No Local Double Exponential Gradient Growth in Hyperbolic Flow for the Euler equation

Abstract

We consider smooth, double-odd solutions of the two-dimensional Euler equation in $[-1,1{)}^2$ with periodic boundary conditions. It is tempting to think that the symmetry in the flow induces possible double-exponential growth in time of the vorticity gradient at the origin, in particular when conditions are such that the flow is "hyperbolic". This is because examples of solutions with $C^{1,\gamma }$-regularity were already constructed with exponential gradient growth by A. Zlatos. We analyze the flow in a small box around the origin in a strongly hyperbolic regime and prove that the compression of the fluid induced by the hyperbolic flow alone is not sufficient to create double-exponential growth of the gradient.

44 pages, 4 figures. Fixed some typos, improved heuristic discussion, some remarks added

References in corpus (2)

• Infinite superlinear growth of the gradient for the two-dimensional Euler equation
• A model for studying double exponential growth in the two-dimensional Euler equations

Cited by in corpus (1)

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