Existence and regularity for perturbed Stokes system with critical drift in 2D

TL;DR

This paper establishes existence and regularity results for a perturbed Stokes system with critical divergence-free drift in 2D, extending previous higher-dimensional work and covering large drift scenarios.

Contribution

It extends prior results to two dimensions, proving existence and regularity of solutions for large divergence-free drift in critical spaces.

Findings

Unique existence of q-weak solutions for large drift in weak L^2 space.

Existence of W^{1,2} solutions for arbitrarily large drift in L^2 in 2D.

Results applicable to scalar equations with divergence-free drifts.

Abstract

We consider a perturbed Stokes system with critical divergence-free drift in a bounded Lipschitz domain in $R^2$, with sufficiently small Lipschitz constant L. It extends our previous work in $\Bbb R^n, n\ge 3$, to two-dimensional case. For large drift in weak $L^2$ space, we prove unique existence of q-weak solutions for force in $L^q$ with q close to 2. Moreover, for drift in $L^2(\Bbb R^2)$ we prove the unique existence of $W^{1,2}$ solutions for arbitrarily large L. Using similar methods we can also prove analogous results for scalar equations with divergence-free drifts in weak $L^2$ space.