The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Li, Jinkai; Titi, Edriss S.; Yuan, Guozhi

doi:10.1016/j.jde.2021.10.048

Cited by 23 publications

(21 citation statements)

References 49 publications

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“…In order to prove the Theorem 2.2, we need the following proposition, which is a direct consequence of Lemma 2.2 with exponent (6, 6, 2) in [34].…”

Section: Strong Convergence For H 1 Initial Data With Additional Regu...mentioning

confidence: 99%

“…The small aspect ratio limit from the Navier-Stokes equations to the primitive equations was studied first by Azérad-Guillén [1] in a weak sense, then by Li-Titi [32] in a strong sense with error estimates, and finally by Furukawa et al [15] in a strong sense but under relaxing the regularity on the initial condition. Subsequently, the strong convergence of solutions of the scaled Navier-Stokes equations to the corresponding ones of the primitive equations with only horizontal viscosity was obtained by Li-Titi-Yuan [34]. Furthermore, the rigorous justification of the hydrostatic approximation from the scaled Boussinesq equations to the primitive equations with full viscosity and diffusivity was obtained by Pu-Zhou [39].…”

Section: Introductionmentioning

confidence: 99%

“…The rigorous mathematical derivation for the governed equations describing the motion of stable stratified fluid, i.e., the viscous primitive equations with density stratification, is due to the work of Pu-Zhou [40]. Based on the ideas that follow from Li-Titi [32], Li-Titi-Yuan [34], and Pu-Zhou [40], We study the hydrostatic approximation of the Boussinesq equations with rotation in a thin domain.…”

Section: Introductionmentioning

confidence: 99%

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The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

Pu¹,

Zhou²

2022

Preprint

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In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is obtained under the assumption of initial data (v0, T0) ∈ H 1 with ∂zv0 ∈ L 4 . Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, in the cases of H 1 initial data, H 1 initial data with additional regularity ∂zv0 ∈ L 4 and H 2 initial data, respectively, as the aspect ration parameter λ goes to zero, and the rate of convergence is of the order O(λ η/2 ) with η = min{2, β − 2, γ − 2}(2 < β, γ < ∞). The convergence result implies a rigorous justification of the hydrostatic approximation.

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“…In order to prove the Theorem 2.2, we need the following proposition, which is a direct consequence of Lemma 2.2 with exponent (6, 6, 2) in [34].…”

Section: Strong Convergence For H 1 Initial Data With Additional Regu...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

Pu¹,

Zhou²

2022

Preprint

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show abstract

“…The small aspect ratio limit from the Navier-Stokes equations to the primitive equations was studied first by Azérad-Guillén [1] in a weak sense, then by Li-Titi [28] in a strong sense with error estimates, and finally by Furukawa et al [14] in a strong sense but under relaxing the regularity on the initial condition. Subsequently, the strong convergence of solutions of the scaled Navier-Stokes equations to the corresponding ones of the primitive equations with only horizontal viscosity was obtained by Li-Titi-Yuan [30]. Furthermore, the rigorous justification of the hydrostatic approximation from the scaled Boussinesq equations to the primitive equations with full viscosity and diffusivity was obtained by Pu-Zhou [34].…”

Section: Introductionmentioning

confidence: 99%

On the rigorous mathematical derivation for the viscous primitive equations with density stratification

Pu¹,

Zhou²

2022

Preprint

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In this paper, we rigorously derive the governed equations describing the motion of stable stratified fluid, from the mathematical point of view. Specially, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ration parameter goes to zero, and the rate of convergence is of the same order as the aspect ratio parameter. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

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“…As one will see later, the Coriolis force induces linear rotation waves with rotating rate |Ω|. The 3D viscous PEs can be derived as the asymptotic limit of the small aspect ratio between the vertical and horizontal length scales from the Boussinesq system, which is justified rigorously first in [1] in a weak sense, then later in [39] in a strong sense with error estimates (see also a recent paper [40] for the Date: March 8, 2022.…”

Section: Introductionmentioning

confidence: 99%

On the effect of fast rotation and vertical viscosity on the lifespan of the $3D$ primitive equations

Lin,

Liu,

Titi

2022

Preprint

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We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation |Ω|, we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L 2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large |Ω|, we show that the existence time of the solution can be prolonged, with "well-prepared" initial data. Finally, in the case of two spatial dimensions with Ω = 0, we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.

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The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Cited by 23 publications

References 49 publications

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

On the rigorous mathematical derivation for the viscous primitive equations with density stratification

On the effect of fast rotation and vertical viscosity on the lifespan of the $3D$ primitive equations

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