# American Institute of Mathematical Sciences

February  2018, 38(2): 749-789. doi: 10.3934/dcds.2018033

## Dispersive effects of weakly compressible and fast rotating inviscid fluids

 1 Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, 76801 Saint-Etienne du Rouvray, France 2 IMB, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France 3 Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain

* Corresponding author: Van-Sang Ngo

Received  November 2016 Revised  August 2017 Published  February 2018

Fund Project: The research of the second author was partially supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.

We consider a system describing the motion of an isentropic, inviscid, weakly compressible, fast rotating fluid in the whole space $\mathbb{R}^3$, with initial data belonging to $H^s \left( \mathbb{R}^3 \right), s>5/2$. We prove that the system admits a unique local strong solution in $L^\infty \left( [0,T]; H^s\left( \mathbb{R}^3 \right) \right)$, where $T$ is independent of the Rossby and Mach numbers. Moreover, using Strichartz-type estimates, we prove the longtime existence of the solution, i.e. its lifespan is of the order of $\varepsilon^{-\alpha}, \alpha >0$, without any smallness assumption on the initial data (the initial data can even go to infinity in some sense), provided that the rotation is fast enough.

Citation: Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033
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