Global solvability of the rotating NavierStokes equations with fractional Laplacian in a periodic domain
Abstract.
We consider existence of global solutions to equations for threedimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with largescale phenomena. In the spatially decaying case, Koh, Lee and Takada (2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the main subject in this case is to show existence of global solutions to the resonant equation. Research in this direction was initiated by Babin, Mahalov and Nicolaenko (1999) who treated the rotating NavierStokes equations on general periodic domains. On the other hand, Golse, Mahalov and Nicolaenko (2008) considered bursting dynamics of the resonant equation in the case of a cylinder with no viscosity. Thus we may not expect to show global existence of solutions to the resonant equation without viscosity in the periodic case. In this paper we show existence of global solutions for fractional Laplacian case (with its power strictly less than the usual Laplacian) in the periodic domain with the same period in each direction. The main ingredient is an improved estimate on resonant threewave interactions, which is based on a combinatorial argument.
Key words and phrases:
NavierStokes equations, Coriolis force, global regularity, resonant equation, divisor bound2010 Mathematics Subject Classification:
76U05,42B051. Introduction
We consider the rotating threedimensional NavierStokes equations with the fractional Laplacian:
(1.1)  
where is the unknown velocity vector field and is the unknown scalar pressure at the point in space and time while is the given initial velocity field. Here is the Coriolis parameter, which is twice the angular velocity of the rotation around the vertical unit vector , and is the kinematic viscosity coefficient. By we denote the exterior product, and hence, the Coriolis term is represented by with the corresponding skewsymmetric matrix , namely,
The Coriolis force plays a significant role in the large scale flows considered in meteorology and geophysics. In 1868 Kelvin 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 (see [9] for references). After that, Taylor [25] and Proudmann [24] did important contributions. Mathematically it was investigated by Poincaré [24], more recently, by Babin, Mahalov and Nicolaenko [2, 3] using the fully NavierStokes equations in a periodic domain.
Throughout this paper we essentially use the spatial Fourier transform denoted by or :
Let us define the inhomogeneous Sobolev spaces as follows:
The homogeneous version can be defined as
We will assume that all the vector fields in this paper are meanzero. This assumption is valid from the following observation: Let be an average of the solution to (1.1) at , that is, the solution to the following ODE:
Then the following invertible transforms
preserve the equation (1.1), and the new velocity field has zero mean for all time. We therefore do not distinguish homogeneous and inhomogeneous Sobolev spaces.
Let us recall the result of Babin et al. as the starting point of our work:
Theorem 1.1 ([3]).
Let and be a torus with arbitrary period (distinguish from ). Let be a divergencefree vector field. Then there exists a positive depending on and the period of torus such that for all , there is a unique global solution
to the equation (1.1) with .
As shown in [2, 3], it turns out that the estimates on the obtained global solutions depend crucially on the period of the torus. For instance, the global a priori bound is independent of the viscosity coefficient for generic periods ([2]), whereas exponentialin dependence may occur in the “worst case” ([3]). In this paper, we will focus on the special torus , which is among the “worst case” as we will see in Section 4.4 below. We remark that the above result was extended to the critical case in [7, Theorem 6.2]; see also [7, Theorem 5.7] for an analogous result on .
We next recall previous results in the inviscid case. In the spatially decaying setting, combining the Strichartz estimates with BealeKatoMajda’s blowup criterion, Koh, Lee and Takada [20] showed long time existence of solutions to the Euler equations provided by large Coriolis parameter. The periodic case may be more difficult due to the appearance of the resonant equation. In [2], Babin et al. initially considered long time solvability of the rotating Euler equations (see also [22] in a cylinder case). However they set specific periodic domains (specific aspect ratios) and eliminate “nontrivial resonant part” which is essentially related to the Rossby wave in physics (see [19, 27] for example). For domains with other periods we need to deal with “nontrivial resonant part”, and it has been an open problem. On the other hand, in a cylinder case, Golse, Mahalov and Nicolaenko [16] considered bursting dynamics of the inviscid resonant equation. Thus we may not expect to show existence of inviscid smooth global flow in general periodic cases. Nevertheless, by a refined estimate on “nontrivial resonant part” based on elementary number theory (Lemma 5.1 below), we can progress a less viscosity effect case (fractional Laplacian case) in the periodic domain . A fractional Laplacian (superviscosity), though it has little physical meaning, has been employed in many numerical works instead of the usual viscosity; see [26] for example.
First we state the following local existence theorem, which is obtained by a standard argument:
Theorem 1.2.
We give its proof in the next section. Since we consider the subcritical problem with respect to the scaling (see Section 7.2), time of local existence is bounded from below in terms of the size of initial data. Note also that the result is uniform in the Coriolis parameter .
We now state the main theorem. For , denotes any positive constant depending on with as . Our main result is as follows.
Theorem 1.3.
Let and . For any , there exists depending on , and such that for any with and any realvalued divergencefree meanzero initial vector field with , there exists a unique global smooth solution of (1.1). Moreover, can be taken as
Remark 1.4.
For , consider the torus
We say regular if or rational if . Although we will mainly focus on the case of , our result can be extended to the case of any rational periodic domains with a slight modification. See Remark 5.5 below.
We also remark that for periodic domains with , global regularity under fast rotation follows immediately from the above theorem and a scaling argument. In fact, by a suitable scaling transformation, any solution to (1.1) on with can be transformed into a solution on with rescaled initial data and rotation speed.
Remark 1.5.
Remark 1.6.
Let us define the function space as follows:
It is well known that is continuously embedded in the space of continuous functions (in the nonperiodic case it is embedded in , the space of bounded uniformly continuous functions). In the spatially almost periodic case, framework seems to be one of the most suitable (see [10, 11, 12, 13, 14, 15, 28] for example). On the other hand, to control the nontrivial resonant part, the energy method is one of the most powerful tools. Note that, up to now, we can use the energy method only in the periodic case, thus, controlling the nontrivial resonant part in the spatially almost periodic case is open. See also Section 7.2.
In the rest of this section, we outline the proof of the main theorem (Theorem 1.3). We basically follow the previous argument in [3, 7].
The Poincaré propagator is defined as the unitary group associated with the linear problem
where denotes the HelmholtzLeray projection onto divergencefree vector fields; acts as multiplication by the matrix in the Fourier space:
We see that the matrix
has eigenvalues , and for each , the vectors defined by
are eigenvectors corresponding to and form an orthonormal basis of
Therefore, the Poincaré propagator acts on a divergencefree and meanfree vector field as
where denotes the inner product of . (Note that whereas , where stands for the complex conjugate.) It is also easy to see that
(1.4) 
Using this, we obtain another representation of :
In fact, for divergencefree and meanzero ,
Next, we set . If solves (1.1), then (formally) solves
(1.5) 
where
so that
Now we decompose into the resonant and the nonresonant parts as
(1.6) 
where
so that
It is expected (and actually proved in Section 6) that only the resonant part contributes in the limit . Therefore, we need to consider the following limit equation (resonant equation):
(1.7) 
We remark that similar local existence results to Theorem 1.2 for the equation (1.5) and for the limit equation (1.7) can be obtained with the identical proof.
The main task is to show existence of global regular solutions to the resonant equation (1.7). More precisely we will show the following:
Proposition 1.7.
To prove the above proposition, we make further decomposition of into the 2D part and the nontrivial resonance part. For a 3D3C (threedimensional threecomponent) vector field , we define

