assumption on the coefficient matrix by treating $A(t,x)=S(t,x)+D(t,x)$ with $S∈L^{∞}$ and Garding, $D$ real skew-symmetric in $BMO$, yielding higher integrability results that extend ABES and related work. The appendix strengthens the Gehring lemma and provides a coefficient-extension tool, broadening the applicability of the local estimates to more general parabolic systems.

Abstract

In this paper we investigate the improved Caccioppoli inequality and the reverse Hölder inequality for gradients of weak solutions to nonhomogeneous parabolic systems whose coefficients can be split into a complex-valued and bounded part, which also satisfies the uniform G$\mathring{a}$rding inequality, and a real and anti-symmetric part in BMO. In particular, unbounded coefficients are allowed.

On regularity for nonhomogeneous parabolic systems with a skew-symmetric part in BMO

TL;DR

The paper addresses regularity for nonhomogeneous parabolic systems with a skew-symmetric BMO part in the coefficient, allowing unbounded . It adopts a global variational method that splits time via to reveal hidden coercivity and derives an improved Caccioppoli inequality and a reverse Hölder inequality for gradients. A key contribution is removing the boundedness assumption on the coefficient matrix by treating with and Garding, real skew-symmetric in , yielding higher integrability results that extend ABES and related work. The appendix strengthens the Gehring lemma and provides a coefficient-extension tool, broadening the applicability of the local estimates to more general parabolic systems.

Abstract

In this paper we investigate the improved Caccioppoli inequality and the reverse Hölder inequality for gradients of weak solutions to nonhomogeneous parabolic systems whose coefficients can be split into a complex-valued and bounded part, which also satisfies the uniform Grding inequality, and a real and anti-symmetric part in BMO. In particular, unbounded coefficients are allowed.
Paper Structure (6 sections, 17 theorems, 185 equations)

This paper contains 6 sections, 17 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and basic notations
  3. The global variational setup
  4. Local regularity estimates for weak solutions in $L^{2}$
  5. Proofs of Theorem \ref{['thm: 4.1']} and Theorem \ref{['thm: 4.2']}
  6. Appendix

Key Result

Theorem 1.1

If $u$ is a locally weak solution to eq: 1.1 with $f$ only belonging to $L^{2_{*}}_{loc}(\Omega)$ ($2_{*}<2$) and $F\in L^{2}_{loc}(\Omega)$, then in time $u$ is bounded and Hölder continuous locally with values in spatial $L^{p}_{loc}(Q_{0})$ for some $p>2.$

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Remark 3.7
  • ...and 12 more