Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity

Cao, Chongsheng; Li, Jinkai; Titi, Edriss S.

doi:10.1016/j.jde.2014.08.003

Cited by 87 publications

(81 citation statements)

References 38 publications

(67 reference statements)

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“…The continuous dependence on the initial data, in particular the uniqueness, are straightforward corollary of Proposition 2.4 in [6]. This completes the proof.…”

Section: A Priori H 2 Estimates and Global Well-posednesssupporting

confidence: 61%

“…Due to the same reasons to those explained in [5,6], system (1.8)-(1.14) defined on Ω 0 is equivalent to the following system defined on Ω := M × (−h, h): Not that the restriction on the sub-domain Ω 0 of a solution (v, w, p, T ) to system (1.15)-(1.21) is a solution to the original system (1.8)-(1.14). Because of this, throughout this paper, we mainly concern on the study of system (1.15)-(1.21) defined on Ω, while the well-posedness results for system (1.8)-(1.14) defined on Ω 0 follow as a corollary of those for system (1.15)-(1.21).…”

Section: Introductionmentioning

confidence: 81%

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Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Cao

Titi

2015

Comm Pure Appl Math

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In this paper, we consider the initial boundary value problem of the three‐dimensional 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 the strong solution is established for any H2 initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L∞‐norm in terms of the Lq‐norms up to a logarithm of the Lp‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H2 estimates for global regularity.© 2016 Wiley Periodicals, Inc.

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“…The continuous dependence on the initial data, in particular the uniqueness, are straightforward corollary of Proposition 2.4 in [6]. This completes the proof.…”

Section: A Priori H 2 Estimates and Global Well-posednesssupporting

confidence: 61%

Section: Introductionmentioning

confidence: 81%

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Cao

Titi

2015

Comm Pure Appl Math

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“…According to the weight in the vertical diffusion terms, we also introduce the weighted norms 4 be nonnegative and let the given velocity field (v h , ω) satisfy the above assumptions (2.8)-(2.11). Then, for any (arbitrarily large) T ∈ (0, ∞), there exists a unique nonnegative solution (T, q v , q c , q r ), with initial value (T 0 , q v0 , q c0 , q r0 ), of the boundary value problem (1.25)-…”

Section: 2mentioning

confidence: 99%

“…Proposition 3.2. Let T ∈ (0, ∞) and (T, q v , q c , q r ) be a solution to (1.25)-(1.30) in M × (0, T ) subject to the boundary conditions (2.5)-(2.7) with non-negative intial data 4 , satisfying the regularities stated in Proposition 3.1 by replacing T 0 with T . Then for every t…”

Section: 2mentioning

confidence: 99%

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes

Hittmeir

Klein

et al. 2017

Nonlinearity

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Abstract. We study a moisture model for warm clouds that has been used by Klein and Majda in [21] as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler [20] and Grabowski and Smolarkiewicz [15], in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation.In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos [1] to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms.

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“…ν H > 0 and ν 3 > 0) and only partial anisotropic vertical diffusion (i.e. κ H = 0 and κ 3 > 0) which stands for the vertical eddy heat diffusivity turbulence mixing coefficient; see also [8] for the case when ν H > 0, ν 3 > 0, κ H > 0 and κ 3 = 0, and [9] for the case when ν H > 0, κ H > 0 and ν 3 = 0, κ 3 = 0 (see, e.g., [16], [17], for the geophysical justification). In the above results, the Coriolis forcing term, with rotation parameter R, did not play any role in proving the global regularity.…”

Section: Introductionmentioning

confidence: 99%

Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

Cao

Ibrahim

Nakanishi

et al. 2015

Commun. Math. Phys.

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107

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Abstract. 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, if they exist, they blow up in finite time. Specifically, we consider the three-dimensional 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.MSC Subject Classifications: 35Q35, 65M70, 86-08,86A10.

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Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity

Abstract: Abstract. In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H 2 initial data.

Cited by 87 publications

References 38 publications

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes

Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

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