Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

This paper is devoted to the study of the modified quasi-geostrophic equation \begin{document}$ \partial_t\theta+u\cdot\nabla\theta+\nu\Lambda^\alpha\theta = 0 \ \ \mbox{ with } \ \ u = \Lambda^\beta\mathcal{R}^\perp\theta $\end{document} in \begin{document}$ \mathbb{R}^2 $\end{document} . By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case \begin{document}$ \beta = \alpha-1 $\end{document} and the existence of regular solutions for large time \begin{document}$ t>T $\end{document} with respect to the supercritical case \begin{document}$ \beta >\alpha -1 $\end{document} in Besov spaces. Earlier results for the equation in Hilbert spaces \begin{document}$ H^s $\end{document} spaces are improved.

An interesting question arises as to whether the result of [7] still holds true for α ∈ (1, 2). This choice of α implies that the velocity u = Λ α−1 R ⊥ θ is more singular than θ. Therefore, u is not Hölder continuous when θ ∈ L ∞ (R 2 × (0, ∞)). What is more, the regularity of the solution cannot be improved by the bootstrap argument presented in [7]. To answer this question, Miao and Xue [28,29] proved the global existence of regular solutions to the critical and supercritical modified quasi-geostrophic equation respectively in the following sense.
The purpose of the present paper is to investigate the existence of regular solutions to (1) in critical homogeneous Besov spaces with respect to the invariance of (1) under the scaling transformation (2). The well-posedness and the regularity of fluid flow motions in scaling critical spaces have been extensively studied (see, for example, [5,25,34] for incompressible Navier-Stokes equations and [1,4,6,17,20,30] for (3). This is partially due to the fact that the solutions are expected to be regular in scaling critical spaces [15,25].
For the readers' convenience for the comparison with the existing results, the main results of the present paper read respectively as follows.
Theorem 1.4. For p, q ∈ [2, ∞), θ 0 ∈ B 2 p +1+β−α p,q (R 2 ) and the supercritical situation β ∈ (0, 1 2 + 1 p ) and α ∈ (2β, β + 1), assume that θ is a Leray-Hopf weak solution of (1). Then there exist positive constants T 1 < T 2 dependent upon the quantities p, q, α, β, ν and θ 0 such that For the significance of the present study, Theorem 1.1 and the improved result of May [27] are extended by Theorem 1.3, while Theorem 1.2 is covered by Theorem 1.4, since, for m > 1, On the other hand, Theorem 1.3 also improves the result of Yamazaki [35], where the global well-posedness of Equation (1) with β = α − 1 is proved by a different method, which is independent of the work of May [27]. This paper is organized as follows. Section 2 contains basic properties of Littlewood-Paley theory and Besov spaces. In Section 3, we develop techniques from [6,29,30,34] to derive local existence and uniqueness of solutions to (1) in critical Besov spaces. This local solution result generalizes the counterpart of [28,29] in Hilbert spaces. In Section 4, by following the techniques of [29] with more careful manipulation, we show that the local solutions initially in Besov spaces given in Section 3 are smooth for t > 0. This result is an extension of the smoothing effect result of [16] in Hilbert spaces. Moreover, compared with the examination of Dong and Li [17] on the global existence of regular solutions to (3), our proof is delivered in a more simplified manner by skipping the use of an L ∞ estimate of ∇θ and ∇u. This estimate plays a crucial role in [17]. Finally, in Section 5, we first prove Theorems 1.3; then Theorem 1.4 is derived in a straight-forward manner from Theorem 1.2 and the smoothing effect result obtained in Section 4 because the local solution in the Besov space B σ p,q (R 2 ) is smooth for t > 0 and hence is in the Hilbert spaces H m (R 2 ).
2. Preliminaries. Throughout this paper c represents a generic positive constant which is independent of the quantities t, T , x, f , g, u, θ, j, k and q 1 . For simplicity, A B denotes the inequality A cB, and A B stands for the combination of B A and A B.
S(R 2 ) denotes the Schwartz space, S (R 2 ) represents the space of tempered distributions, F is the Fourier transform and Λ α = F −1 |ξ| α F . To define Besov spaces, we use the Littlewood-Paley dyadic decomposition (see, for example, [2]) by taking a positive function φ ∈ S(R 2 ) such that and φ(ξ) = 1 for |ξ| < 3 4 and the dyadic block symbols Thus Littlewood-Paley dyadic blocks are defined as Thus we have Littlewood-Paley unit decompositions where P(R 2 ) denotes the set of all polynomials over R 2 .
Note that B s p,q (R 2 ) =Ḃ s p,q (R 2 ) ∩ L p (R 2 ) for s > 0. For this case, the norm · B s p,q is equivalent to · Ḃs p,q LARGE-TIME REGULAR SOLUTIONS 3753 Definition 2.3. For T > 0, s ∈ R and 1 p, q, ρ ∞. The homogeneous timespace Besov space is defined by By the Minkowski inequality, it is easy to clarify the imbedding relations: We will also use the following Bernstein inequalities.
Proposition 2.5. For s 0, 1 p q ∞, j ∈ Z and constants K > 0, K 2 > K 1 > 0, there hold the inequalities: if To treat the fractional Laplacian in L p spaces, we need the following modified Bernstein's inequalities.
The following pointwise multiplier estimate is a consequence of Bony's decomposition (see, for example, [6]).
3. Local well-posedness in critical Besov spaces. This section is devoted to the following existence and uniqueness of local solutions to (1). Our analysis is developed from [6,29,30,34].
. Then there exists a positive constant T 0 such that (1) admits a unique solution θ satisfying Proof. This theorem is shown through four steps.
Firstly, we show an a priori estimate for the solution θ in the space L q1 (0, T ;Ḃ , applying the operator ∆ j to (1) and using Bony's decomposition of the nonlinear term, we have Multiplying (14) by p|∆ j θ| p−2 ∆ j θ and then integrating in R 2 , we use Proposition 2.6 and the divergence free condition ∇ · u = 0 to produce By Hölder inequality, (15) becomes the inequality with the common factor ∆ j θ p−1 Lp , which can be removed. Thus (15) By the L p commutator estimate [34], Hölder and Bernstein inequalities, the righthand side of (16) is bounded by Since the last term can be absorbed by the third term in (17), thus (16) is formulated as

