Global Existence and Long Time Asymptotics for Rotating Fluids in a 3D layer

Time Asymptotics

Description

Niveau: Supérieur, Doctorat, Bac+8
Global Existence and Long-Time Asymptotics for Rotating Fluids in a 3D layer Thierry Gallay Institut Fourier (UMR CNRS 5582) Universite de Grenoble I B.P. 74 38402 Saint-Martin-d'Heres, France Violaine Roussier-Michon Institut de Mathematiques de Toulouse (UMR CNRS 5219) INSA Toulouse 135 av Rangueil 31077 Toulouse cedex 4, France December 5, 2008 Abstract The Navier-Stokes-Coriolis system is a simple model for rotating fluids, which allows to study the influence of the Coriolis force on the dynamics of three-dimensional flows. In this paper, we consider the NSC system in an infinite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocity parameter is sufficiently large, depending on the initial data, we prove the existence of global, infinite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb-Oseen vortices as t? ∞. 1

• velocity ?


• u˜ ?


• large initial


• velocity field


• sufficiently large


• nsc system


• dimensional vector


• navier-stokes system


• has no physical

Sujets

• Gallagher
• Asymptotic analysis
• Institut de Mathématiques de Toulouse
• Flow velocity
• Sufficiently large

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 33
• Langue: English

Extrait

Global Existence and Long-Time Asymptotics for Rotating Fluids in a 3D layer

Thierry Gallay Institut Fourier (UMR CNRS 5582) UniversitedeGrenobleI B.P. 74 38402Saint-Martin-d’Heres,France [e-mail]

Violaine Roussier-Michon InstitutdeMathematiquesdeToulouse(UMRCNRS5219) INSA Toulouse 135 av Rangueil 31077 Toulouse cedex 4, France [e-mail]

December 5, 2008

Abstract

TheNavier-Stokes-Coriolissystemisasimplemodelforrotatinguids,whichallowsto study the in uence of the Coriolis force on the dynamics of three-dimensional o ws. In this paper, we consider the NSC system in an innite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocityparameterissucientlylarge,dependingontheinitialdata,weprovetheexistence of global, innite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb-Oseen vortices ast→ ∞.

1

1

Introduction

Inrecentyearsalotofactivityhasbeendevotedtothemathematicalstudyofgeophysicalows, andinparticulartovariousmodelsofrotatinguids.Takingadvantageofthestraticationeect due to the Coriolis force, signican t results have been obtained which are still out of reach for the usual Navier-Stokes system, such as global existence of solutions for large initial data [1, 3] and stability of boundary layers for small viscosities [10, 15]. We refer the interested reader to therecentmonograph[4]whichcontainsageneralintroductiontogeophysicalows,anoverview of the mathematical theory, and an extensive bibliography. In this article we study the so-called Navier-Stokes-Coriolis (NSC) system in a three-dimen- sionallayerdelimitedbytwoinnitehorizontalplanes,assumingasusualthattherotationvector is constant and aligned with the vertical axis. This is a reasonably simple model for the motion of the ocean in a small geographic zone at mid-latitude, where the variation of the Coriolis force due to the curvature of Earth can be neglected. More realistic systems exist which take into accountthevariationsoftemperatureandsalinityinsidetheocean,andincludeboundaryeects modelling the in uence of coasts, the topography of the bottom, or the action of the wind at thefreesurface,see[9,17].Nevertheless,keepingonlytheCoriolisforceismeaningfulinarst approximation, because its eect is very important on the ocean’s motion at a global scale due to the fast rotation of Earth compared to typical velocities in the ocean. Our main goal is to investigate the long-time behavior of the solutions to the NSC system foraxed,buttypicallylarge,valueoftherotationspeed.Asin[1,3]weshallusetheeect of the Coriolis force to prove global existence of solutions for large initial data, but the long- time asymptotics of those solutions turn out to be essentially two-dimensional and are therefore not aected by the rotation. Thus we shall recover as a leading term in our expansion the Lamb-Oseen vortex which plays a similar role for the usual Navier-Stokes system in the plane R 2 [8] or the three-dimensional layerR 2 (0 To1) [19]. avoid all problems related to boundary layers,weshallalwaysassumethattheuidmotionisperiodicin the vertical direction. This hypothesishasnophysicaljusticationandisonlyaconvenientmathematicalwaytodisregard the in uence of the boundaries. Although boundary conditions do play an important role in the problem we study and will have to be considered ultimately, in this paper we chose to focus on the motion of the uid in the bulk. We thus consider the Navier-Stokes-Coriolis system in the three-dimensional layerD=R 2  T 1 , whereT 1 =R/Z points of Theis the one-dimensional torus.Dwill be denoted by (x z), wherex= (x 1  x 2 )∈R 2 is the horizontal variable andz∈T 1  Theis the vertical coordinate. system reads ∂ t u+ (u
r)u+ e 3 ∧u
u rp divu= 0(1) = whereu=u(t x z)∈R 3 ehevitstyelociatnhdeludido,fp=p(t x z)∈Ris the pressure eld. Here and in what follows, it is understood that dieren tial operators such asror
act on all spatial variables (x z), unless otherwise indicated. System (1) diers from the usual incompressible Navier-Stokes equations by the presence of the Coriolis term e 3 ∧u, where ∈Ris a parameter ande 3 = (001) t  term is Thisis the unit vector in the vertical direction. due to the fact that our reference frame rotates with constant angular velocity /2 around the verticalaxis.Notethat(1)doesnotcontainanycentrifugalforce,becausethiseectcanbe included in the pressure termrpikehamenvcitocsitysithofuieasdh.Forsimplicity,tebne rescaled to 1, and the uid density has been incorporated in the denition of the pressurep. As in the ordinary Navier-Stokes system, the role of the pressure in (1) is to enforce the incompressibility condition divu= 0. eliminate the pressure, one can apply to both sides the To Leray projectorP, which is just the orthogonal projector inL 2 (D) 3 onto the space of divergence-

2