Global existence for the primitive equations with small anisotropic viscosity

Charve, Frédéric; Ngo, Van-Sang

doi:10.4171/rmi/629

Cited by 24 publications

(41 citation statements)

References 33 publications

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“…For the viscid case in the periodic setting I. Gallagher in [25] achieved the result thanks to some normal form techniques introduced by S. Schochet in [40] which we will exploit as well in the following. We mention at last the work of F. Charve and V.-S. Ngo in [13] for the primitive equation in the whole space for F = 1 and anisotropic vanishing (horizontal) viscosity. We recall that the primitive equations and the rotating fluid system…”

Section: Introductionmentioning

confidence: 99%

Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

Scrobogna

2018

Rev. Mat. Iberoam.

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We prove that the primitive equations without vertical diffusivity are globally well-posed (if the Rossby and Froude number are sufficiently small) in suitable Sobolev anisotropic spaces. Moreover if the Rossby and Froude number tend to zero at a comparable rate the global solutions of the primitive equations converge globally to the global solutions of a suitable limit system. The space domain considered has to belong to a class of tori which is general enough to include all non-resonant tori and many resonant tori as well.IMB, UNIVERSITÉ DE BORDEAUX 351, cours de la Libération,

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Section: Introductionmentioning

confidence: 99%

Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

Scrobogna

2018

Rev. Mat. Iberoam.

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“…We refer to Remark 6 for the notion of well/ill-prepared initial data.Remark 2 As explained in [5,11] two distinct regimes have to be considered regarding the eigenvalues of the linearized system: the case F ∈]0, 1[ where the system features dispersive properties, and the case F = 1, with simpler operators but where no dispersion occurs. In the dispersive case (see [6] for weak solutions, [5] for strong solutions), using the approach developped by Chemin, Desjardins, Gallagher and Grenier in [14,15,16] for the rotating fluids system, we manage to filter the fast oscillations (going to zero in some norms thanks to Strichartz estimates providing positive powers of the small parameter ε) and prove the convergence to the solution of System (QG) below (even for blowing-up ill-prepared initial data as in [7,11], less regular initial data as in [8] or with evanescent viscosities as in [9]). On the contrary when F = 1 no dispersion is available and only well-prepared initial data are considered.…”

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confidence: 99%

Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion

Charve

2018

Communications in Mathematical Sciences

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J.-Y. Chemin proved the convergence (as the Rossby number ε goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity ν and the thermal diffusivity ν ′ are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure. 1 2 (global for small initial data). We refer to Remark 6 for the notion of well/ill-prepared initial data.Remark 2 As explained in [5,11] two distinct regimes have to be considered regarding the eigenvalues of the linearized system: the case F ∈]0, 1[ where the system features dispersive properties, and the case F = 1, with simpler operators but where no dispersion occurs. In the dispersive case (see [6] for weak solutions, [5] for strong solutions), using the approach developped by Chemin, Desjardins, Gallagher and Grenier in [14,15,16] for the rotating fluids system, we manage to filter the fast oscillations (going to zero in some norms thanks to Strichartz estimates providing positive powers of the small parameter ε) and prove the convergence to the solution of System (QG) below (even for blowing-up ill-prepared initial data as in [7,11], less regular initial data as in [8] or with evanescent viscosities as in [9]). On the contrary when F = 1 no dispersion is available and only well-prepared initial data are considered. In addition, in [13] the asymptotics are obtained only when ν and ν ′ are very close, in [20] is dealt the inviscid case. We refer also refer to [21,22,24] for results in other context such as periodic domains for example where there is no dispersion, and resonences have to be studied.

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“…This paper focuses on determining leading order asymptotics of (1.7). We consider the stably-stratified Boussinesq system (1.7) above, posed on D = R 2 × [0, 1] with either stress-free boundary conditions 13) or periodic boundary conditions…”

Section: Our Resultsmentioning

confidence: 99%

Vortices in stably-stratified rapidly rotating Boussinesq convection

Goh

Wayne

2019

Nonlinearity

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We study the Boussinesq approximation for rapidly rotating stably-stratified fluids in a three dimensional infinite layer with either stress-free or periodic boundary conditions in the vertical direction. For initial conditions satisfying a certain quasi-geostrophic smallness condition, we use dispersive estimates and the large rotation limit to prove global-in-time existence of solutions. We then use self-similar variable techniques to show that the barotropic vorticity converges to an Oseen vortex, while other components decay to zero. We finally use algebraically weighted spaces to determine leading order asymptotics. In particular we show that the barotropic vorticity approaches the Oseen vortex with algebraic rate while the barotropic vertical velocity and thermal fluctuations go to zero as Gaussians whose amplitudes oscillate in opposite phase of each other while decaying with an algebraic rate.(1.7) with∇ = (∂ 1 , ∂ 2 , ∂ 3 , 0) T , and PJ Ω,Γ P is the operator formed by the Coriolis, buoyancy, and gravity effects with,J Ω,Γ = ΩJ 0 0 ΓJ , J = 0 −1 1 0 .As discussed in [28] (and reviewed below), there is one physically relevant eigendirection, which we call the "quasi-geostrophic mode", which undergoes no dispersive smoothing. However, the other eigendirections correspond to rapidly oscillating waves known as "inertial" or "Poincaré waves". While these do not decay in L 2 they do decay in L p spaces with p > 2. This can be quantified with the

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Global existence for the primitive equations with small anisotropic viscosity

Cited by 24 publications

References 33 publications

Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion

Vortices in stably-stratified rapidly rotating Boussinesq convection

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