Nonlinear Potential Estimates for Generalized Stokes System

Abstract

In this paper, we consider the generalized stationary Stokes system with $p$-growth and Dini-$\mathrm{BMO}$ 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.