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Stability of Nontrivial Solutions of the Navier–Stokes System on the Three Dimensional Torus

✍ Scribed by Piotr Bogusław Mucha

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
156 KB
Volume
172
Category
Article
ISSN
0022-0396

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✦ Synopsis

This paper examines the stability of nontrivial regular solutions to the Navier Stokes equations on a three dimensional torus. It is shown that the W 2, 1 r -norm of the perturbation can be controlled if its initial data are small enough in the L 2 -norm. A key element of the proof is to apply the Marcinkiewicz theorem to obtain the estimate for the Stokes system. In particular we prove the stability of unforced two dimensional flows.

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The problem of existence of regular (continuous, Hiilder continuous) solutions of the nonstationary Navier-Stokes system is an important one in modern mathematical physics. It is closely connected with two main issues: the uniqueness of the solution and the possibility to apply approximate methods i