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

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 ${\mathbb{R}}^3$. More specifically, we suppose there exists an open set $G\subset {\mathbb{R}}^3$ containing the origin such that the velocity field $V\in C_s^1{C}_y^2({\mathbb{R}}^{3+1})$ 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 $(y,s)\in {\mathbb{R}}^{3+1}$. Similar property also holds for the inviscid magnetohydrodynamic system

16 pages. arXiv admin note: text overlap with arXiv:1308.1051

References in corpus (2)

• Nonexistence of self-similar singularities for the 3D incompressible Euler equations
• On formation of a locally self-similar collapse in the incompressible Euler equations

Cited by in corpus (1)

• Discretely self-similar singular solutions for the incompressible Euler equations