Strichartz estimates for the Euler equations in the rotational framework

Koh, Youngwoo; Lee, Sanghyuk; Takada, Ryo

doi:10.1016/j.jde.2013.09.017

Cited by 47 publications

(79 citation statements)

References 19 publications

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“…In the rest of this section, we follow the ideas in , , and give the proof of Lemma . First, since the phase function

p_{μ} false(ξ false)

is homogeneous of degree 0, by the Littlewood–Paley decomposition and scaling, the matter is reduced to the frequency localized case.…”

Section: Dispersive Estimatesmentioning

confidence: 99%

“…Note that

M_{μ} false(ξ false)

and

M_{Ω, N} false(ξ false)

appear due to the effect of the constant stratification. Indeed, if we consider only the rotational effect (

N = θ = 0

), the phase is

ξ_{3} / false| ξ false|

and its Hessian matrix is given by

R_{1, 0} false(ξ false)

(see [, page 731]). We remark that all columns of the matrix

M_{μ} false(ξ false)

are linearly dependent.…”

Section: Dispersive Estimatesmentioning

confidence: 99%

“…If the surface S is given by the graph of some function p , the number of non‐vanishing principal curvatures is equal to the number of non‐zero eigenvalues of

H p

, that is, the rank of the symmetric matrix

H p

(see also [, Lemma 7.6]).…”

Section: Dispersive Estimatesmentioning

confidence: 99%

See 2 more Smart Citations

Global solutions for the incompressible rotating stably stratified fluids

Iwabuchi

Mahalov

Takada

2016

Mathematische Nachrichten

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We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle.

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“…In the rest of this section, we follow the ideas in , , and give the proof of Lemma . First, since the phase function

p_{μ} false(ξ false)

is homogeneous of degree 0, by the Littlewood–Paley decomposition and scaling, the matter is reduced to the frequency localized case.…”

Section: Dispersive Estimatesmentioning

confidence: 99%

“…Note that

M_{μ} false(ξ false)

and

M_{Ω, N} false(ξ false)

appear due to the effect of the constant stratification. Indeed, if we consider only the rotational effect (

N = θ = 0

), the phase is

ξ_{3} / false| ξ false|

and its Hessian matrix is given by

R_{1, 0} false(ξ false)

(see [, page 731]). We remark that all columns of the matrix

M_{μ} false(ξ false)

are linearly dependent.…”

Section: Dispersive Estimatesmentioning

confidence: 99%

See 1 more Smart Citation

Global solutions for the incompressible rotating stably stratified fluids

Iwabuchi

Mahalov

Takada

2016

Mathematische Nachrichten

Self Cite

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show abstract

“…In particular, Strichartz estimates can then be used to construct global solutions once the dispersion is sufficiently strong, and to study limiting systems as the dispersion tends to infinity. For the rotating Navier-Stokes equations this was done by Chemin et al [5][6][7] and Koh, Lee, and Takada [18].…”

Section: Related Work On Rotating Fluidsmentioning

confidence: 99%

Long Time Stability for Solutions of a β‐Plane Equation

Elgindi¹,

Widmayer

2016

Comm Pure Appl Math

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We prove stability for arbitrarily long times of the zero solution for the so‐called β‐plane equation, which describes the motion of a two‐dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the two‐dimensional incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, L1:=∂1/| ∇ |2, exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.© 2016 Wiley Periodicals, Inc.

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“…Also, Dutrifoy [13] and Charve [10] obtained analogous results for quasigeostrophic systems. Recently, for s > s 0 = 3 2 +1 and u 0 ∈ H s (R 3 ), Koh, Lee and Takada [23] proved that there exists a unique local in time solution u for (1.1)…”

Section: Introductionmentioning

confidence: 99%

On the 3D Euler equations with Coriolis force in borderline Besov spaces

Angulo-Castillo

Ferreira

2018

Communications in Mathematical Sciences

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We consider the 3D Euler equations with Coriolis force (EC) in the whole space. We show long-time solvability in Besov spaces for high speed of rotation Ω and arbitrary initial data. For that, we obtain Ω-uniform estimates and a blow-up criterion of BKM type in our framework. Our initial data class is larger than previous ones considered for (EC) and covers borderline cases of the regularity. The uniqueness of solutions is also discussed.

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Strichartz estimates for the Euler equations in the rotational framework

Cited by 47 publications

References 19 publications

Global solutions for the incompressible rotating stably stratified fluids

Global solutions for the incompressible rotating stably stratified fluids

Long Time Stability for Solutions of a β‐Plane Equation

On the 3D Euler equations with Coriolis force in borderline Besov spaces

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