Long-time solvability in Besov spaces for the inviscid 3D-Boussinesq-Coriolis equations

We investigate the long-time solvability in Besov spaces of the initial value problem for the inviscid 3D-Boussinesq equations with Coriolis force. First we prove a local existence and uniqueness result with critical and supercritical regularity and existence-time \begin{document}$ T $\end{document} uniform with respect to the rotation speed \begin{document}$ \Omega $\end{document} . Afterwards, we show a blow-up criterion of BKM type, estimates for arbitrarily large \begin{document}$ T $\end{document} , and then obtain the long-time existence and uniqueness of solutions for arbitrary initial data, provided that \begin{document}$ \Omega $\end{document} is large enough.

1. Introduction. We analyse the long-time solvability in the full inviscid case for the Boussinesq-Coriolis system in the whole R 3 which describes the evolution of an incompressible fluid under the effects of convection and Earth's rotation. In its complete form, including thermal diffusivity and kinematic viscosity, that system reads as ∂ t u + Ωe 3 × u + ∇p + (u · ∇)u − ν∆u = gθe 3 in R 3 × (0, ∞), ∂ t θ + (u · ∇)θ − κ∆θ = 0 in R 3 × (0, ∞), ∇ · u = 0 in R 3 × (0, ∞), u| t=0 = u 0 , θ| t=0 = θ 0 in R 3 , where u = u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) is the velocity field, θ = θ(x, t) is the temperature and p = p(x, t) stands for the pressure. The initial velocity u 0 = (u 0,1 (x), u 0,2 (x), u 0,3 (x)) satisfies the condition ∇ · u 0 = 0 and θ 0 = θ 0 (x) is the initial temperature. The kinematic viscosity is represented by ν ≥ 0, the thermal diffusivity is denoted by κ ≥ 0 and g is the gravity. The Coriolis parameter Ω ∈ R plays the role of twice the rotation speed around the vector unit e 3 = (0, 0, 1). The term gθe 3 represents the buoyancy force that arises from the Boussinesq approximation in which density variations are proportional to temperature variations and assumed to exist only in the gravitational term. The reader is referred to the book [30] for more details about the model (1.1).
Among the problems in partial differential equations that are of great interest to the mathematical community, there are the equations in fluid mechanics, especially those dealing with atmospheric and oceanographic phenomena where heat transfer and rotation effects play an important role. Such is the case of the system (1.1) and related models on which different theoretical studies have been carried out in the last decades, analyzing various aspects such as the local and global well-posedness, regularity and stability of solutions, blow-up criteria, among others. In what follows, we will briefly review some results about these aspects.
For the 3D-Boussinesq equations (i.e., (1.1) with Ω = 0), the global smoothness of solutions is a challenging open problem for any viscosity case, that is inviscid, partially and totally viscous cases. Some results about the local and global wellposedness and blow-up criteria have been obtained in different function spaces for this system. In particular, for the viscous case (i.e., ν > 0 and κ > 0), see, e.g., the works [2,16,18,20,23] and their references. More precisely, Ferreira and Villamizar-Roa [20] showed well-posedness and asymptotic self-similarity of small solutions in the framework of weak-L p spaces; Hmidi and Rousset [23] proved the global well-posedness with axisymmetric initial data in Sobolev spaces; de Almeida and Ferreira [2] considered homogeneous critical Morrey spaces and proved results on well-posedness, self-similarity and large-time behavior of small solutions. Deng and Cui [18] showed the well-posedness of small solutions with initial data belonging to Besov spaces of negative regularity index s = −1; and Dai et al. [16] established a blow-up criterion of weak solutions in terms of the pressure in the homogeneous Besov spacesḂ 0 ∞,∞ (R 3 ). We also refer the reader to the references contained in the papers mentioned in the present paragraph.
