A minimum critical blowup rate for the high-dimensional Navier-Stokes equations

Abstract

We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$, $d\ge 4$ when the solution lies in the critical space $L_t^{\infty }{L}_x^d$. Explicit subcritical bounds on the solution are obtained in terms of the critical norm. A consequence is that $\Vert u(t){\Vert }_{L_x^d({\mathbb{R}}^d)}$ grows at a minimum rate of $(\log \log \log \log (T_{\ast }-t{)}^{-1}{)}^c$ along a sequence of times approaching a hypothetical blowup at $T_{\ast }$. These results quantify a theorem of Dong and Du and extend the three-dimensional work of Tao.

arXiv comment32 pages, 2 figures. Revised version to appear in JMFM

References in corpus (8)

• A certain necessary condition of potential blow up for Navier-Stokes equations
• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• The Navier-Stokes equations in the critical Lebesgue space
• Quantitative regularity for the Navier-Stokes equations via spatial concentration
• Global weak Besov solutions of the Navier-Stokes equations and applications
• Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces
• Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces
• Quantitative bounds for critically bounded solutions to the Navier-Stokes equations