Global existence for the primitive equations with small anisotropic viscosity

Frederic Charve?

Description

Global existence for the primitive equations with small anisotropic viscosity Frederic Charve?, Van-Sang Ngo† Resume: Dans cet article, nous considerons le systeme des equations prim- itives dans le cas ou la viscosite verticale et la diffusivite thermique verticale sont nulles, la viscosite horizontale et la diffusivite thermale horizontale sont de taille ?? avec 0 < ? < ?0. Nous demontrons l'existence globale et l'unicite d'une so- lution forte pour des donnees grandes a condition de supposer que le nombre de Rossby est petit (c'est-a-dire que la rotation est rapide et la stratification verticale de la densite est forte). Classification AMS: 35Q35, 35S30, 76D05, 76U05 Mots cles: Equations primitives, systeme quasi-geostrophique, anisotropie, dispersion, estimations de Strichartz. Abstract: In this paper, we consider the primitive equations with zero vertical viscosity, zero vertical thermal diffusivity, and the horizontal viscosity and horizontal thermal diffusivity of size ?? where 0 < ? < ?0. We prove the global existence of a unique strong solution for large data provided that the Rossby number is small enough (the rotation and the vertical stratification are large). AMS Classification: 35Q35, 35S30, 76D05, 76U05 Key words: Primitive equations, quasi-geostrophic system, anisotropy, disper- sion, Strichartz estimates.

• u05 key words


• viscosite horizontale


• horizontal viscosity


• vertical thermal


• ams classification


• ∂tu˜qg ?


• existence globale


• global weak


• ∂1∆f ?1?

Sujets

• Charles de Gaulle
• maths

Informations

• Publié par: pefav
• Nombre de lectures: 24
• Langue: English

Extrait

Global existence for the primitive equations with small anisotropic viscosity Fr´ed´ericCharve, ∗ Van-Sang Ngo †

Resume´:nolssesynois´dre´equatiot`emedesrpsn-minsDacitrateccsuon,el ´ itivesdanslecasou`laviscosite´verticaleetladiffusivite´thermiqueverticalesont nulles,laviscosit´ehorizontaleetladiffusivite´thermalehorizontalesontdetaille ε α avec 0< α < α 0 -rtnome´dixe’lsnousNo.ic´tu’inenosdeu’ceglsteneetlobal lutionfortepourdesdonne´esgrandesa`conditiondesupposerquelenombrede Rossbyestpetit(c’est-a`-direquelarotationestrapideetlastratificationverticale deladensite´estforte). Classification AMS:35Q35, 35S30, 76D05, 76U05 Motscle´s:osrta,inqieuorhp,opierisptimis,vestsyeme`sauq´g-itsoeqEauitno dispersion, estimations de Strichartz. Abstract:In this paper, we consider the primitive equations with zero vertical viscosity, zero vertical thermal diffusivity, and the horizontal viscosity and horizontal thermal diffusivity of sizeε α where 0< α < α 0  prove the. We global existence of a unique strong solution for large data provided that the Rossby number is small enough (the rotation and the vertical stratification are large). AMS Classification:35Q35, 35S30, 76D05, 76U05 Key words:Primitive equations, quasi-geostrophic system, anisotropy, disper- sion, Strichartz estimates.

1 Introduction In this paper, we consider the primitive equations with no vertical viscosity and no vertical thermal diffusivity and we also suppose that the horizontal viscosity and thermal diffusivity go to zero when the rotation goes to infinity. ∗ vinU)CL,U(EPetlirCe´-Estarist´ePersieeyseMtdh´atatemrobaiota’derlanAqieus Applique´es(UMR8050),61AvenueduGe´ne´raldeGaulle,94010Cre´teilCedex(France). E-mail: [e-mail] † sAnalyseeatoired’metaqieudtMeta´h´eCrst-EisarePt´robaL,)CEPU(lietreisnUvi ees Appliqu´(UMR8050),61AvenueduG´ene´raldeGaulle,94010Cr´eteilCedex(France). E-mail: [e-mail] 1

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We prove the convergence towards the quasi-geostrophic system and the global existence of a unique strong solution when the rotation is fast enough. The primitive equations describe the hydrodynamical flow in a large scale (of order of hundreds or thousands of kilometers) on the earth, typically an ocean or the atmosphere, under the assumptions that vertical motion is much smaller than horizontal motion and that the fluid layer depth is small compared to the radius of the earth. Concerning the difference between the vertical and horizontal scales, it is also observed that for geophysical fluids, the vertical component of the diffusion term (viscosity or thermal diffusiv- ity in the case of primitive equations) is much smaller than the horizontal component. In the case of a rotating fluid between two planes for example (see [26] and [16]), the viscosity has the form (−ν h Δ h −βε∂ 32  is then). It relevant to consider a zero vertical diffusivity in the primitive equations. In the studies of the fluids of that scale, two important phenomena have to be considered: the earth rotation and the vertical stratification of the density due to the gravity. When the movement is observed in a frame located at the surface of the Earth, the rotation of the earth, defined by the Rossby numberRo the, induces two additional force terms in the equations: Coriolis force and the centrifugal force. The latter is included in the gravity gradient term in the right-hand side of the equations and has no important influence in our work. The Coriolis force, on the other hand, induces a vertical rigidity in the fluid. Under a fast rotation, the velocity of each particle of an homogeneous fluid, which has the same horizontal coordinates, is the same. This is called the phenomenon of Taylor-Proudman columns. Gravity forces the fluid masses to have a vertical structure: heavier layers lay under lighter ones. Internal movements in the fluid tend to destroy this structure and the gravity basically tries to restore it, which gives a horizontal rigidity (to be opposed to the vertical rigidity induced by the rotation). In order to formally estimate the importance of this rigidity, we also compare thetypicaltimescaleofthesystemwiththeBrunt-V¨ais¨al¨afrequencyand define the Froude numberF r. We will not give more details here and, for more physical considerations, we refer to [17], [18], [39], and [6] for example. The primitive equations are obtained with moment, energy, mass conser- vations and scale simplifications by choosing the same scale for the rotation and stratification (see Embid and Majda [21]). In what follows, we denote byεthe Rossby number and we setF r=εF, whereF ris the Froude num- ber, andε > the following, we call In0 will go to zero.εandFthe Rossby and Froude numbers. Then, the anisotropic primitive equations are given