In the inviscid case (i.e., ν = κ = 0), Cui et al. [15] showed the local wellposedness and a blow-up criterion in Hölder spaces in R n for n ≥ 2; Xiang and Yan [35] proved the well-posedness in the Triebel-Lizorkin-Lorentz spaces F s,r p,q (R n ) for 1 < r ≤ p < ∞, 1 < q < ∞ and s > 1 + n p , and also obtained a blow-up criterion in the Triebel-Lizorkin space F 1 ∞,∞ (R n ). Bie et al. [8] obtained the local well-posedness and a blow-up criteria in Besov-Morrey spaces N s p,q,r (R 3 ) for the supercritical case s > 1 + n p , 1 < q ≤ p < ∞, 1 ≤ r ≤ ∞, and the critical case s = 1 + n p , 1 < q ≤ p < ∞, r = 1. In the rotating case (i.e., Ω = 0) and with an additional stratification term −N 2 u 3 in the R.H.S of the temperature equation in (1.1), results about global well-posedness in Sobolev spaces have been obtained in the viscous case (ν > 0 and κ > 0), see e.g. [6,12,13,14,25,26] and references therein. These results require some conditions such as small initial data, sufficiently large stratification parameter N , sufficiently large Ω (w.r.t the initial data norm), or a mixture thereof. Still considering stratification effects, Sun and Yang [32] obtained a result in the homogeneous hybrid Besov space B 1 2 ,−1+ 3 p 2,p,N whose norm depends on the parameter N > 0. For the viscous stably stratified 3D Boussinesq equations (i.e., Ω = 0, N > 0, and ν, κ = 0), Lee and Takada [28] proved global well-posedness in the framework of Sobolev spaces with N sufficiently large w.r.t to initial data, after obtaining sharp dispersive estimate for the linear propagator related to the stable stratification. Recently, in the rotating viscous case of (1.1) (i.e., Ω = 0, ν, κ = 0, and without the additional stratification term −N 2 u 3 ), Sun, Liu and Yang [31] showed global well-posedness with Ω sufficiently large w.r.t the norm of (u 0 , θ 0 ) in the Besov spacesḂ s1 p,q ×Ḃ s2 p,q , where 3/2 < p < 2, 1 ≤ q ≤ ∞, s 1 = −1 + 3/p + 2/δ, s 2 = −1 + 3/p + 2/ρ, δ ∈ (2, ∞) and ρ ∈ (1, δ]. It is worth mentioning that the first results of long-time solvability for Euler equations and global well-posedness for Navier-Stokes equations, under high rotation speed, were obtained by Babin, Mahalov and Nicolaenko [4,5] by considering the framework of Sobolev spaces and periodic boundary conditions. In view of the previous results, it is natural to wonder about the long-time solvability (i.e., arbitrarily large existence-time) of system (1.1) in the non-viscous (partially or inviscid) cases and without stratification effects. Inspired by the papers for Euler [10,38] and Euler-Coriolis [3,27] equations, we show long-time solvability of (1.1) in the inviscid case κ = ν = 0 for initial data in the Besov space B s 2,q × B s 2,q with critical and supercritical values of the regularity s, 1 ≤ q ≤ ∞ and |Ω| sufficiently large w.r.t to the initial data and existence-time T . The critical regularity sense is that used in the context of Euler equations since we are working in the totally inviscid case. To be more precise, we consider the inviscid 3D-Boussinesq-Coriolis system where P = (δ i,j +R i R j ) 1≤i,j≤3 is the Leray-Helmholtz projector and {R j } 1≤j≤3 are the Riesz transforms. In comparison to the 3D viscous rotating Boussinesq with or without stratification and to 3D viscous stably stratified Boussinesq, system (1.2) is more difficult to handle in the sense that there is only a dispersive effect in the first equation and no dissipation in any equation.
