2016/06/27 by Dongho Chae, Jihoon Lee, Chae, Dongho +1
Computer Science · Mathematics · #35Q30 #76D03 #76D05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP #msc:35Q30 #msc:76D03 #msc:76D05
paper · pdf · doi:10.48550/arxiv.1606.08126
12 pages
openalex publication_date 2016/06/27 · openalex created_date 2016/07/22 · arxiv created 2016/08/30 · arxiv updated 2016/08/31 · openalex updated_date 2026/07/28
We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote \f\+=max\f, 0\ , QT=\Bbb R3× (0, T). If \( v × \fracω|ω| )⋅ (Λβv)/(|Λβv|)\+ ∈ Lγ, αx,t (QT) with 3/γ+2/α≤ 1 for some γ>3 and 1 ≤ β≤ 2, then the local smooth solution v of the Navier-Stokes equations on (0,T) can be continued to (0, T+δ) for some δ>0. We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier-tokes equations in a space-time domain containing z0= (x0, t0), let Qz0, r=Bx0, r × (t0-r2, t0) be a parabolic cylinder in the domain. We show that if either \( v × \fracω|ω|) ⋅ (∇ × ω)/(|∇ × ω|)\+ ∈ Lγ, αx,t(Qz0, r) with \frac3γ+\frac2α ≤ 1, or \((v)/(|v|) × ω) ⋅ (∇ × ω)/(|∇ × ω|)\+ ∈ Lγ, αx,t(Qz0, r) with \frac3γ+\frac2α ≤ 2, (γ≥ 2, α≥ 2), then z0 is a regular point for v. This improves previous local regularity criteria for the suitable weak solutions.