Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation

Abstract

For any initial datum ${\theta }_0\in L_x^{\frac{4}{3}}$ it is proved the existence of a global-in-time weak solution $\theta \in L_t^{\infty }{L}_x^{\frac{4}{3}}$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the ${\dot{H}}_x^{-\frac{1}{2}}$ norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the $L_x^{\frac{4}{3}}$ norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.

20 pages. Final version accepted on Arch. Ration. Mech. Anal