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

For any initial datum ${\theta }_0\in L_x^{\frac{4}{3}}$ it is proved the existence of a global-in-time weak solution $\theta \in L_t^{\mathrm{\infty }}{L}_x^{\frac{4}{3}}$ 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. Outside the classical strong compactness setting, 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 strong compactness in the Hamiltonian norm is deduced. General no anomalous dissipation results under minimal Onsager supercritical assumptions are also obtained.