Optimality of Serrin type extension criteria to the Navier-Stokes equations

Advances in Nonlinear Analysis

article

Reinhard Farwig¹  Ryo Kanamaru² 

[1] Department of Mathematics, Darmstadt University of Technology;Department of Pure and Applied Mathematics, School of Fundamental Science and Engineering Waseda University

关键词: Serrin type extension criterion;    Navier-Stokes equations;    bilinear estimate;    logarithmic interpolation inequality;   

DOI  :  10.1515/anona-2020-0130

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We prove that a strong solution u to the Navier-Stokes equations on (0, T ) can be extended if either u ∈ L θ (0, T ; U˙ ∞ ,1/θ ,∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$) for 2/ θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T ; V˙ ∞ ,∞ ,20 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), where U˙ p,β ,σ s $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V˙ p,q,θ s $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space B˙ p,qs $\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

【 授权许可】

CC BY