A Semi Discrete Dynamical System for a 2D Dissipative Quasi Geostrophic Equation

A semi-discretization in time, according to a full implicit Euler scheme, for a 2D dissipative quasi geostrophic equation, is studied. We prove existence, uniqueness and regularity results of the solution to the predicted discretization, in the subcritical case for any initial data in ${\dot{L}}^2$. Hence, we define an infinite semi-discrete dynamical system, then we prove the existence and the regularity of the corresponding global attractor, for a source term $f$ in ${\dot{L}}^{p_{\alpha }}$, for a fixed $p_{\alpha }=\frac{2}{1-\alpha }$.