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} $.