LONG-TIME SOLVABILITY FOR THE 2D DISPERSIVE SQG EQUATION WITH IMPROVED REGULARITY

. In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter A . Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large A and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes.


(Communicated by Eduard Feireisl)
Abstract. In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter A. Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large A and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes. where θ = θ(x, t) is a scalar function, which represents the potential temperature of the fluid (or a buoyancy field), u stands for the velocity field and A > 0 is the amplitude parameter. The Riesz transforms R l , l = 1, 2, are defined by means of the Fourier transform as Equation (1.1) models the evolution of a surface temperature or surface buoyancy field in a rapidly rotating stratified fluid, which plays the same role as the conserved potential vorticity driving the interior dynamics. The presence of an environmental horizontal gradient −R 1 θ = ∂ x1 Λ −1 θ, where Λ := √ −∆ , represents the meridional advection of a large-scale buoyancy, which comes from the meridional variation of the Coriolis parameter [26].
In recent years, the unforced SQG (i.e. A = 0) has spurred a lot of mathematical research since the nondissipative (inviscid) SQG equation have properties that are similar to those of the 3D Euler (E) system in vorticity form [14], although the former equation has been shown to possess global finite energy weak solutions [29]. Moreover, in the presence of a fractional dissipation given by ( √ −∆ ) α , 0 < α ≤ 2, the issue of global regularity has been the object of numerous studies. In particular, the critical case (α = 1) is challenging since the balance between the nonlinearity and the dissipative term is the same no matter the scale at which one zooms in, so that, in this sense, this case is the 2D analogue of the 3D Navier-Stokes (NS) system. The global regularity of the critical dissipative SQG was an outstanding open problem until the independent breakthroughs in [6] and [23]. Unlike, this problem is still open for the supercritical dissipative SQG (0 < α < 1). For extensive details and further issues related to the SQG, the reader is referred to, e.g., [8], [10], [11], [15], [16], [17], [21], [29], [35] and their references.
Just as for the 3D Euler equations, and the supercritical dissipative SQG and Navier-Stokes equations, the long-time existence and uniqueness (i.e., solvability) of the inviscid SQG equation is an outstanding open problem in several settings. In fact, the problem is subtle since the 3D Euler equations can exhibit ill-posedness in Sobolev and Besov spaces, even locally-in-time, depending on the regularity index (see [5] for further details). However, the dispersive forcing in (1.1) has a crucial effect on the dynamics and makes this equation analogous to the 3D Euler equation with Coriolis forcing term Ω (e 3 × u), i.e. the Euler-Coriolis (EC) system ∂ t u + u · ∇u + ∇p + Ω (e 3 × u) = 0 , where the parameter |Ω| −1 is the Rossby number. The viscous case of (EC) then forms the 3D Navier-Stokes-Coriolis (NSC) system and both these systems are basic oceanographic and atmospheric models dealing with large-scale phenomena (cf. [13]). At the same time, the dispersive forcing in (1.1) does not contribute to energy decay (see Remark 2.6) similarly to the Coriolis forcing in (EC), so that the 2D analogues of (EC) and (NSC) are, respectively, equation (1.1) and the dissipative dispersive SQG (viscous case of (1.1)). The main advantage of these systems is that in the limit of vanishing Rossby number (Ω → ∞) a stabilization effect arising from the Coriolis term yields the long-time well-posedness of strong solutions for arbitrary initial data [2,7,19,24].
In particular, Koh, Lee and Takada [24] recently showed that (EC) has a unique local-in-time solution u in the space . Also, assuming more regularity on u 0 , s > s 1 = 5 2 +1, they proved that the solution u can be uniquely continued up to an arbitrarily large time T Ω for speeds of rotation |Ω| sufficiently large; the proof relies crucially on their Strichartz-type estimates. For the viscous case (NSC), similar estimates are used to show global well-posedness in Sobolev spaces (see [2], [3], [13], [19]). Bearing in mind the previous results for (EC) (and also those for 3D Euler equations, see [5], [9], [27], [36]), the cases s 0 and s 1 emerge as important values of regularity for local and long-time solvability of (EC), respectively, in finite L 2energy spaces of Sobolev and Besov types, i.e., H s p and B s p,q -spaces with p = 2. These cases were treated in [1], where the results of [24] were extended by employing LONG-TIME SOLVABILITY 2D DISPERSIVE SQG EQUATION 1413 the framework of Besov spaces. There, the local solvability were proved for initial data in the borderline Besov space B s0 2,1 and small existence-time independent of Ω ∈ R. It is then also shown that, for large |Ω|, it is possible to obtain long-time solvability of (EC) in B s 2,1 with the improved regularity s = s 1 . In view of the results for (EC) in H s p and B s p,q -spaces, and making a dimensional analysis, a suitable regularity index for the long-time solvability of (1.1) would be s 1 = n 2 +2 for p = 2. Thus, exploring the dispersive effect (i.e., for large values of A), our main aim is to show the long-time solvability of (1.1) in Besov spaces with the improved regularity s = s 1 . Our starting point are the results developed for (1.1) which are related to those mentioned above for (EC), mainly the papers [18] and [33]. In [18], Elgindi and Widmayer proved sharp dispersive estimates and applied it to show existence of strong solutions with A = 1 and existence-time In fact, by means of a scaling argument, their result works well for arbitrary T > 0 and The dispersive and corresponding Strichartz estimates were later generalized by Wan and Chen [33] to show long-time solvability of strong solutions for (1.1) with an improved regularity taking θ 0 ∈ H s and s > 3, as well as for a Boussinesqtype system derived from it. Moreover, in a follow-up to [33], Wan [32] refined the condition on the size of the dispersive forcing in [33] to obtain long-time solvability and showed that it suffices that A ≥ C T 3 θ 0 4 H s , for s > 3 and some universal constant C > 0. We remark that analogous results to those of (NSC) have been shown for dissipative dispersive SQG, particularly dealing with the issues of global regularity [22], as well as the global well-posedness and asymptotic behavior of solutions with large dispersive forcing [7].
