Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities
| Autor: | Tobias Barker, Christophe Prange |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Unit sphere
Physics Pure mathematics Mechanical Engineering 010102 general mathematics Singular point of a curve Type (model theory) 35A99 35B44 35B65 35D30 35Q30 76D05 Space (mathematics) 01 natural sciences 010101 applied mathematics Mathematics - Analysis of PDEs Mathematics (miscellaneous) FOS: Mathematics Besov space [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Gravitational singularity 0101 mathematics QA Navier–Stokes equations Analysis Energy (signal processing) Analysis of PDEs (math.AP) |
| Zdroj: | Archive for Rational Mechanics and Analysis Archive for Rational Mechanics and Analysis, Springer Verlag, 2020 |
| ISSN: | 0003-9527 1432-0673 |
| Popis: | 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 $\dot B^{-1+\frac3p}_{p,\infty}$, $p\in(3,\infty)$. Comment: 47 pages |
| Databáze: | OpenAIRE |
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