Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner

TL;DR

This paper investigates the growth of vorticity gradients in 2D Euler flows around a corner, establishing that under certain symmetry conditions, the vorticity's Lipschitz estimate grows at most exponentially near stagnation points.

Contribution

It provides a new bound on vorticity gradient growth for 2D Euler flows with symmetry in a corner domain, improving understanding of flow regularity near stagnation points.

Findings

Vorticity Lipschitz estimate grows at most exponentially near stagnation points.

Symmetry conditions influence vorticity behavior in corner flows.

The result applies to flows in a specific square domain with hyperbolic structure.

Abstract

In this paper, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_1,x_2):0<x_1+x_2<\sqrt{2},0<-x_1+x_2<\sqrt{2}\}$ is considered. It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.