Euler's equations and the maximum principle

TL;DR

This paper employs the maximum principle to exclude discretely self-similar blow-up solutions in the Euler equations and MHD system under specific decay conditions, providing new a priori estimates for vorticity at infinity.

Contribution

It introduces explicit decay conditions near infinity that prevent discretely self-similar blow-up in Euler and MHD equations, extending previous results.

Findings

Discretely self-similar blow-up is excluded under certain decay conditions.

Triviality of discretely self-similar solutions in MHD is established.

A priori vorticity estimates are derived for the Euler equations.

Abstract

In this paper we use maximum principle in the far field region for the time dependent self-similar Euler equations to exclude discretely self-similar blow-up for the Euler equations of the incompressible fluid flows. Our decay conditions near spatial infinity of the blow-up profile are given explicitly in terms the coefficient in the equations. We also deduce triviality of the discretely self-similar solution to the magnetohydrodynamic system(MHD), under suitable decay conditions near spatial infinity than the previous one. Applying similar argument directly to the Euler equations, we obtain a priori estimate of the vorticity in the far field region.