The Green function for the Stokes system with measurable coefficients

2015/03/25 by Choi, Jongkeun, Lee, Ki-Ahm
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1503.07290

Abstract

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain Ω⊂ ℝⁿ, n≥ 3. We construct the Green function in Ω under the condition (\bfA1) that weak solutions of the system enjoy interior Hölder continuity. We also prove that (\bfA1) holds, for example, when the coefficients are VMO. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption (\bfA2) that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori L^q-estimates for Stokes systems with BMO coefficients on a Reifenberg domain, we verify that (\bfA2) is satisfied when the coefficients are VMO and Ω is a bounded C¹ domain.

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• Fundamental solutions for stationary Stokes systems with measurable coefficients
• Green functions for pressure of Stokes systems
• Green functions of mixed boundary value problems for stationary Stokes systems in two dimensions
• Green functions for stationary Stokes systems with conormal derivative boundary condition in two dimensions
• L_q-estimates for stationary Stokes system with coefficients measurable in one direction