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

2018/12/21 by Tobias Barker, Barker, Tobias, Christophe Prange +1 · 2 citations
Mathematics · #35A99 #35B44 #35B65 #35D30 #35Q30 #76D05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP #msc:35A99 #msc:35B44 #msc:35B65 #msc:35D30 #msc:35Q30 #msc:76D05

paper · pdf · doi:10.48550/arxiv.1812.09115

47 pages

openalex publication_date 2018/12/21 · arxiv created 2019/01/09 · arxiv updated 2019/01/10 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

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 L³, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Šverák, 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(⋅,t)‖_L³(B_R(0))≥ γ_univ, R=O(√(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,∞ and the Besov space B^-1+\frac3p_p,∞, p∈(3,∞).

Citations

• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions
• Sur le mouvement d'un liquide visqueux emplissant l'espace
• Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces
• Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces
• Short time regularity of Navier-Stokes flows with locally L³ initial data and applications

Cited by

• Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes
• Full-Tail Dynamical Rigidity Forced by Atomic Navier-Stokes Energy Concentration

Related

• Localized smoothing and concentration for the Navier-Stokes equations in the half space
• The localized characterization for the singularity formation in the Navier-Stokes equations
• From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations
• Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations
• Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary