On the Effect of Rotation on the Life-Span of Analytic Solutions to the 3D Inviscid Primitive Equations

Ghoul, Tej-Eddine; Ibrahim, Slim; Lin, Quyuan; Titi, Edriss S.

doi:10.1007/s00205-021-01748-y

Cited by 19 publications

(47 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This suggests that the suitable space for the well-posedness of the inviscid PEs (with or without rotation) is Gevrey class of order s = 1, which is the space of analytic functions. This is consistent with positive results in [28,38]. Notably, for the Prandtl equations, which has some similarity in its structure with the PEs, is shown in [27] that its linearization around a special background flow has unstable solutions of similar form, but with ℜσ k ∼ λ √ k for k ≫ 1 arbitrarily large and some positive λ ∈ R + .…”

Section: Introductionsupporting

confidence: 88%

“…Furthermore, we also establish a lower bound on the life-span of the solution that grows to infinity with the rotation rate for "well-prepared" initial data. In a sense the results reported in this paper furnish a solid justification and motivation for our study in [28]. Specifically, the purpose of this paper is to establish the finite-time blowup and the ill-posedness in Sobolev spaces of the 3D inviscid PEs with rotation, and to investigate the effect of rotation on the blowup time.…”

Section: Introductionmentioning

confidence: 73%

“…In addition, we refer to [3,30,36,46] for simple examples demonstrating the above mechanism. In [28], we improve the results in [38] by establishing the local in time well-posedness in the space of analytic functions for a time interval that is independent of the rate of rotation. Furthermore, we also establish a lower bound on the life-span of the solution that grows to infinity with the rotation rate for "well-prepared" initial data.…”

Section: Introductionmentioning

confidence: 91%

“…Due to the linear ill-posedness in Sobolev spaces and Gevrey class of order s > 1, it is natural to consider the question of well-posedness of the inviscid PEs (with or without rotation) in Gevrey class of order s = 1, which is the space of analytic functions (cf. [28] and [38]). By virtue of the finite-time blowup results, one can conclude that there is no hope to show the global well-posedness of the 3D inviscid PEs, even with fast rotation.…”

Section: Introductionmentioning

confidence: 99%

“…The optimal result one can expect is that fast rotation prolongs the life-span of the 3D inviscid PEs (cf. [28]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation

Ibrahim¹,

Lin²,

Titi³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. First, we construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s > 1, and suggests that a suitable space for the well-posedness is Gevrey class of order s = 1, which is exactly the space of analytic functions.

show abstract

Section: Introductionsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

“…The optimal result one can expect is that fast rotation prolongs the life-span of the 3D inviscid PEs (cf. [28]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation

Ibrahim¹,

Lin²,

Titi³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

On the stabilizing effect of rotation in the 3d Euler equations

Guo

Huang

Pausader

et al. 2023

Comm Pure Appl Math

View full text Add to dashboard Cite

While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in with a fixed speed of rotation. We show that for any , axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.

show abstract

On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations

Lin

Liu

Titi

2022

J. Math. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

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 $$|\Omega |$$ | Ω | , we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only $$L^2$$ 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 $$|\Omega |$$ | 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 $$\Omega =0$$ 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.

show abstract

On the Effect of Rotation on the Life-Span of Analytic Solutions to the 3D Inviscid Primitive Equations

Cited by 19 publications

References 49 publications

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation

On the stabilizing effect of rotation in the 3d Euler equations

On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations

Contact Info

Product

Resources

About