Global Lorentz estimates for nonlinear parabolic equations on nonsmooth 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.