Discretely self-similar singular solutions for the incompressible Euler equations

TL;DR

This paper investigates discretely self-similar singular solutions of the incompressible Euler equations, analyzing their asymptotic behaviors, proving nonexistence results, and characterizing pressure profiles in relation to velocity profiles.

Contribution

It introduces new nonexistence results for certain self-similar solutions and provides a representation formula for pressure profiles with non-decaying asymptotics.

Findings

Proved nonexistence of some discretely self-similar solutions.

Characterized energy behavior of nontrivial velocity profiles.

Derived pressure representation formula for non-decaying asymptotics.

Abstract

In this article we consider the discretely self-similar singular solutions of the Euler equations, and the possible velocity profiles concerned not only have decaying spatial asymptotics, but also have unconventional non-decaying asymptotics. By relying on the local energy inequality of the velocity profiles and the bootstrapping method, we prove some nonexistence results and show the energy behavior of the possible nontrivial velocity profiles. For the case with non-decaying asymptotics, the needed representation formula of the pressure profile in terms of velocity profiles is also given and justified.