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 @math : we prove that it is well-posed in @math for any @math , 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, @math and @math 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.
The proof of higher regularity energy estimates (namely, for @math ) relies on a paralinearization of the convective term and a special decomposition for treating it, which has already been used in @cite_9 : in particular, thanks to this approach we are able to reproduce the special cancellations in the transport operator, up to some remainders; now, a careful analysis of these remainder terms allows us to control them by the @math -energy of the solution @math , so that one can conclude by an application of the Gronwall lemma. On the contrary, propagation of intermediate regularities (namely for @math ) is surprisingly more involved, since now the special cancellations concern only the lower order item in the convective term, and no more @math , which hence needs to be controlled very carefully. This can be done by resorting once again to a paralinearization of the transport term and to delicate estimates concerninig the commutators involved in the computations; eventually, we manage to bound all the terms, and Gronwall lemma allows us to close the estimates as before.
@cite_9
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Abstract In [18] Paicu and Zarnescu have studied an order tensor system which describes the flow of a liquid crystal. They have proven the existence of weak solutions, the propagation of higher regularity, namely H s with s > 1 and the weak-strong uniqueness in dimension two. This paper is devoted to fill the gap of their results, namely to propagate the low regularity, namely H s for 0 s 1 and to prove the uniqueness of the weak solutions. For the completeness of this research, we also propose an alternative approach in order to prove the existence of weak solutions.