On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

Cao, Chongsheng; Lin, Quyuan; Titi, Edriss S.

doi:10.1007/s00021-020-00495-6

Cited by 15 publications

(12 citation statements)

References 42 publications

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“…We remark that the pressure term is not really an unknown and can be determined as functions of (u Θ , T Θ ) as we did for the hydrostatic limit system (see also [6] for more details). In what follows, we recall that we use "C" to denote a generic positive constant which can change from line to line.…”

Section: L2mentioning

confidence: 96%

“…The difficulty here consists in the presence of the unknown pressure term ∂ x p in the first equation of (1.4). However, as in [6], we can reformulate the problem by writing v and ∂ x p as functions of u and T . First, we remark that the Dirichlet boundary condition (u, v)| t=0 = (u, v)| t=1 = 0 and the incompressibility condition div u…”

Section: Global Wellposedness Of the Hydrostatic Limit Systemmentioning

confidence: 99%

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Hydrostatic approximation of the 2D primitive equations in a thin strip

Aarach¹,

Ngo

2020

Preprint

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Section: L2mentioning

confidence: 96%

Section: Global Wellposedness Of the Hydrostatic Limit Systemmentioning

confidence: 99%

Hydrostatic approximation of the 2D primitive equations in a thin strip

Aarach¹,

Ngo

2020

Preprint

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“…It has been shown that the PEs with ν h = 0 are linearly ill-posed in any Sobolev spaces and Gevrey class of order s > 1 [41] (see also [27,29]). With only vertical viscosity, to overcome the ill-posedness, one can consider additional weak dissipation [11], assume Gevrey regularity with some convex condition [20], or take the initial data to be analytic in the horizontal direction and only Sobolev in the vertical direction without any special structure [38,40]. For the inviscid case, the ill-posedness can be overcomed by assuming either some special structures (local Rayleigh condition) on the initial data or real analyticity in all directions for general initial data [4,5,21,25,32,33,39].…”

Section: Introductionmentioning

confidence: 99%

Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations

Hu,

Lin

2022

Preprint

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We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as t > 0 and the vertical analytic radius increases as long as the solutions exist.

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“…[32]. This ill-posedness can be overcome by considering additional linear (Rayleigh-like friction) damping [8], or Gevrey regularity and some convex conditions on the initial data [16].…”

Section: Introductionmentioning

confidence: 99%

“…Differentiating (7) with respect to X, then injecting (6) and (8), and taking X = 0, one obtains the following closed evolution equation for a:…”

Section: Introductionmentioning

confidence: 99%

Stable Singularity Formation for the Inviscid Primitive Equations

Collot,

Ibrahim,

Lin

2021

Preprint

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The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.

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On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

Cited by 15 publications

References 42 publications

Hydrostatic approximation of the 2D primitive equations in a thin strip

Hydrostatic approximation of the 2D primitive equations in a thin strip

Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations

Stable Singularity Formation for the Inviscid Primitive Equations

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