However, the methods of [32,33] only treat the cases for regularity s > 3 and do not reach the case s = 3. Notice that for s > 3 we have the continuous inclusion H s (R 2 ) → B 3 2,1 (R 2 ). Thus, our main goal is to improve these results to obtain long-time solvability for s = 3, although we borrow from their Strichartz estimates. We prove the following results. Theorem 1.1. Let s and q be real numbers such that s > 2 with 1 ≤ q ≤ ∞ or s = 2 with q = 1.
(i) (Local solvability) Let θ 0 ∈ B s 2,q (R 2 ). There exists . Our proofs also provide a relation between the strength of the dispersive forcing and the increased time of existence (see (4.17)), where in order to deal with the critical case s = 3 we start from the local well-posedness in item (i) of Theorem 1.1 and later use a blow-up criterion. In general lines then, as in [1,24], we employ the following basic steps: construction of approximate solutions {θ ε } ε∈(0,1) ; a priori estimates uniformly w.r.t the amplitude parameter A; passing to the limit for obtaining local-in-time solvability; blow-up criterion; and finally long-time solvability.
Here we carry out estimates in Besov spaces with suitable regularity for (1.1). In order to obtain a solution as the limit of {θ ε } ε∈(0,1) , we control the approximations θ ε via estimates based on localizations and B s 2,1 -norms (see, e.g., Proposition 3.3 and the proof of Theorem 1.1) and using the embedding B 1 2,1 (R 2 ) → L ∞ (R 2 ) which is not verified for H 1 (R 2 ). Then, for large values of A and s = 3, we obtain the long-time solvability by showing a blow-up criterion and controlling globally the integral t 0 ∇u(τ ) L ∞ dτ by using Besov-norms of θ. We remark that Chae-Lee [12] showed the global existence and uniqueness for the dissipative SQG in the Besov space B 2−γ 2,1 (0 < γ < 1) for small initial data. This result also applies to the case γ = 0 and is thus linked to our local result. However, the approximation used in [12] is different than our parabolic regularization approach, which we used specifically to obtain a local result with existence time independent of the dispersion parameter A. This independence allows us to show the extension to long-time solvability without restriction on the size of initial data.
In this regard our work is also motivated by Vishik's result [31] where the longtime solvability for the 2D Euler equation was shown in Besov spaces of borderline regularity B 2/p+1 p,1 with 1 < p < ∞. See also Chae [9] for local solvability of ndimensional Euler equations in borderline Besov spaces B n/p+1 p,1 .
Given that the dimensionality of the inviscid SQG is analogous to that of the 3D Euler equation it is very hard to obtain long-time solvability for arbitrary initial data in the same borderline Besov spaces of [31] and in that of [12] with γ = 0. For this reason we consider it relevant to show that the stabilization effect from large dispersive forcing allows the long-time solvability for arbitrary initial data with the improved regularity s = s 1 = 3, as is the case in the 3D Euler-Coriolis equations. Nevertheless, it is open whether this regularity is sharp. This paper is organized as follows. In Section 2 we give some preliminaries about Besov spaces and Strichartz estimates. Section 3 is devoted to the parabolic approximation scheme {θ ε } ε>0 where we show its local existence uniformly with respect to the parameters ε and A. The proof of Theorem 1.1 is performed in Section 4 where items (i) and (ii) are addressed in subsections 4.1 and 4.2, respectively.

2.
