Global large solutions for the Navier-Stokes equations with the Coriolis force

In this paper, we construct a class of global large solution to the three-dimensional Navier-Stokes equations with the Coriolis force in critical Fourier-Besov space $\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)$. In fact, our choice of special initial data $u_0$ can be arbitrarily large in $\dot{FB}^{s}_{p,r}(\mathbb{R}^3)$ for any $s\in\mathbb{R}$ and $1\leq p,r\leq \infty$.


Introduction and main result
Rotating fluid equations have important applications in meteorology and oceanography, particularly in the models describing large-scale ocean and atmosphere flows. The Coriolis force, arising from the rotation of the Earth, plays a significant role in such systems.
In 1868, Kelvin first observed that a sphere, moving along the axis of uniformly rotating water, takes with it a column of liquid as if this were a rigid mass, and pioneered the research on the motion of rotating fluid, see [10]. Later on, Taylor [28] and Proudman [25] strictly proved that high-speed rotation brings about a vertical rigidity in the fluid described by the Taylor-Proudman theorem: Under a fast rotation the velocity of all particles located on the same vertical line is horizontal and constant.
Mathematically, the Coriolis forces give rise to the so-called Poincaré waves, which are dispersive waves. Poincaré waves propagate in both directions with extremely fast speed in the propagation domain, and the waves with different wavenumbers move at different speeds. This makes that the nonlinear interactions between different modes are typically less significant.
On the other hand, Poincaré waves are a kind of high frequency wave, whose particles not only have vibrations parallel to the propagation direction, but also have vibrations perpendicular to the propagation direction. Therefore, one of the major difficulties encountered in understanding dynamics of rotating fluid is the influence of the oscillations generated by Coriolis forces.
In the paper, we consider the following Cauchy problem of three-dimensional incompressible Navier-Stokes equations with the Coriolis force: where the unknown functions u = (u 1 , u 2 , u 3 ) and p denote velocity field and pressure, respectively; Ω ∈ R is the Coriolis parameter, which is twice angular velocity of the rotation around the vertical unit vector e 3 = (0, 0, 1), and Ωe 3 × u represents the so-called Coriolis force; u 0 is the given initial velocity.
The behavior of fluid flows in rapidly rotating environments is fundamentally different from that of non-rotating flows.
When Ω = 0, (1.1) reduces to the problem of classical three-dimensional incompressible Navier-Stokes equations, which have been widely studied during the past seventy years. It has been proved that (1.1) with Ω = 0 is globally well-posed for small initial data, see [3,11,16,17,20,21]. For more results of large initial data with special structures in various scaling invariant spaces which generate unique global solutions to (1.1), we refer the reader to see [5,6,7,8,22,23,24] and the references therein. We note that the global regularity or global well-posedness issue of the three-dimensional incompressible Navier-Stokes equations for arbitrarily large initial data is still a challenging open problem.
When Ω = 0, it is a very remarkable fact that (1.1) admits a global solution for arbitrary large initial data, provided that the speed of rotation is fast enough. More precisely, when Ω is large enough, by taking the full exploration of dispersive effects of the Coriolis forces, the existence and uniqueness of global solution has been proved for the periodic large data in [1,2], for the spatially almost periodic large data in [29], and for the decay large data see [4,14,18,26]. For any given and fixed Ω, we refer to [12,13,15,19] for the global well-posedness of (1.1) with uniformly small initial data u 0 . Especially, it has been proved that (1.1) is globally well-posed [15,19]. It is now a natural question to ask whether there exists a unique global solution to (1.1) for any given and fixed Ω, if the initial data is not small inḞ B We first recall the definition of the Fourier-Besov spacesḞ B s p,r (R 3 ). As usual we denote by S (R 3 ) the space of Schwartz functions on R 3 , and by S ′ (R 3 ) the space of tempered distributions on R 3 . Choose radial function ψ ∈ S (R 3 ) such that its Fourier transformψ satisfies the following properties: = ||û|| L p .
Let U satify the following linear system: According to [13], we can show that U have the following explicit form: and it is easy to check that The main result of this paper reads as follows:

5)
then the system (1.1) has a unique global solution.
Therefore, we can conclude that for any s ∈ R and 1 ≤ p, r ≤ ∞

Proof of the main results
Proof of Theorem 1.3 Introduce the quantity u = U + v, we can show that v satisfies the following Cauchy problem: By the Duhamel principle, this problem is equivalent to the integral equation where P = (δ ij + R i R j ) 1≤i,j≤3 denotes the Helmholtz projection onto the divergence free vector fields, and {T Ω (t)} t≥0 denotes the Stokes-Coriolis semigroup given explicitly in [13]. By the similar argument of Lemma 2.2 in [27], we have for all t ∈ [0, T ] that where we have used Remark 1.2 and the fact || ab|| L 1 ≤ ||â|| L 1 ||b|| L 1 in the last inequality. Now, we define where η is a small enough positive constant which will be determined later on. Then, it yields From Gronwall's inequality, we have Choosing η = 2Cδ, thus we can get So if Γ < T * , due to the continuity of the solutions, we can obtain that there exists 0 < ǫ ≪ 1 such that which is contradiction with the definition of Γ.
Proof of Corollary 1.4 Notice that divU=0, we have Using the fact ||ab||Ḟ B 0