Global regularity for the hyperdissipative Navier-Stokes equation below\n the critical order

2019/11/06 by Maria Colombo, Colombo, Maria, Silja Haffter +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1911.02600

openalex publication_date 2019/11/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We consider solutions of the Navier-Stokes equation with fractional\ndissipation of order \α\≥ 1. We show that for any divergence-free\ninitial datum u₀ such that ||u₀||_H^\δ \≤ M, where M is\narbitrarily large and \δ is arbitrarily small, there exists an explicit\n\ε=\ε(M, \δ)>0 such that the Navier-Stokes equations with\nfractional order \α has a unique smooth solution for \α \∈\n(\(5)/(4)-\ε, \(5)/(4)]. This is related to a new stability result\non smooth solutions of the Navier-Stokes equations with fractional dissipation\nshowing that the set of initial data and fractional orders giving rise to\nsmooth solutions is open in H^5/4 \× ( frac 34, \(5)/(4)].\n

Cited by

• On Bridging Analyticity and Sparseness in Hyperdissipative Navier-Stokes Systems
• Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades

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