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

2014/05/30 by Vu Hoang, Maria Radosz, Hoang, Vu +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1405.7756

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

arxiv created 2016/01/15 · arxiv updated 2016/01/19

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, γ}-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.

Related

• Maximal Double-Exponential Growth for the Euler Equation on the Half-Plane
• Double exponential growth of the vorticity gradient for the two-dimensional Euler equation
• Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner
• Ill-posedness for the incompressible Euler equations in critical Sobolev spaces
• Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow