On the strong solutions of the primitive equations in 2D domains

Guillén-González, Francisco; Rodríguez-Bellido, María Ángeles

doi:10.1016/s0362-546x(01)00773-8

Cited by 59 publications

(119 citation statements)

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“…With respect to regularity results, the existence of strong solutions (with H 2 (Ω)-regularity for the horizontal velocity) was treated in [Zia95] for the linear stationary case. For the nonlinear time-dependent problem, the existence and uniqueness of local in time strong solutions is given in [GMR01]. Finally, in [CT07] it is shown that this strong solution is global in time, imposing interesting physical boundary conditions which include wind driven stress on the surface and slip friction on the bottom.…”

Section: Preliminaries and Problem Settingmentioning

confidence: 99%

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

Guillén-González

Galván

2014

Numer. Math.

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The stability of velocity and pressure mixed finite-element approximations in general meshes of the hydrostatic Stokes problem is studied, where two "inf-sup" conditions appear associated to the two constraints of the problem; namely incompressibility and hydrostatic pressure. Since these two constraints have different properties, it is not easy to choose finite element spaces satisfying both. From the analytical point of view, two main results are established; the stability of an anisotropic approximation of the velocity (using different spaces for horizontal and vertical velocities) with piecewise constant pressures, and the unstability of standard (isotropic) approximations which are stable for the Stokes problem, like the mini-element or the Taylor-Hood element. Moreover, we give some numerical simulations, which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes.

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Section: Preliminaries and Problem Settingmentioning

confidence: 99%

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

Guillén-González

Galván

2014

Numer. Math.

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“…We give here an schedule of the result (see [9] for the details): Under the hypothesis of Theorem 4.…”

Section: Application To the Non-stationary Nonlinear Primitive Equationsmentioning

confidence: 99%

“…If we use the usual estimates in the Sobolev spaces, as for the Navier-Stokes equation, this lack of regularity of the vertical velocity does not allow to obtain strong regularity for (32). Searching for an alternative method, we separate the regularity in x and z, considering the anisotropic spaces and estimates that were introduced in [9] and that we overview here:…”

Section: There Exists a Unique Solution U Of Problem (30) Either mentioning

confidence: 99%

“…This model has been extensively studied from a mathematical point of view by several authors [12,13,3,2,10,8,9], who in particular have established existence and uniqueness results for the stationary and non-stationary problems.…”

Section: Introductionmentioning

confidence: 99%

“…-In Section 4 we apply the results obtained in Section 2 in order to obtain strong solutions for the non-stationary hydrostatic Stokes Problem, weakening the hypothesis on the time derivative for the wind stress tensor imposed in [9] (here, a very weak solution will be used to lift the time derivative of the Neumann data). Finally, the application to the nonlinear non-stationary problem (or non-stationary Primitive Equations) is sketched, rewriting the arguments used in [9].…”

Section: Introductionmentioning

confidence: 99%

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Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions

GUILLENGONZALEZ¹

2004

Annales de l?Institut Henri Poincare (C) Non Linear Analysis

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The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions.In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem.As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001) Guillén-González et al. / Ann. I. H. Poincaré -AN 21 (2004) [807][808][809][810][811][812][813][814][815][816][817][818][819][820][821][822][823][824][825][826] Dans ce papier, on introduit le problème dual qui est aussi un système de Stokes hydrostatique mais avec une condition de divergence non nulle. D'abord, on obtient la régularité forte pour le problème dual (ce résultat peut être consideré comme une généralisation des résultats de régularité pour le système de Stokes hydrostatique avec la condition de divergence nulle, obtenus par Ziane [Appl. Anal. 58 (1995)]). On montre ensuite l'existence et unicité de solution très faible pour le problème primal.Comme conséquence de ce résultat, on montre l'existence de solution forte pour le problème de Navier-Stokes hydrostatique non-stationnaire, avec une hypothèse sur la dérivée par rapport au temps du tenseur du vent plus faible que celle qui était imposée par Guillén-González, Masmoudi et Rodríguez-Bellido [Differential Integral Equations 50 (2001)].

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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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On the strong solutions of the primitive equations in 2D domains

Abstract: Comisión Interministerial de Ciencia y Tecnologí

Cited by 59 publications

References 8 publications

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions

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

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