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

TL;DR

This paper investigates locally self-similar solutions to the incompressible Euler equations, deriving pressure representations and proving nonexistence results for certain velocity profiles with non-decaying asymptotics.

Contribution

It introduces a new pressure representation formula and establishes nonexistence results for specific self-similar solutions with non-decaying velocity profiles.

Findings

Derived a meaningful pressure profile representation in terms of velocity profiles.

Proved nonexistence of certain nontrivial solutions with non-decaying asymptotics.

Analyzed energy behavior of potential velocity profiles.

Abstract

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