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

Abstract

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ Lθ(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 ∈ L2(0, T; V˙∞,∞,20$\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), where U˙p,β,σs$\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V˙p,q,θs$\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ 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.

Keywords

• serrin type extension criterion
• navier-stokes equations
• bilinear estimate
• logarithmic interpolation inequality
• 35q30 (primary)
• 35b65
• 46e35
• 76d05