On well-posedness theory of very weak solutions to Navier-Stokes equations on irregular domains with nonhomogeneous Dirichlet boundary data

2026/07/25 by Xiaojin Bai, Siran Li, Xiangxiang Su
#math.AP

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Abstract

The well-posedness theory of very weak solutions is a central topic in mathematical hydrodynamics, especially in the regularity theory for Navier-Stokes equations. It has been fully developed for incompressible fluid flows on bounded domains in R³ of C^2,1-regularity. In this paper, based on the analytic theories in [D. Breit and A. Gaudin, ArXiv Preprint: 2511.19091 (2025)] and [V.G. Maz'ya and T.O. Shaposhnikova, Vol.337, Grundlehren der mathematischen Wissenschaften (2009)], we establish the well-posedness theory of very weak solutions to the Navier-Stokes equations on bounded Lipschitz domains whose boundary has local graphing functions with sufficiently small Sobolev multiplier norm, which contain the bounded Lipschitz domains with sufficiently small Lipschitz constants as a special case.

Citations

• Optimal regularity results for the Stokes--Dirichlet problem
• Partial boundary regularity for the Navier-Stokes equations in irregular domains
• Regularity results in 2D fluid-structure interaction
• On the L^p-theory of the Navier--Stokes equations on three-dimensional bounded Lipschitz domains

Related

• Partial boundary regularity for the Navier-Stokes equations in irregular domains
• Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary
• Non-uniqueness of solutions for 3D Navier-Stokes equations in bounded domains
• Wellposedness and stability results for the Navier-Stokes equations in \mathbf R³
• Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density