Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics (original) (raw)
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Communications on Pure and Applied Mathematics, 2015
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any H 2 initial data. An N-dimensional logarithmic Sobolev embedding inequality, which bounds the L ∞ norm in terms of the L q norms up to a logarithm of the L p-norm, for p > N , of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori H 2 estimates for the global regularity.
Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
arXiv (Cornell University), 2014
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any H 2 initial data. An N-dimensional logarithmic Sobolev embedding inequality, which bounds the L ∞ norm in terms of the L q norms up to a logarithm of the L p-norm, for p > N , of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori H 2 estimates for the global regularity.
Journal of Functional Analysis, 2017
In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data (v 0 , T 0) ∈ H 1 , by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data (v 0 , T 0) ∈ H 1 ∩ L ∞ , such that ∂ z v 0 ∈ L m , for some m ∈ (2, ∞), by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in [Cao, C.; Li, J.; Titi, E. S.: Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., 69 (2016), 1492-1531.], where the initial data (v 0 , T 0) was assumed to have H 2 regularity.
SIAM Journal on Mathematical Analysis, 2019
We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.
arXiv (Cornell University), 2017
In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity v 0 ∈ H 2 (Ω) and the initial temperature T 0 ∈ H 1 (Ω)∩L ∞ (Ω) with ∇ H T 0 ∈ L q (Ω), for some q ∈ (2, ∞). Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to H 2 (Ω). The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the " mismatching " of regularities between the horizontal momentum and temperature equations. To handle this "mismatching" of regularities, we introduce several auxiliary functions, i.e., η, θ, ϕ, and ψ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity v or its vertical derivative ∂ z v. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some L 1 t (W 1,∞ x)type a priori estimates on v, we decompose the velocity into the "temperatureindependent" and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Brézis-Gallouet-Wainger and Beale-Kato-Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work [12], is used, and a new logarithmic type Gronwall inequality is exploited.
Journal de Mathématiques Pures et Appliquées, 2008
In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be nonlocal. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.
Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics
Communications in Mathematical Physics, 2015
In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the threedimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.
Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations
Archive for Rational Mechanics and Analysis, 2021
This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.