On the Global Existence of a Class of Strongly Coupled Parabolic Systems

TL;DR

This paper proves the global existence of certain strongly coupled parabolic systems by developing new regularity estimates and assuming boundedness of solutions in large $L^p$ norms, bypassing the need for maximum principles.

Contribution

Introduces a novel approach to regularity theory for parabolic systems, establishing global solutions under large $L^p$ norm bounds instead of maximum principles.

Findings

Established global existence for a class of strongly coupled parabolic systems.

Developed new $W^{1,p}$ estimates for solutions with integrable data.

Provided a practical criterion based on $L^p$ bounds for verifying solution boundedness.

Abstract

We establish the global existence of a class of strongly coupled parabolic systems. The necessary apriori estimates will be obtained via our new approach to the regularity theory of parabolic scalar equations with integrable data and new $W^{1,p}$ estimates of their solutions. The key assumption here is that the $L^p$ norms of solutions are uniformly bounded for some sufficiently large $p\in (1,\infty)$, an assumption can be easily affirmed for systems with polynomial growth data. This replaces the usual condition that the solutions are uniformly bounded which is very hard to be verified because maximum principles for systems are generally unavailable.