LARGE-TIME REGULAR SOLUTIONS 3755
By Gronwall inequality, the previous equation yields Taking the L q1 (0, T ) norm of (18), multiplying by 2 j(τ + α q 1 ) to the resultant equation and then taking l q (Z) norm, we obtain the required estimate in the space with .
Secondly, we consider the case q 1 ∈ [2, ∞) to obtain the estimate for τ > β − σ − 2α q1 given in the first step. It is readily seen that the first term on the righ-hand side of (19) is estimated as To consider the other terms on the right-hand side of (19), we use Young inequality and the condition q 1 2 to obtain

This yields that
The last term on the right-hand side of (19) can also be estimated in a similar way. By Young inequality, Hölder inequality and the condition q 1 2, we find that This together with Young inequality gives that where we have used the condition τ + σ − β + 2α q1 > 0.
Hence we obtain (20) by combining (19) and (21)- (24). Equation (20) with q 1 → ∞ implies the estimate: (25) However, the second term on the right-hand side of (25) does not vanish as T tends to 0. Thus, to show the local existence result of (1) in the space L ∞ (0, T ;Ḃ σ p,q ), it is necessary to derive an additional estimate in the next step analysis.
Thirdly, for q 1 ∈ [1, ∞] and τ + σ − β + α > 0, we show the a priori estimate We begin with q 1 = 1. Similar to the derivation of (20), we use Young inequality, Hölder inequality and the condition α > 2β to obtain Collecting the above terms and (21) with q 1 = 1, we obtain from (19) that In the same way, we have , for q 1 = ∞, . Therefore, (26) is obtained after the use of the interpolation inequality between L 1 and L ∞ .
Finally, we use the Banach contraction mapping principle to show the local existence and the uniqueness assertion for 1 q 1 ∞ and θ 0 ∈ B σ p,q (R 2 ).
is the local solution of (1) derived from Theorem 3.1. Then The result stated in Theorem 4.1 is also valid for α = 0. For this case, the index q can be extended to q ∈ [1, ∞).
Proof. It is easy to see that t γ θ is a solution of the following equation: Applying the operator ∆ j to the both sides of (34) and using Bony decomposition, we have Similar to the estimate of the Bony decomposition in the proof of Theorem 3.1, multiplying (35) by p|∆ j (t γ θ)| p−2 ∆ j (t γ θ) and then integrating over R 2 , we obtain after the use of Proposition 2.6 and the divergence free condition. Then multiplying the above inequality by q2 j(σ+γα)q ∆ j (t γ θ) q−1 Lp and summing over j, we find that After the use of Young inequality, we have To estimate the previous equation, we use the L p commutator estimate [34] and Bernstein inequality to obtain Hence, due to q > 1 and α > 0, we have Since α > 2β and q 2 implies α > (1 − 1 q ) −1 β, by Young inequality, we obtain Furthermore, observe that q > 1, β > 0, α < 2 p + 1 and γ 0 yield α < (2 − 1 q ) −1 (β + 2( 2 p + 1) + γα). Hence by Young inequality, we find that Similarly, we have Therefore, inserting (37)-(40) into (36), we have Let t ∈ (0, T 0 ). Integrating the previous inequality over the interval (0, t) yields Then using Gronwall's inequality, we have It remains to estimate the term t γ θ in time-space Besov spaces. In order to do this, it is sufficient to bound the L p -norm of t γ θ. Multiplying (34) by p|t γ θ| p−2 t γ θ, applying the divergence free condition and using the fact that ν R 2 Λ α (t γ θ)|t γ θ| p−2 t γ θdx 0, we deduce that Now we are going to prove that the right-hand side of (43) is finite. By Theorem 3.1, we have θ For the general case of γ > 0, it can be shown that (45) remains true by interpolation.
The proof of Theorem 4.1 is completed.
Remark 4.2. Dong and Li [17] studied the smoothing effect of the solution θ of (1) with p ∈ [2, ∞), q ∈ [1, ∞), β = 0 and α ∈ (0, 1]. More precisely, they first proved that there exists a T 1 ∈ (0, ∞) such that, for γ ∈ (0, 1/2), for some constant c independent of γ and q 1 . Then they showed that for any δ ∈ (0, T 1 ), q 1 ∈ [1, ∞] and γ > 0, θ L∞(δ,T 1 ;B α p,q ) < ∞. Their proof is divided into these two steps because of the need of the L ∞ estimate for ∇θ. In view of the work of [17], it is an interesting question whether or not the inequality (33 ) holds for the modified quasi-geostrophic equation (1). Similar to the proofs of Theorems 3.1 and 4.1, we point out that a result which is even stronger than (33 ) can be obtained for the equation (1). That is, there exists a constant T 1 ∈ (0, T 0 ) such that for any γ > 0 and q 1 ∈ [1, ∞], However, to avoid a tedious computation, we chose to state and prove (33), which is sufficient for us to deduce that θ is smooth on any closed interval contained in (0, T 0 ). We end this remark by pointing out that our proof of the infinite differentiability of (1) with β = 0 is simpler than the method presented in [17] because of the skip of the L ∞ estimate for ∇θ.