Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities
| dc.contributor.author | BARKER, Tobias | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| hal.structure.identifier | Centre National de la Recherche Scientifique [CNRS] | |
| dc.contributor.author | PRANGE, Christophe | |
| dc.date.accessioned | 2024-04-04T02:58:56Z | |
| dc.date.available | 2024-04-04T02:58:56Z | |
| dc.date.issued | 2020-03-07 | |
| dc.identifier.issn | 0003-9527 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/192701 | |
| dc.description.abstractEn | This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $L^3$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and \v{S}ver\'{a}k, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if $(0, T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{R}(0))}\geq \gamma_{univ},\qquad R=O(\sqrt{T^*-t}).$$ This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely $L^{3,\infty}$ and the Besov space $\dot B^{-1+\frac3p}_{p,\infty}$, $p\in(3,\infty)$. | |
| dc.language.iso | en | |
| dc.publisher | Springer Verlag | |
| dc.title.en | Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.identifier.arxiv | 1812.09115 | |
| bordeaux.journal | Archive for Rational Mechanics and Analysis | |
| 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-02374652 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02374652v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Archive%20for%20Rational%20Mechanics%20and%20Analysis&rft.date=2020-03-07&rft.eissn=0003-9527&rft.issn=0003-9527&rft.au=BARKER,%20Tobias&PRANGE,%20Christophe&rft.genre=article |
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