Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

TL;DR

This paper establishes global Lorentz space estimates for solutions to nonlinear parabolic equations on nonsmooth Reifenberg domains, extending previous results to less regular nonlinearities and more general function spaces.

Contribution

It proves Calderón-Zygmund estimates for weak solutions under minimal regularity assumptions on the nonlinearity and domain, in Lorentz spaces.

Findings

Global estimates in Lorentz spaces including Lebesgue spaces

Extension to equations with less regularity on nonlinearity

Applicable to nonsmooth Reifenberg domains

Abstract

Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad \Omega\times(0,T), $$ where $T>0$ and $\Omega$ is a Reifenberg domain. We suppose that the nonlinearity $\mathbf{a}(\xi,x,t)$ has a small BMO norm with respect to $x$ and is merely measurable and bounded with respect to the time variable $t$. In this paper, we prove the global Calder\'on-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calder\'on-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity $\mathbf{a}(\xi,x,t)$ and to more general setting of Lorentz spaces.