Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the smallness of mean oscillations and regularity of weak solutions to regular/degenerate strongly coupled parabolic systems

Dung Le

Abstract

It will be established that the mean oscillation of bounded weak solutions to strongly coupled parabolic systems is small in small balls. If the systems are regular elliptic then their bounded weak solutions are Hölder continuous. Further assumptions on the systems will even prove that these solutions exist globally. Weak solutions to degenerate systems of porous media type are also studied.

On the smallness of mean oscillations and regularity of weak solutions to regular/degenerate strongly coupled parabolic systems

Abstract

It will be established that the mean oscillation of bounded weak solutions to strongly coupled parabolic systems is small in small balls. If the systems are regular elliptic then their bounded weak solutions are Hölder continuous. Further assumptions on the systems will even prove that these solutions exist globally. Weak solutions to degenerate systems of porous media type are also studied.
Paper Structure (5 sections, 8 theorems, 32 equations)

This paper contains 5 sections, 8 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. BMOsmall) of bounded weak solutions
  3. Regular strongly coupled parabolic systems
  4. Degenerate strongly coupled parabolic systems
  5. Uniqueness of weak solutions

Key Result

Theorem 2.1

Let $K\ge1$ and $\Omega$ be a domain in ${\rm I{\rm R}}^{N+K}$ and $\mathbf{B}_R$ be a cube in $\Omega$ with sides parallel to the axes of ${\rm I{\rm R}}^{N+K}$. We write ${\rm I{\rm R}}^{N+K}=\{(x,y_*)\,:\, x\in{\rm I{\rm R}}^N, y_*\in{\rm I{\rm R}}^K\}$. If for small $R_0,\varepsilon>0$ Then, for $R,\varepsilon$ as in (DxNW1), $\mathbf{B}_R=B'_R\times B_R(s_*)$, and for some constant $C$

Theorems & Definitions (14)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 4 more