Global well-posedness and long-time dynamics for a higher order Quasi-Geostrophic type equation
In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier-Stokes-Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate. The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space $\R^2$: we prove that it is well-posed in $H^s$ for any $s\geq3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existence and uniqueness of weak solutions with different regularities (namely, $H^3$ and $H^4$ regularities); this fact can be viewed as a remainder of the so called BD-entropy structure of the original system. In the second part of the paper we investigate the long-time behaviour of these solutions. We show that they converge to the solution of the corresponding linear parabolic type equation, with same initial datum and external force. Our proof is based on dispersive estimates both for the solutions to the linear and non-linear problems.