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

Abstract For any initial datum $$\theta _0\in L^{\frac{4}{3}}_x$$ θ 0 ∈ L x 4 3 it is proven that the existence of a global-in-time weak solution $$\theta \in L^\infty _t L^{\frac{4}{3}}_x$$ θ ∈ L t ∞ L x 4 3 to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $$\dot{H}^{-\frac{1}{2}}_x$$ H ˙ x - 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 $$L^{\frac{4}{3}}_x$$ L x 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.