BARKER, Tobias; PRANGE, Christophe

doi:10.1007/s00220-021-04122-x

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BARKER, Tobias

PRANGE, Christophe
Institut de Mathématiques de Bordeaux [IMB]

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Ce document a été publié dans

Communications in Mathematical Physics. 2021-06-14

Springer Verlag

Résumé en anglais

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 $\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$ is a first blow-up time and $(0,T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ 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 $$\sup_{n}\|u(\cdot,t_{(n)})\|_{L^{3}(\mathbb{R}^3)}<\infty\,\,\,\textrm{and}\,\,\,t_{(n)}\uparrow 1,$$ 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 \v{S}ver\'{a}k (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019).< Réduire

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https://oskar-bordeaux.fr/handle/20.500.12278/190357

DOI

http://dx.doi.org/10.1007/s00220-021-04122-x

Project ANR

Bords, oscillations et couches limites dans les systèmes différentiels - ANR-16-CE40-0027
Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure - ANR-18-CE40-0027

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Institut de Mathématiques de Bordeaux (IMB) - UMR 5251