Quantitative regularity for the Navier-Stokes equations via spatial concentration
| dc.contributor.author | BARKER, Tobias | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | PRANGE, Christophe | |
| dc.date.accessioned | 2024-04-04T02:32:07Z | |
| dc.date.available | 2024-04-04T02:32:07Z | |
| dc.date.issued | 2021-06-14 | |
| dc.identifier.issn | 0010-3616 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/190357 | |
| dc.description.abstractEn | 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). | |
| dc.description.sponsorship | Bords, oscillations et couches limites dans les systèmes différentiels - ANR-16-CE40-0027 | |
| dc.description.sponsorship | Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure - ANR-18-CE40-0027 | |
| dc.language.iso | en | |
| dc.publisher | Springer Verlag | |
| dc.title.en | Quantitative regularity for the Navier-Stokes equations via spatial concentration | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1007/s00220-021-04122-x | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.identifier.arxiv | 2003.06717 | |
| bordeaux.journal | Communications in Mathematical Physics | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-02509766 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02509766v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Communications%20in%20Mathematical%20Physics&rft.date=2021-06-14&rft.eissn=0010-3616&rft.issn=0010-3616&rft.au=BARKER,%20Tobias&PRANGE,%20Christophe&rft.genre=article |
Fichier(s) constituant ce document
| Fichiers | Taille | Format | Vue |
|---|---|---|---|
|
Il n'y a pas de fichiers associés à ce document. |
|||