Uniform Local Solvability for the Navier-Stokes Equations with the Coriolis Force

Giga, Yoshikazu; Inui, Katsuya; Mahalov, Alex; Matsui, Susumu

doi:10.4310/maa.2005.v12.n4.a2

Cited by 38 publications

(47 citation statements)

References 10 publications

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“…The significance of the space FM 0,σ (R 3 , C 3 ) for the Navier-Stokes equations with rotation in the whole space R 3 was first noticed in [2] and [3]. In [2] a local-in-time existence result is proved with an existence interval independent of Ω.…”

Section: Description and Main Resultsmentioning

confidence: 93%

“…Adopting the ideas from [2] and [3] we can prove a corresponding local-in-time existence result uniformly in Ω for the Ekman boundary layer problem, i.e., for system (1). The result reads as follows.…”

Section: Description and Main Resultsmentioning

confidence: 99%

“…In [2] a local-in-time existence result is proved with an existence interval independent of Ω. In [3] global-in-time solvability and exponential stability is derived for initial data u 0 ∈ FM 0,σ (R 3 , C 3 ) such that supp F u 0 is contained in a sum-closed frequency set.…”

Section: Description and Main Resultsmentioning

confidence: 99%

“…Here L(X) denotes the class of all bounded operators on a Banach space X, and F and F −1 the Fourier transform and its inverse respectively. Hence, in order to obtain results on stability, besides the natural requirements (i) and (ii), a potential class of initial data should also admit a multiplier result as (2). A suitable class satisfying these requirements and which is still large enough is the class…”

Section: Description and Main Resultsmentioning

confidence: 99%

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On the stability of the Ekman boundary layer

Giga

Saal

2007

Proc Appl Math and Mech

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We present a result on well-posedness and stability of the Ekman boundary layer problem in the space FM(R 2 , L 2 (R+) 3 ), i.e., in the space of L 2 (R+) 3 -valued Fourier transformed finite Radon measures. In particular we obtain stability in the angle velocity of rotation, which is important in the analysis of fast oscillating singular limits. Description and main resultThe Ekman boundary layer problem is a meteorological model for the motion of a rotating fluid (atmosphere) inside a boundary layer, appearing in between a uniform geostrophic flow (wind) and a solid boundary (earth) at which the no slip condition applies. Mathematically this situation is described by the Navier-Stokes equations with Coriolis forceHere the unknowns u and p denote velocity and pressure of the fluid respectively, whereas e 3 = (0, 0, 1) and the parameters ν and Ω correspond to viscosity and angle velocity of the rotation around the x 3 -axis.There is a famous stationary and exact solution to (1) called Ekman spiral and which is given by the vectorHere δ = ν/|Ω| denotes the layer thickness and U ∞ the velocity of the geostrophic flow away from the boundary pointing in x 1 direction. The corresponding pressure to U E is given by p E (x 2 ) = −ΩU ∞ x 2 . Remarkable persistent stability of the Ekman spiral in atmospheric and oceanic boundary layers has been noticed in geophysical literature. Here we are interested in stability in the parameters t, Ω, ν, and δ. In particular in the existence of solutions with norms uniformly bounded in Ω in spaces including functions nondecaying at infinity. Results of this type are essential in studies of statistical properties of turbulence, see e.g. [6,7], and in the analysis of fast oscillating singular limits for system (1), see e.g. [1,5].The observation that U E depends on the x 3 variable only, i.e., that it has infinite energy, and thatleads to two natural requirements on a potential class E of initial data:The class E should include functions nondecreasing at infinity in tangential direction.(A first result on well-posedness for system (1) is obtained in [4] for u 0 in the classNote that the spaceḂ3 -valued almost periodic functions. Thus, E 1 satisfies (i) and (ii). However, the class E 1 seems to be inappropriate for stability investigations. This relies essentially on the fact that the Poincaré-Riesz semigroup (e −tBΩ ) t≥0 generated by the Coriolis operator B Ω u := 2Ωe 3 × u proved to be unbounded in t *

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Section: Description and Main Resultsmentioning

confidence: 93%

Section: Description and Main Resultsmentioning

confidence: 99%

Section: Description and Main Resultsmentioning

confidence: 99%

Section: Description and Main Resultsmentioning

confidence: 99%

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On the stability of the Ekman boundary layer

Giga

Saal

2007

Proc Appl Math and Mech

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“…The idea to use the space of Fourier transformed Radon measures for the treatment of the Navier-Stokes equations with Coriolis force, first appeared in [22]. There the local-in-time well-posedness in the whole space R 3 , i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Giga

Saal

2012

J. Math. Fluid Mech.

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ABSTRACT. In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.

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Global solutions to the rotating Navier–Stokes equations with large data in the critical Fourier–Besov spaces

Fujii

2023

Mathematische Nachrichten

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We consider the initial value problem for the 3D incompressible Navier–Stokes equations with the Coriolis force. The aim of this paper is to prove the existence of a unique global solution with arbitrarily large initial data in the scaling critical Fourier–Besov spaces (, ), provided that the size of the Coriolis parameter is sufficiently large. Moreover, if the initial data additionally belong to the scaling sub‐critical spaces, we obtain an explicit relationship between the initial data and the Coriolis force, which ensures the existence of a unique global solution.

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Uniform Local Solvability for the Navier-Stokes Equations with the Coriolis Force

Cited by 38 publications

References 10 publications

On the stability of the Ekman boundary layer

On the stability of the Ekman boundary layer

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Global solutions to the rotating Navier–Stokes equations with large data in the critical Fourier–Besov spaces

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