Dynamics of Partial Differential Equations

Volume 12 (2015)

Number 4

On the locally self-similar singular solutions for the incompressible Euler equations

Pages: 321 – 342

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n4.a2

Author

Liutang Xue (School of Mathematical Sciences, Beijing Normal University, Beijing, China; Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, China; and Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, Marne-la-Vallée, France)

Abstract

In this paper we consider the locally backward self-similar solutions for the Euler system, and specially focus on the case that the possible nontrivial velocity profiles have non-decaying spatial asymptotics. We derive the representation formula of the pressure profile in terms of the velocity profiles in such a situation, and by using it and the local energy inequality of the profiles, we prove some nonexistence results and show the energy behavior concerning these possible velocity profiles.

Keywords

Euler equations, backward self-similar solutions, nonexistence, energy behavior

2010 Mathematics Subject Classification

35B99, 35L67, 35Q35, 76Bxx

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Published 10 December 2015