Long time existence of classical solutions for the 3D incompressible rotating Euler equations

Takada, Ryo

doi:10.2969/jmsj/06820579

Cited by 13 publications

(11 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…u ω solves the rotating Euler equations with speed of rotation ω, on a time interval [0, ω −1 T ]. While our assumptions on the initial data are more restrictive than those of [21,26] (and in particular require axisymmetry), this rescaling shows that for initial data of size ε in (1.1) the increase in the lifespan of solutions due to the above works is of order log(ε), whereas our result in Theorem 1.1 gives an improvement to any polynomial scale ε −M , M > 0.…”

Section: Introductionmentioning

confidence: 88%

See 1 more Smart Citation

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

Guo¹,

Huang²,

Pausader³

et al. 2020

Preprint

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 R 3 with a fixed speed of rotation. We show that for any M > 0, axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least ε −M 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

Section: Introductionmentioning

confidence: 88%

“…In the rotating Euler equations (1.1), decay and Strichartz estimates have been used to treat the case of fast rotation [8,21,26,27,1]. Here the (inverse) rotation speed plays the role of a small parameter that can be used to suppress the truly nonlinear dynamics.…”

Section: Introductionmentioning

confidence: 99%

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

Guo¹,

Huang²,

Pausader³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Let us also mention works in the periodic case where resonences have to be studied (see for example [24,40,41,43]), in the rotating fluids system case (see [16,17,18,26,28,36]) or in the inviscid case (see [22,23,37,47,48]).…”

Section: Going Back To Systemmentioning

confidence: 99%

Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data

Charve

2019

Preprint

View full text Add to dashboard Cite

In this article we obtain, for a stratified, rotating, incompressible Navier-Stokes system, generalized asymptotics as the Rossby number ε goes to zero (without assumptions on the diffusion coefficients). For ill-prepared, less regular initial data with large blowing-up norm in terms of ε, we show global well-posedness and improved convergence rates (as a power of ε) towards the solution of the limit system, called the 3D quasi-geostrophic system. Aiming for significant improvements required us to avoid as much as possible to resort to classical energy estimates involving oscillations. Our approach relies on the use of structures and symmetries of the limit system, and of highly improved Strichartz-type estimates.

show abstract

“…In the setting of fast rotation, the (inverse) speed of rotation introduces a parameter of smallness that can be used to prolong the time of existence of solutions. For Euler-Coriolis (1.2), this has been done in [1,8,15,45,49,59,60] via Strichartz estimates associated to the linear semigroup, based on work in the viscous setting [9,23]. Such results do not require axisymmetry and apply for sufficiently smooth initial data without size restrictions.…”

Section: Introductionmentioning

confidence: 99%

Global axisymmetric Euler flows with rotation

2022

View full text Add to dashboard Cite

We construct a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform “rigid body” rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.

show abstract

Long time existence of classical solutions for the 3D incompressible rotating Euler equations

Cited by 13 publications

References 21 publications

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

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

Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data

Global axisymmetric Euler flows with rotation

Contact Info

Product

Resources

About