Throughout the paper we denote spaces of scalar and vector functions in the same way, e.g., we write u 0 ∈ B s 2,q (R 3 ) in place of u 0 ∈ B s 2,q (R 3 ) 3 . Our results read as follows: Theorem 1.1. Let s and q such that s = 3/2 + 1 with q = 1 or s > 3/2 + 1 with 1 ≤ q ≤ ∞. Then, the following assertions hold: Let us briefly discuss about some technical aspects in the proof of the theorem. To show item (i), we use an approximate linear iteration system similar to the one that Zhou [38] used to study the Euler equations. Here the sequence (u n , θ n ) n∈N satisfying this approximate system converges to the unique solution of the system (1.2). Afterwards, in order to show item (ii) we prove a criterion of blow-up and employ a Strichartz estimate due to Koh et al. [27]. This paper is organized as follows. Section 2 is devoted to some preliminaries about Besov spaces together with product and commutator estimates in these spaces, as well as projection operators linked to the Coriolis term. The main result is proved in Section 3: first we prove the local-in-time solvability corresponding to item (i) on [0, T ] for some T > 0 independent of Ω (see subsection 3.1); second, we obtain a blow-up criterion to (1.1) in subsection 3.2; and finally we conclude with the proof of global solvability in subsection 3.3.

2.
Preliminaries. In this section we give some preliminaries about Besov spaces. The reader is referred to the book [7] for further details on these spaces.
We begin by recalling the Schwartz class of rapidly decreasing smooth functions S(R 3 ) and the space of tempered distributions S (R 3 ). For f ∈ S , we denote the Fourier transform of f by f .
Next take a function φ 0 ∈ S( For each k ∈ Z, consider the function S k ∈ S defined in Fourier variables as The Littlewood-Paley operator ∆ j is defined by Let P stand for the set of polynomials with 3 variables. For the indexes s ∈ R and 1 ≤ p, q ≤ ∞, we define the homogeneous Besov space aṡ and its inhomogeneous version as where S 0 is given by (2.2). The pairs (Ḃ s p,q , · Ḃs p,q ) and (B s p,q , · B s p,q ) are Banach spaces.
Now we recall the Bernstein inequality.
Using (2.1)-(2.2), it is not difficult to see the equivalence of norms Moreover, Bernstein inequality implies the equivalence For s > 3/p with 1 ≤ p, q ≤ ∞ or s = 3/p with 1 ≤ p ≤ ∞ and q = 1, we have the continuous embedding (see, e.g., [7, p.153-154 and 163]) In the lemma below we recall some product estimates in the Besov setting (see [10]).
There exists a positive universal constant C such that f g Ḃs . We finish this section with estimates in the framework of Besov spaces for the [10,33]).
Then, there exists a universal constant C > 0 such that 3. Proof of Theorem 1.1. This section is devoted to the proof of Theorem 1.1. We start with the local-in-time existence result in item (i).
3.1. Local solvability. We consider the following linear approximation scheme for Next, in order to build the local solution we are going to obtain uniform estimates for the sequence (u n , θ n ) n∈N .
Uniform estimates. First, we deal with estimates for u n and θ n in the norm · Ḃs 2,q . For that, we apply the operator ∆ j in the system (3.1), and then, we take the L 2 -norm product with ∆ j u n+1 and ∆ j θ n+1 to the first and second equation, respectively, to get Here, we have used the skew-symmetry of e 3 × and the condition ∇ · ∆ j u n = 0. Thus, by the Cauchy-Schwarz inequality we obtain Integrating over (0, t) we have that Multiplying by 2 sj and applying the l q -norm, we arrive at the estimate Using (i) of Lemma 2.3 and the inclusion B s 2,q →Ḃ s 2,q , it follows that Now, we deal with estimates for u n and θ n in the norm · L 2 . Taking the L 2 -product with u n+1 and θ n+1 to the first and second equation in (3.1), respectively, and using the skew-symmetry of e 3 ×, the condition ∇ · ∆ j u n = 0 and the Cauchy-Schwarz inequality, we obtain Since u n+1 (0) B s 2,q ≤ C u 0 