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.