Remarks on regularity conditions of the Navier-Stokes equations

2007/05/16 by Dongho Chae, Chae, Dongho
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0705.2285

13 pages

openalex publication_date 2007/05/16 · arxiv created 2007/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let v and ø be the velocity and the vorticity of the a suitable weak solution of the 3D Navier-Stokes equations in a space-time domain containing z₀ =(x₀, t₀), and Q_{z0, r} =B_{x0, r}× (t₀-r², t₀) be a parabolic cylinder in the domain. We show that if v× \fracø|ø|∈ L^{γ, α}_x,t (Q_{z0, r}) or ø× (v)/(|v|)∈ L^{γ, α}_x,t (Q_{z0, r}), where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for v. This refines previous local regularity criteria for the suitable weak solutions.

Related

• On the Geometric Regularity Conditions for the 3D Navier-Stokes Equations
• On the regularity of weak solutions of the 3D Navier-Stokes equations in B^-1_∞,∞
• Boundary ε-regularity criteria for the 3D Navier-Stokes equations
• On the possible time singularities for the 3D Navier-Stokes equations
• On the interior regularity criterion and the number of singular points to the Navier-Stokes equations