Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity

2004/06/15 by Isabelle Gallagher, Gallagher, Isabelle, Thierry Gallay +1 · 4 citations
Engineering · Mathematics · Physics and Astronomy · #35Q30 #76D03 #76D05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math-ph #math.AP #math.MP #msc:35Q30 #msc:76D03 #msc:76D05

paper · pdf · doi:10.48550/arxiv.math/0406297

32 pages

arxiv created 2004/06/15 · openalex publication_date 2004/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L¹(R²) for positive times is entirely determined by the trace of the vorticity at t = 0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa, and Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R² is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.

Citations

• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions

Cited by

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• Harnack inequalities for nonlocal operators with supercritical drifts and their applications
• Long time evolution of a pair of 2D viscous point vortices
• Existence and stability of Sadovskii vortices: from patch to smooth vortices

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