Dispersive effects of weakly compressible and fast rotating inviscid fluids

Van-Sang Ngo; Stefano Scrobogna

doi:10.3934/dcds.2018033

Article Contents

2018, Volume 38, Issue 2: 749-789. Doi: 10.3934/dcds.2018033

This issue Previous Article On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs Next Article N-barrier maximum principle for degenerate elliptic systems and its application

Dispersive effects of weakly compressible and fast rotating inviscid fluids

Van-Sang Ngo^1, , and
Stefano Scrobogna^2,3,

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
* Corresponding author: Van-Sang Ngo

Received: November 2016

Revised: August 2017

Published online: February 2018

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.

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Abstract

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.

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Mathematics Subject Classification: 35A01, 35A02, 35Q31, 76N10, 76U05.

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