Abstract
Uniform well-posedness and stability for fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces
Highlights
I n this paper, we consider the initial value problem of the fractional Navier-Stokes equations with the Coriolis force in R3, ut + μ(−∆)αu + Ωe3 × u + (u.∇)u + ∇π = 0(t, x) ∈ R+ × R3, ∇.u = 0, (1) u(0, x) = u0(x)x ∈ R3, where u = u(t, x) = (u1(t, x), u2(t, x), u3(t, x)) represents the unknown velocity vector, the scalar function π = π(t, x) denotes the unknown scalar pressure and u0 is a divergence free vector field
When α = 1, the Equation (1) corresponds to the usual Navier-Stokes equation with Coriolis force, which receives some attention for its importance in geophysical flow applications
Iwabuchi and Takada [5] proved the existence of global solutions for the Navier-Stokes equations with Coriolis force in Sobolev spaces Hs(R3) with 1/2 < s < 3/4 if the speed of rotation Ω is large enough compared with the norm of initial data u0 Hs ; they obtained the global existence and the uniqueness of the mild solution for small initial data in the Fourier-Besov spaces FB 1−,21, and proved the ill-posedness in the space FB 1−,q1, 2 < q ≤ ∞ for all Ω ∈ R
Summary
Iwabuchi and Takada [5] proved the existence of global solutions for the Navier-Stokes equations with Coriolis force in Sobolev spaces Hs(R3) with 1/2 < s < 3/4 if the speed of rotation Ω is large enough compared with the norm of initial data u0 Hs ; they obtained the global existence and the uniqueness of the mild solution for small initial data in the Fourier-Besov spaces FB 1−,21, and proved the ill-posedness in the space FB 1−,q1, 2 < q ≤ ∞ for all Ω ∈ R (see [6]). Inspired by the works [6,10–13], the aim of this paper is to prove the global existence and the decay property and the stability of the global solutions of the fractional Navier-Stokes equations with Coriolis force (1) in the Fourier-Besov-Morrey space
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