Besov spaces and Strichartz estimates. In this section we review some facts about Besov spaces. For further details on these spaces, the reader is referred to [4]. Moreover, we give some suitable estimates of Strichartz type.
2.1. Besov spaces. We denote the Schwartz class and its dual (space of tempered distributions) respectively by S(R 2 ) and S (R 2 ). For f ∈ S (R 2 ), f stands for the Fourier transform of f . Let φ 0 be a radial function in S( and set ψ = S 0 . Let P denote the set of polynomials with two variables. For each j ∈ Z, consider the Fourier localization operator ∆ j : which is called the Littlewood-Paley decomposition of f. For s ∈ R and 1 ≤ p, q ≤ ∞, the homogeneous and inhomogeneous Besov spaces are defined respectively bẏ The spacesḂ s p,q and B s p,q endowed with · Ḃs p,q and · B s p,q are Banach spaces. For s > 0, the following equivalence of norms holds: In view of (2.2), without loss of generality, we can assume that In the next lemma, we recall the Bernstein inequality.
Some Leibniz-type rules in Besov spaces are the subject of the lemma below (see [9]).
. We will obtain estimates in a setting involving Besov norms of the terms in the right-side hand of the approximate problem (3.2). In this direction, we will need some estimates for the heat semigroup.
Lemma 2.4. (see [25]) Let 1 ≤ p, q ≤ ∞ and s 0 ≤ s 1 . We have the estimate Commutator estimates inḂ s p,q and B s p,q will also be useful to obtain convergence of our approximate solutions. Recall the commutator operator We have the following estimates (see [9,30,34]): 2.2. Strichartz estimates. We will employ the Strichartz estimates of [33], linked to the dispersive term Au 2 of (1.1), which will allow us to obtain long-time solvability for (1.1).
Firstly, we note that the term Au 2 does not contribute to the energy decay, which is in direct analogy with the Coriolis term in the rotating Euler equations.
The lemma below contains the Strichartz estimate of [33].
Then, there holds , for all f ∈Ḃ From Lemma 2.7 and the change of variable t → At, we get .
Then, there holds and φ is a compactly supported smooth function in R 2 .
Proof. Proceeding similarly as Lemma 3.1 in [33] and using the change of variable t → At, it follows the result. Then, Proof. As in [33], we use the change of variable ξ = 2 j η to get Multiplying by 2 sj and applying the l q (Z)-norm, we have . Since γ ≤ q, we apply L γ (0, ∞) and use Lemma 2.8 to get Finally, we use the change of variable t → At to obtain (2.6).
3. Regularized problem and uniform estimates. In order to prove the item (i) of Theorem 1.1, we will first show the local existence and uniqueness of mildsolutions of the regularized problem constructed from (1.1) with viscous approximations. For this, we consider for each 0 < ε < 1 the following problem is the initial data. Considering the heat semigroup, we can represent the above equation in the following integral formulation We will prove that the above problem possesses a unique solution for each ε > 0 in a suitable class involving Besov spaces. For the second term in the right-side hand of (3.2), we have the following estimates.
(i) There exists C > 0 (independent of T ) such that
For the third term in the right-side hand of (3.2), we get the following estimates.
There exists C > 0 (independent of T ) such that the following inequalities hold
Proof. Considering s, p and q as in the hypotheses and using Lemma 2.4 and Lemma 2.3, it follows that Applying the supremum over [0, T ], we arrive at (3.5). On the other hand, notice that which gives (3.6).
Now, we prove that (3.1) has a local-in-time solution in Besov spaces with existence time independent of A ∈ R and ε ∈ (0, 1). Proposition 3.3. Let ε ∈ (0, 1) and A ∈ R. Assume that s > 2 with 1 ≤ q ≤ ∞ or s = 2 with q = 1, and that θ 0 ∈ B s 2,q (R 2 ). Then, there exists a time T = T ( θ 0 B s 2,q ) > 0 such that (3.1) has a unique strong solution θ ε satisfying Proof. We divide the proof of Proposition 3.3 in three parts.
First part (Local existence and uniqueness of θ ε ). In the first part, we will prove that there exist T ε,A = T ε,A (ε, A, θ 0 B s 2,q ) > 0 and a unique solution (3.1). For this, we recall the mild formulation for (3.1) Using Lemma 2.4, we obtain the following estimate and the complete metric space We will show that the map Γ is a contraction on W T for some T > 0. By Lemma 3.1, Lemma 3.2 and the continuity of the Riesz transforms R l 's in Besov spaces, we get constants C 1 > 0 and C 2 > 0 such that for all θ ε , θ ε ∈ W T . On the other hand, using Lemma 2.4 and (3.8), we can estimate for all θ ε , θ ε ∈ W T ε,A . Thus, we can applied the Banach Fixed Point Theorem in order to obtain a unique solution θ ε ∈ W T ε,A for (3.1).