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by  ∂ v 1 +v ε .rv ε 1 −νΔ h v ε 1 − ε 1 v ε 2 =− ∂ 1 ε Φ ε inR + ×R 3 t ε ∂ t v ε 2 +v ε .rv ε 2 −νΔ h v ε 2 1 +v ε ε 1 =− ∂ 2 ε Φ ε inR + ×R 3 (AP E ε )   ∂ t v ε 3 +v ε .rv ε 3 −νΔ h v ε 3 + F 1 θ ε ε =− ∂ 3 ε Φ ε inR + ×R 3 ∂ t θ ε +v ε .rθ ε −ν 0 Δ h θ ε − F 1 ε v ε 3 = 0 inR + ×R 3 div v ε = 0 inR + ×R 3   (v ε , θ ε ) | t=0 = (v 0 , θ 0 ) inR 3 , where Δ h =∂ x 2 1 +∂ x 2 2 is the horizontal Laplacian, and whereνandν 0 are the horizontal viscosity and the horizontal thermal diffusivity, which are of orderε α , whereαis a positive constant which will be made more precise in the last section. The vector fieldv ε (t, x) is the fluid velocity, θ ε (t, xis the (scalar) density fluctuation and Φ) ε (t, x) is the geopotential term, containing the pressure and the centrifugal force. Notice that, in the case of meteorology problems,θ ε is only a function of the temperature whereas, in oceanography,θ ε depends on both temperature and salinity. For more details on the physical meanings of the quantities, we refer the reader to [6], [18], [25] and [39]. DenotingU ε the pair (v ε , θ ε ), we can write the system (AP E ε ) in the more compact form  rU ε −LU ε + 1 AU ε ( 1 =−rΦ ε ,0) (AP E ε )   di ∂ t U v ε v ε = +U ε 0 εε .   U ε | t=0 =U 0,ε . where 3 U ε .rU ε =v ε .rU ε = X v ε i .∂ i U ε , i=1 the matrixAis defined by  0− 01 0 A=   0 0 0 0 0 1 −F 0 0 −1 F 0 0 −1    andLdenotes the linear operator LU ε = (νΔ h v ε , ν 0 Δ h θ ε ).

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We introduce the potential vorticity: Ω(U ε ) d = ef ∂ 1 v ε 2 −∂ 2 v ε 1 −F ∂ 3 θ ε . Then we define the orthogonal decomposition ofU ε into its quasi-geostrophic part, and its oscillating part : U ε =U ε,QG + (U ε −U ε,QG ) d ≡ ef U ε,QG +U ε,osc , where U ε,QG d = ef    − ∂ ∂ 1 2 Δ Δ F F − − 1 1 Ω( Ω( U U ε ) ε )   , U ε,osc d = ef    θ v v ε ε ε 1 2 + + − ∂ ∂ ∂ F 3 2 1 Δ Δ v Δ F F ε 3 F − − −1 1 1 (Ω Ω( Ω( U U U ε ε ε ) ) )    , 0  −F ∂ 3 Δ F −1 Ω(U ε ) and Δ F d = ef ∂ 2 1 +∂ 2 2 +F 2 ∂ 3 3 . Remark 1.1We refer to [8] to [10] for general properties of the potential vorticity, but we recall that Ω(U ε,osc ) = 0 and Ω(U ε,QG ) = Ω(U ε ). In[39],assumingthattheBrunt-V¨ais¨al¨afrequencyisconstant,theau- thors have shown that, whenεgoes to zero, the formal limit of the system (AP E ε ) is the following quasi-geostrophic system: (QG)    ∂ t U g QG −ΓU g QG =−    − − F ∂ 0 ∂ 1 ∂ 3 2    Δ F −1  U QG ∙ rΩ(U QG )  g g g divU QG = 0 g   U QG | t=0 =U 0,QG , whereU g QG ∙ r d = ef v Q g G ∙ r, Γ is the operator: Γ d = ef Δ h Δ F −1  ν∂ 1 2 +ν∂ 2 2 +ν 0 F 2 ∂ 2 3  , andU 0,QG is the quasi-geostrophic part of the initial dataU 0 . In real observations, the rotation is usually not fast enough and the fluid not homogeneous enough, but this quasi-geostrophic system (which consid- ers not only rotation terms but also vertical stratification terms) is still a very good approximation of the behavior of real geophysical fluids. From the mathematical point of view, the quasi-geostrophic system has an “almost”