Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems

TL;DR

This paper investigates the regularity of solutions to linear parabolic systems within generalized Morrey spaces, establishing conditions under which the gradient of solutions belongs to these spaces and providing gradient estimates.

Contribution

It introduces new regularity results for linear parabolic systems in generalized Morrey spaces, extending previous understanding to systems with coefficients in continuous and VMO classes.

Findings

Gradient of solutions in $L_^{2,}$ under continuous coefficients

Gradient of solutions in $L_^{2,}$ with VMO coefficients

Derived estimates for the solutions' gradients

Abstract

The purpose of this paper is to study the parabolic system $u_t^i-D_\alpha(a_{ij}^{\alpha\beta}D_\beta u^j)=-div f^i$ in the generalized Morrey Space $L_{\phi}^{2,\lambda}$ . We would like to understand the regularity of the solutions of this system. It will be shown that 1: if $a_{ij}^{\alpha\beta}\in C(\bar{Q_{T}})$ then $Du\in L_\phi^{2,\lambda}$, and 2: if $a_{ij}^{\alpha\beta}\in VMO(Q_{T})$ then $Du\in L_\phi^{2,\lambda}$. Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions.