Continuation of the zero set for discretely self-similar solutions to the Euler equations

TL;DR

This paper proves a unique continuation property for discretely self-similar solutions to the 3D Euler equations, showing that if the velocity vanishes on an open set, it must vanish everywhere, with implications for related MHD systems.

Contribution

The paper establishes a novel unique continuation theorem for discretely self-similar solutions of the Euler equations, extending understanding of solution behavior and regularity.

Findings

Velocity vanishing on an open set implies it vanishes everywhere.

The result applies to discretely self-similar solutions in 3D Euler equations.

Similar continuation property holds for the inviscid MHD system.

Abstract

We are concerned on the study of the unique continuation type property for the 3D incompressible Euler equations in the self-similar type form. Discretely self-similar solution is a generalized notion of the self-similar solution, which is equivalent to a time periodic solution of the time dependent self-similar Euler equations. We prove the unique continuation type theorem for the discretely self-similar solutions to the Euler equations in $\Bbb R^3$. More specifically, we suppose there exists an open set $G\subset \Bbb R^3$ containing the origin such that the velocity field $V\in C_s^1C^{2}_y (\Bbb R^{3+1})$ vanishes on $G\times (0, S_0)$, where $S_0 > 0$ is the temporal period for $V(y,s)$. Then, we show $V(y,s)=0$ for all $(y,s)\in \Bbb R^{3+1}$. Similar property also holds for the inviscid magnetohydrodynamic system