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Existence and regularity for perturbed Stokes system with critical drift

Misha Chernobai, Tai-Peng Tsai

TL;DR

This work analyzes the existence and gradient regularity of solutions to the perturbed Stokes system $- abla^2 u + b\cdot\nabla u + \nabla\pi = \operatorname{div}\mathbb G$ in a bounded Lipschitz domain with a divergence-free drift $b$ in the critical Lorentz space $L^{n,\infty}$. It develops three main results: (i) gradient estimates in $W^{1,q}$ under smallness of $b$ in $L^n$ or weak $L^n$; (ii) existence with $W^{1,q}$-control when $b$ is small in $L^{n,\infty}$ for a wide range of $q$; and (iii) gradient estimates for exponents $q$ near 2 without smallness by leveraging Wolf's local pressure projection to manage the pressure term. The key novelty is the use of Wolf's pressure projection to obtain uniform $L^2$-based control of the pressure and to propagate higher integrability of the velocity gradient via Gehring-type arguments, even in the presence of a critical drift. The results extend known scalar-DRIFT regularity theory to the vector Stokes setting and provide a framework for boundary-value problems with critical drift terms.

Abstract

We consider the existence and $L^q$ gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in $\mathbb{R}^n$, $n \ge 3$. The first two results assume the drift is either in $L^n$ or sufficiently small in weak $L^n$. The third result assumes the drift is in weak $L^n$ without smallness, and obtain results for $q$ close to 2.

Existence and regularity for perturbed Stokes system with critical drift

TL;DR

This work analyzes the existence and gradient regularity of solutions to the perturbed Stokes system in a bounded Lipschitz domain with a divergence-free drift in the critical Lorentz space . It develops three main results: (i) gradient estimates in under smallness of in or weak ; (ii) existence with -control when is small in for a wide range of ; and (iii) gradient estimates for exponents near 2 without smallness by leveraging Wolf's local pressure projection to manage the pressure term. The key novelty is the use of Wolf's pressure projection to obtain uniform -based control of the pressure and to propagate higher integrability of the velocity gradient via Gehring-type arguments, even in the presence of a critical drift. The results extend known scalar-DRIFT regularity theory to the vector Stokes setting and provide a framework for boundary-value problems with critical drift terms.

Abstract

We consider the existence and gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in , . The first two results assume the drift is either in or sufficiently small in weak . The third result assumes the drift is in weak without smallness, and obtain results for close to 2.
Paper Structure (7 sections, 17 theorems, 112 equations)

This paper contains 7 sections, 17 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Proposition \ref{['main1']} and Theorem \ref{['main2']}
  4. Wolf's local pressure projection
  5. A priori bound for large drift and $p$ close to $2$
  6. Proof of Theorem \ref{['main3']}
  7. Proof of Proposition \ref{['main4']}

Key Result

Proposition 1.3

Let $1<q<\infty$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\ge 3$, with sufficiently small Lipschitz constant $L>0$$($i.e., $L \le L_0$ where $L_0=L_0(n,q)>0)$. Assume $b\in L^n(\Omega)^n$, $\mathop{\rm div}\nolimits b=0$. Then for any $\mathbb{G}\in L^q(\Omega)^{n\times n}$, ,

Theorems & Definitions (34)

  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6: Scalar equations
  • Lemma 2.1: Hölder inequality in Lorentz spaces
  • Lemma 2.2: Sobolev inequality in Lorentz spaces
  • Lemma 2.3
  • proof
  • ...and 24 more