Lq -estimates for stationary Stokes system with coefficients measurable in one direction

Hongjie Dong, Doyoon Kim

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori W˙q1-estimates for any q∈ [2 , ∞) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a Wq1-estimate and prove the solvability for any q∈ (1 , ∞) when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

Original languageEnglish
JournalBulletin of Mathematical Sciences
DOIs
Publication statusAccepted/In press - 2018 Jan 1

Fingerprint

Stokes System
Lipschitz Domains
Measurable function
Coefficient
Estimate
Half-space
Bounded Domain
Ball
Variable Coefficients
Lipschitz
Solvability
Oscillation

Keywords

  • Boundary value problem
  • Measurable coefficients
  • Stokes systems

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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N2 - We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori W˙q1-estimates for any q∈ [2 , ∞) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a Wq1-estimate and prove the solvability for any q∈ (1 , ∞) when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

AB - We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori W˙q1-estimates for any q∈ [2 , ∞) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a Wq1-estimate and prove the solvability for any q∈ (1 , ∞) when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

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