# Global Marcinkiewicz estimates for nonlinear parabolic equations with nonsmooth coefficients

## TL;DR

This paper establishes global Marcinkiewicz estimates for solutions to nonlinear parabolic equations with measure data, under minimal regularity assumptions on coefficients and domain, advancing the understanding of regularity in complex PDEs.

## Contribution

It proves a global Marcinkiewicz estimate for SOLA solutions to nonlinear parabolic equations with nonsmooth coefficients and Reifenberg flat domains, under small BMO conditions.

## Key findings

- Established global Marcinkiewicz estimates for solutions.
- Extended regularity results to nonsmooth coefficients and domains.
- Provided new bounds for solutions with measure data.

## Abstract

Consider the parabolic equation with measure data \begin{equation*} \left\{ \begin{aligned} &u_t-{\rm div} \mathbf{a}(D u,x,t)=\mu&\text{in}& \quad \Omega_T, &u=0 \quad &\text{on}& \quad \partial_p\Omega_T, \end{aligned}\right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $\Omega_T=\Omega\times (0,T)$, $\partial_p\Omega_T=(\partial\Omega\times (0,T))\cup (\Omega\times\{0\})$, and $\mu$ is a signed Borel measure with finite total mass. Assume that the nonlinearity ${\bf a}$ satisfies a small BMO-seminorm condition, and $\Omega$ is a Reifenberg flat domain. This paper proves a global Marcinkiewicz estimate for the SOLA (Solution Obtained as Limits of Approximation) to the parabolic equation.