The centrally symmetric V-states for active scalar equations. Two-dimensional Euler with cut-off

Abstract

We consider the truncated equation for the centrally symmetric $V$-states in the 2d Euler dynamics of patches and prove the existence of solutions $y(x,\lambda ),x\in [-1,1],\lambda \in (0,{\lambda }_0)$. These functions $y(x,\lambda )\in C^{\infty }(-1,1)$ and $\Vert y(x,\lambda )-|x|{\Vert }_{C[-1,1]}\to 0,\lambda \to 0$.

Several typos are fixed and explanation is improved