No local double exponential gradient growth in hyperbolic flow for the 2d Euler equation

We consider smooth, double-odd solutions of the two-dimensional Euler equation in [-1, 1)^2 with periodic boundary conditions. This situation is a possible candidate to exhibit strong gradient growth near the origin. 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.