Quantitative regularity for the Navier-Stokes equations via spatial concentration

Abstract

This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound $\Vert u{\Vert }_{L_t^{\infty }{L}_x^{3,\infty }}\le M$. Namely, we show that if $T^{\ast }$ is a first blow-up time and $(0,T^{\ast })$ is a singular point then $$\Vert u(\cdot ,t){\Vert }_{L^3(B_0(R))}\ge C(M)\log \left(\frac{1}{T^{\ast }-t}\right),R=O((T^{\ast }-t{)}^{\frac{1}{2}-}).$$ We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (2012), which says that if $u$ is a smooth finite-energy solution to the Navier-Stokes equations on ${\mathbb{R}}^3\times (0,1)$ with $$\underset{n}{\sup }\Vert u(\cdot ,t_{(n)}){\Vert }_{L^3({\mathbb{R}}^3)}<\infty \text{and}{t}_{(n)}\uparrow 1,$$ then $u$ does not blow-up at $t=1$. To prove our results we develop a new strategy for proving quantitative bounds for the Navier-Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.

69 pages, 1 figure. The current version contains, in addition, quantitative bounds for the number of singular points in a Type I blow-up scenario

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Cited by in corpus (6)

• Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces
• A minimum critical blowup rate for the high-dimensional Navier-Stokes equations
• Remarks on sparseness and regularity of Navier-Stokes solutions
• The localized characterization for the singularity formation in the Navier-Stokes equations
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang
• Mild criticality breaking for the Navier-Stokes equations