Title: Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation Show Chinese title

Title: 全局存在性，哈密顿守恒和粘性消失的表面准地转方程

Abstract: For any initial datum $\theta_0\in L^{\frac{4}{3}}_x$ it is proved the existence of a global-in-time weak solution $\theta \in L^\infty_t L^{\frac43}_x$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $\dot{H}^{-\frac{1}{2}}_x$ 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 $L^{\frac{4}{3}}_x$ 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. Show Chinese abstract

Abstract: 对于任何初始数据$\theta_0\in L^{\frac{4}{3}}_x$，证明了表面准地转方程存在全局时间的弱解$\theta \in L^\infty_t L^{\frac43}_x$，其哈密顿量，即$\dot{H}^{-\frac{1}{2}}_x$范数，在时间上是常数。 该解是作为粘性消失极限得到的。 在经典的强紧性设置之外，主要思想是传播初始数据的$L^{\frac{4}{3}}_x$范数的时间非集中性，从而推导出哈密顿量范数中的强紧性。 在最小 Onsager 超临界假设下，也获得了普遍的无异常耗散结果。