2D3C vector field by ,
or , 
3D3C vector field by ,
or , 
3D2C vector field by .
It is easily verified that for any divergencefree and meanzero vector fields ,
where is the 2D Helmholtz projection, and . Note that if . Moreover, we see (Lemma 3.1 below) that
These properties imply that . Consequently, the limit equation (1.7) can be decomposed into the following three equations:
(1.8) 
(1.9) 
(1.10) 
where .
The energy estimate for the 2D part can be obtained straightforwardly (see Section 4):
(1.11) 
The key is to control the following norm globally in time (see Section 4):
(1.12) 
In order to control the above quantity in the weak viscosity case (), we essentially use a new estimate on nontrivial resonant threewave interactions (Lemma 4.1 below). We prove this estimate in Section 5 by using some tools from elementary number theory, which is the crucial idea in this paper. However, this kind of argument is only available for the case of regular (or rational) torus. As a result, our main theorem is also restricted to that case. Once we get the above energy estimates, we will be able to deduce Proposition 1.7. See Section 4 for details.
To prove the main theorem, we first decompose the time interval into three parts:

: dominant linear part,

: dominant Coriolis force part,

: exponentially decaying part.
What we need for the first time interval is nothing more than the local existence result. By the global existence for small initial data, we can also control the solution to (1.1) in as long as it becomes sufficiently small at . On the other hand, by (4.8) the solution to the resonant equation eventually becomes arbitrarily small. Thus, our main task is to ensure under the large Coriolis parameter assumption that solutions to the original equation and the resonant equation that coincide at stay very close to each other until , which also means that we can obtain the existence theorem in . To this end, it suffices to control the nonresonant part in (1.6). More precisely, our task is to estimate the difference , which satisfies
Let be the global upper bound of in obtained in (4.7), and let be a local existence time of the solution to (1.5) of size . The following lemma enables us to control the nonresonant part :
Lemma 1.8.
For any , there exists such that the following holds for . Let be the positive integer satisfying
and let . Assume that for . Then, we have
for .
Roughly saying, we can control the contribution from the nonresonant forcing term by an arbitrarily small constant . By the above lemma, which will be restated as Lemma 6.1 and proved in Section 6, we can prove the main theorem (Theorem1.3). For the precise argument, see Section 6.
Acknowledgments. The first author was partially supported by GrantinAid for Young Scientists (B), No. 24740086 and No. 16K17626, Japan Society for the Promotion of Science.
2. Proof of local wellposedness
In this section, we shall establish local wellposedness for (1.1), Theorem 1.2. Let us consider the corresponding integral equation:
(2.1) 
Lemma 2.1.
Let , , and . Then, for divergencefree and meanzero vector fields , it holds that
The implicit constant depends only on and is independent of .
Remark 2.2.
In the following argument, we apply this lemma only with . Then, we can take the implicit constant independent of and .
Proof.
We see that
We will apply fixed point argument in the norm
for suitable to be chosen later. By Lemma 2.1, we have
For the Duhamel term, we want to estimate by using Lemma 2.1 as
for , where we have used the divergencefree condition for so that , and we have also used the Sobolev estimate
which holds if and (applying Lemma 7.1 below with and ). Hence, we need to assume
(2.2) 
for the above argument. Similarly,
for , whenever (2.2) holds.
If , we can take to fulfill (2.2). Then, the contraction mapping argument can be applied if we take so that
In particular, we obtain existence, uniqueness, continuous dependence on initial data and regularity for local solutions on by a standard argument.
To show that depends on the initial data only in the norm, we first obtain the solution by the above argument (taking ) and then notice that
where at the second inequality Lemma 7.1 has been applied with , , , for (we also divide by a similar argument as in the proof (Case 2) of Lemma 7.1). Here, the above may be smaller than the local existence time for the