B s 2,q and θ n+1 (0) B s 2,q ≤ C θ 0 B s 2,q , by the Gronwall inequality there exist positive constants C 0 and C 1 such that We are going to prove that there exist T > 0 and positive constants P and Q such that for all 0 ≤ t ≤ T and n ∈ N 0 . For n = 0 and from (3.4), there exists Q > 0 such that And for u 1 and using (3.4), there exists P > 0 such that Proceeding similarly for n = 1, it follows that And for u 2 , we have that Denoting T := min{T 1 , T 2 , T 3 , T 4 } and making the same accounts as in the case n = 2, we obtain that Thus, proceeding inductively we get (3.5) for all n ∈ N 0 . Convergence of the approximation scheme. Now, we show that the sequence {u n } n∈N is a Cauchy sequence in L ∞ (0, T ; B s−1 2,q (R 3 )) for some T ≤ T . So, we consider the difference between the systems (AP n+1 ) and (AP n ), and denoting u n+1 := u n+1 − u n and θ n+1 := θ n+1 − θ n , we obtain the following system Applying both the Leray-Helmholtz projector P and the operator ∆ j to the above system, we get Here, we have used the skew-symmetry of e 3 × and the condition ∇·u n = ∇·u n−1 = 0. By the Cauchy-Schwarz inequality, it follows that and then Integrating over (0, t), we arrive at the estimates Multiplying by 2 (s−1)j and applying the l q (Z)-norm, we obtain (3.7) Next, we can take the L 2 -norm product with u n+1 and θ n+1 to the first and the second equation in (3.6), respectively, to get

Cauchy-Schwarz inequality leads us to
Integrating over (0, t), we have that In view of the embedding (2.4) and equivalence (2.3), we have that provided that the indexes s, p, q satisfy s > 3/p + 1 with 1 ≤ p, q ≤ ∞ or s = 3/p + 1 with 1 ≤ p ≤ ∞ and q = 1.

3.2.
Blow-up criterion. The subject of this subsection is a blow-up criterion of BKM type.
For the first case, we derive an estimate in B 1 ∞,1 for u. Since u = P + u + P − u, it suffices to obtain an estimate in B 1 ∞,1 for P + u and P − u. By (3.19) and (3.20), we remark that Thus, Similarly, we have that Therefore, for all 0 < T < T * , it holds that Using (3.16), it follows that for all 0 < T < T * . Now, we define Hence, there exist positive constants C 2 and C 3 (independent of Ω) such that for all t ∈ [0, T * ). Now, considering for 0 < T < ∞ the set C T defined by } and its supremum T * = sup C T , we are going to show that T * = min{T, T * }. For that, we assume by contradiction that T * < min{T, T * }. Thus we can select T so that T * < T < min{T, T * }, and then, since u ∈ C([0, T ]; B 7/2 2,1 (R 3 )), it follows that U (t) is uniformly continuous on [0, T ] and Choosing |Ω| > 0 such that ) exp (C3U (T )) 2 exp (2C3U (T )) u0 B 7/2 2,1 The latter estimate contradicts the definition of T * , since we can choose L ∈ ( T * , T ) such that U (L) ≤ C 2 T 1− 1 . Therefore, with Ω satisfying (3.24), it follows that T * = min{T, T * }. If T * < T , it holds that T * = T * = sup C T and From the blow-up criterion, it follows the result for the case s = 5/2 with q = 1. Now we turn to the case s > 5/2 with 1 ≤ q ≤ ∞. In this case, we can choose α = α(s) ∈ (0, 1) such that s ≥ 5/2 + α. Also, for each 1 ≤ q ≤ ∞ we consider 2 < r ≤ ∞ such that q ≤ r. We are going to obtain an estimate in B 1+α ∞,∞ for the solution u. Due to the equality u = P + u+P − u, it is suffices to derive an estimate in B 1+α ∞,∞ for P + u and P − u. By the embedding l q → l ∞ , Minkowski inequality (q ≤ r) and (3.20), note that Therefore, there exists a positive constant C = C(r, α) such that Using similar arguments for the other terms in (3.18), it follows that  We recall the definition of U (t) and use (3.17) and (3.16)  for all t ∈ [0, T * ). Then, there exist C 4 and C 5 (independent of Ω) such that for all t ∈ [0, T * ). As before, we also define We are going to prove that T * = min{T, T * } by contradiction. In fact, assuming that T * < min{T, T * }, we can take T so that T * < T < min{T, T * }.
The blow-up criterion concludes the proof for the present case.