Inviscid Limits and Regularity Estimates for the Solutions of the 2-D Dissipative Quasi-geostrophic Equations

We discuss two important topics of turbulence theory: inviscid limit and decay of the Fourier spectrum for the 2-D dissipative quasi-geostrophic (QGS) equations. In the first part we consider inviscid limits for both smooth and weak solutions of the 2-D dissipative QGS equations and prove that the classical solutions with smooth initial data tend to the solutions of the corresponding non-dissipative equations as the dissipative coefficient tends to zero. Here the convergence is in the strong L sense and we give the optimal convergence rate. For the weak solutions of the dissipative QGS equations with L initial data, we obtain weak L inviscid limit results. In the second part we use the methods of Foias-Temam [8] and Doering-Titi [7] developed for the Navier-Stokes equations to establish exponential decay of the spatial Fourier spectrum for the solutions of the dissipative QGS equations, but we treat general norms, and also our method of estimating the nonlinear terms is different.