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Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities

dc.contributor.authorBARKER, Tobias
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
hal.structure.identifierCentre National de la Recherche Scientifique [CNRS]
dc.contributor.authorPRANGE, Christophe
dc.date.accessioned2024-04-04T02:58:56Z
dc.date.available2024-04-04T02:58:56Z
dc.date.issued2020-03-07
dc.identifier.issn0003-9527
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192701
dc.description.abstractEnThis 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.isoen
dc.publisherSpringer Verlag
dc.title.enLocalized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities
dc.typeArticle de revue
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.identifier.arxiv1812.09115
bordeaux.journalArchive for Rational Mechanics and Analysis
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-02374652
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-02374652v1
bordeaux.COinSctx_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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