Abstract
This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound Vert uVert _{L^{infty }_{t}L^{3,infty }_{x}}le M. Namely, we show that if T^* is a first blow-up time and (0,T^*) is a singular point then ‖u(·,t)‖L3(B0(R))≥C(M)log(1T∗-t),R=O((T∗-t)12-).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert u(\\cdot ,t)\\Vert _{L^{3}(B_{0}(R))}\\ge C(M)\\log \\Big (\\frac{1}{T^*-t}\\Big ),\\,\\,\\,\\,\\,\\,R=O((T^*-t)^{\\frac{1}{2}-}). \\end{aligned}$$\\end{document}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 (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on {mathbb {R}}^3times (0,1) with supn‖u(·,t(n))‖L3(R3)<∞andt(n)↑1,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _{n}\\Vert u(\\cdot ,t_{(n)})\\Vert _{L^{3}({\\mathbb {R}}^3)}<\\infty \\,\\,\\,\\text {and}\\,\\,\\,t_{(n)}\\uparrow 1, \\end{aligned}$$\\end{document}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.
Highlights
IntroductionWe consider the three-dimensional incompressible Navier–Stokes equations
In this paper, we consider the three-dimensional incompressible Navier–Stokes equations∂t u − u + u · ∇u + ∇ p = 0, ∇ · u = 0 (1)T
1 u(·, t ) L3(B0(R)) ≥ C(M) log T ∗ − t, R. We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions
Summary
We consider the three-dimensional incompressible Navier–Stokes equations. In this setting ‘1) Backward propagation of vorticity concentration’ still remains valid if a sufficiently well-separated subsequence of t(k) is taken (see Lemma 3.3 and Proposition 2.2) To show this we use energy estimates in [40] for solutions to the Navier– Stokes equations with L3(R3) initial data. For the case of a smooth finite-energy solution u having finite scale-invariant L5(R3 × (0, 1)) norm, one way to obtain quantitative estimates is to consider the vorticity equation (37) with initial vorticity ω0 ∈ L2(R3). The proof of Theorem A relies on the combination (as is showed in Fig. 1) of the quantitative bound in the Type I case (Proposition 2.1) on the one hand, with concentration estimates near a potential singularity for the local L3 norm ( [6]) and for the L∞ norm (Corollary 5.3) on the other hand.
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