Nonlinear Potential Estimates for Generalized Stokes System

TL;DR

This paper develops nonlinear potential estimates for the generalized stationary Stokes system with p-growth and Dini-BMO coefficients, providing pointwise bounds for shear rate and pressure in the plane, and extending results to higher dimensions.

Contribution

It introduces new nonlinear potential estimates for the Stokes system with minimal regularity assumptions, advancing understanding of regularity in fluid mechanics models.

Findings

Pointwise estimates for shear rate and pressure in the plane.

L^ estimates for symmetric gradient without extra coefficient regularity.

Potential estimates for solutions in higher dimensions.

Abstract

In this paper, we consider the generalized stationary Stokes system with $p$-growth and Dini-$\operatorname{BMO}$ regular coefficients. The main purpose is to establish pointwise estimates for the shear rate and the associated pressure to such Stokes system in terms of an unconventional nonlinear Havin-Maz'ya-Wolff type potential of the nonhomogeneous term in the plane. As a consequence, a symmetric gradient $L^{\infty}$ estimate is obtained. Moreover, we derive potential estimates for the weak solution to the Stokes system without additional regularity assumptions on the coefficients in higher dimensional space.