The global attractor for the solutions to the 3D viscous primitive equations

Ju, Ning

doi:10.3934/dcds.2007.17.159

Cited by 113 publications

(142 citation statements)

References 26 publications

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“…The proof of (2.17)-(2.18) can be found in [12]; the higher-order results (2.19) can be found in [23]. Both these works followed [6] and [13].…”

Section: 2mentioning

confidence: 97%

“…8. As proved in [12,13,23], assuming sufficiently smooth forcing, the primitive equations admit a finite-dimensional global attactor. Theorem 2 states that, for ε ≤ ε * (|f | G σ ), the solution will enter, and remain in, an exponentially thin neighbourhood of U * (W 0< , f < ; ε) in L 2 (M) after some time.…”

mentioning

confidence: 99%

“…Alternately, one may view Theorem 2 as an extension to exponential order of the leading-order results of [2] for a closely related model. Finally, our results here put a strong constraint on the nature of the global attractor [12] in the strong rotation limit: the attractor will have to lie within an exponentially thin neighbourhood of the slow manifold.…”

Section: Introductionmentioning

confidence: 99%

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Stability of the Slow Manifold in the Primitive Equations

Témam¹,

Wirosoetisno²

2010

SIAM J. Math. Anal.

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Abstract. We show that, under reasonably mild hypotheses, the solution of the forced-dissipative rotating primitive equations of the ocean loses most of its fast, inertia-gravity, component in the small Rossby number limit as t → ∞. At leading order, the solution approaches what is known as "geostrophic balance" even under ageostrophic, slowly time-dependent forcing. Higherorder results can be obtained if one further assumes that the forcing is timeindependent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional "slow manifold" after some time.

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“…The proof of (2.17)-(2.18) can be found in [12]; the higher-order results (2.19) can be found in [23]. Both these works followed [6] and [13].…”

Section: 2mentioning

confidence: 97%

mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stability of the Slow Manifold in the Primitive Equations

Témam¹,

Wirosoetisno²

2010

SIAM J. Math. Anal.

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“…Modifying the argument in Ju (2007), one can prove that the uniform bound E 1 (U(t)) #Ñ 1 (S, U(0); m, «) also holds for the system (1). For the development in the next section, however, we need a stronger (as yet unproved) bound that is independent of «,…”

Section: Global Bounds and Attractorsmentioning

confidence: 99%

“…Further refinement by Ju (2007) gives us a uniform bound, 4 E 1 (U(t)) #Ñ 1 (S, U(0); m, «), valid for all t $ 0. When the forcing S does not depend on time, following works for the Navier-Stokes equations (see, e.g., Temam 1997), Ju proved that this implies the existence of the global attractor for the primitive equations.…”

Section: Global Bounds and Attractorsmentioning

confidence: 99%

Slow Manifolds and Invariant Sets of the Primitive Equations

Témam

Wirosoetisno

2011

Journal of the Atmospheric Sciences

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. ABSTRACTThe authors review, in a geophysical setting, several recent mathematical results on the forced-dissipative hydrostatic primitive equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence and regularity (smoothness) results and describing their implications for the long-time behavior of the solution. These results are used to show how the solution of the primitive equations in a periodic box comes close to geostrophic balance as t / '. Then a review follows of how geostrophic balance could be extended to higher orders in the Rossby number, and it is shown that the solution of the primitive equations also satisfies a higher-order balance up to an exponentially small error. Finally, the connection between balance dynamics in the primitive equations and its global attractor, which is the only known invariant set (for a sufficiently general forcing), is discussed.

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Pullback attractor for the three dimensional nonautonomous primitive equations of large‐scale ocean and atmosphere dynamics

You

2019

Comp and Math Methods

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The main objective of this paper is concerned with the existence of a finite fractal dimensional pullback attractor for the three‐dimensional nonautonomous primitive equations of large‐scale ocean and atmosphere dynamics. Under some slightly strong assumptions on the heat source, we prove the existence of a pullback attractor in a more regular phase space by asymptotical a priori estimates and provide the upper bound of the fractal dimension of the pullback attractor in V by calculating the Lyapunov exponent.

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The global attractor for the solutions to the 3D viscous primitive equations

Cited by 113 publications

References 26 publications

Stability of the Slow Manifold in the Primitive Equations

Stability of the Slow Manifold in the Primitive Equations

Slow Manifolds and Invariant Sets of the Primitive Equations

Pullback attractor for the three dimensional nonautonomous primitive equations of large‐scale ocean and atmosphere dynamics

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