Third part (Boundedness of θ ε ∈ C([0, T ]; B s 2,q (R 2 ))). Finally, we are going to prove the result. For that, we apply the Littlewood-Paley operator ∆ j to the equation in (3.1) and we take the L 2 -norm product with ∆ j θ ε (t) to obtain Since the second term in the right-hand side of (3.10) is non-negative, and recalling the definition of the commutator [u ε (t) · ∇, ∆ j ], we get By Hölder inequality, (3.12), Lemma 2.5 and the continuity of R l in Besov spaces, we have that d dt θ ε (t) Ḃs (3.13) Also, taking the L 2 -norm product with θ ε (t) in the first equation of (3.1) and using the fact ∇ · u ε = 0, it follows that (3.14) Recalling (2.3) and combining (3.13) and (3.14), we obtain the estimate The existence time T > 0 can be taken independent of ε ∈ (0, 1) and A ∈ R. In fact, if T ε,A < T , using (3.9) and (3.15) we can take T ε,A = T ε,A ( u 0 B s 2,q ) > 0 sufficiently small and obtain a solution for (3.1) with initial data θ ε (T ε,A ) ∈ B s 2,q (R 2 ) on the interval [T ε,A , T ε,A +T ε,A ] . Thus, the solution θ ε can be extended to [0, T ε,A + T ε,A ]. In case of being necessary, the same argument can be repeated in order to extend θ ε to [0, T ε,A + 2T ε,A ], [0, T ε,A + 3T ε,A ] and so on. Therefore, we obtain a solution θ ε for (3.1) on [0, T ] verifying estimate (3.15).

4.2.
Proof of item (ii). In this section we prove the long-time solvability of (1.1) for large values of A. We start with a proposition containing a blow-up criterion.
Proposition 4.1. Let s and q be such that s > 2 with 1 ≤ q ≤ ∞ or s = 2 with q = 1. Assume that θ 0 ∈ B s 2,q (R 2 ) and θ is the corresponding solution of (1.1) in the class C([0, T ); B s 2,q (R 2 )) ∩ C 1 ([0, T ); B s−1 2,q (R 2 )) satisfying ). Proof. By item (i) of Theorem 1.1, we know that the existence time T > 0 is independent of A. Taking the L 2 -inner product and using the fact that the dispersive term does not contribute to the energy estimates (see Remark 2.6), we get the energy equality θ(t) 2 L 2 = θ 0 2 L 2 for all t ∈ [0, T ). (4.11) Applying ∆ j in (1.1), multiplying the result by ∆ j θ and using (u(t) · ∇)∆ j θ, ∆ j θ L 2 = 0, we obtain Integrating over (0, t) and after estimating the L 2 -inner product, we arrive at Now, multiplying (4.12) by 2 sj and taking the l q (Z)-norm, it follows that Estimating the commutator operator via Lemma 2.5 (i), we obtain a constant C > 0 such that (4.13) It follows from (4.11) and (4.13) that By Remark 2.2, (4.6) and the continuity of R l in Besov spaces, there exist positive constants C 3 and C 4 such that By Gronwall's inequality, we get for all t ∈ [0, T ). By standard arguments, using T 0 ∇u(τ ) L ∞ dτ < ∞ and (4.14), we are done.
Here we have used the inequality R l (u(τ ) · ∇)θ(τ ) Ḃs , which can be shown by using Bony's paraproduct and the continuity of R l in Besov spaces. Also, by Lemma 2.8, the continuity of R l in L 2 and the embedding B s+1 2,q → B s 2,q → L 2 , we have t 0 e −AR1(τ −τ ) R l (u(τ ) · ∇)θ(τ ) L γ (τ ,t;L ∞ ) dτ It follows that Thus, for each 0 < t < T * , we have Therefore, for both cases of s and q such that s = 2 with q = 1 or s > 2 with 1 ≤ q ≤ ∞, we have that there exists C 5 > 0 such that (4.15) Next, for each 0 < T < ∞ we defineT = sup D T , where We first show thatT = min{T, T * }. We proceed by contradiction. So assume on the contrary thatT < min{T, T * }. We have that there exists T 1 such that T < T 1 < min{T, T * }. It follows that θ ∈ C([0, T 1 ]; B s+1 2,q (R 2 )), M(t) is uniformly continuous on [0, T 1 ] and (4.16) We now take |A| large enough so that Thus, we can choose T 2 such thatT < T 2 < T 1 with M( This contradicts the definition ofT . It follows thatT = min{T, T * } when A verifies (4.17). If T * < T , we have that T * =T = sup D T and It follows that M(T * ) < ∞ which contradicts the maximality of T * because the blow-up criterion. This concludes the proof.