On global large energy solutions to the viscous shallow water equations

By exploring the smooth effect of the heat flows and the weighted-Chemin-Lerner technique, we obtain the global solutions of large energy to the viscous shallow water equations with initial data in the critical Besov spaces, which improves the previous small energy type arguments [ 5 ], [ 13 ]. Moreover, the method used here is quiet different from [ 5 ], [ 13 ].

The shallow water equations are the simplest form of equation of motion that can be used to describe the horizontal structure of the atmosphere. They describe the evolution of an incompressible fluid in response to gravitational and rotational accelerations. The solutions of the shallow water equations represent many types of motion, including Rossby waves and inertia-gravity waves. One can find more details of the derivation for the viscous shallow water equations in [2,3]. Bui in [12] proved the local existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for the shallow water equations with initial data in C 2+α by using Lagrangian coordinates and Hölder space estimates. Following the energy method of Matsumura and Nishida [8], Kloeden in [7] and Sundbye in [11] independently obtained the global classical solutions to the Cauchy-Dirichlet problem. Recently, Wang and Xu in [13] proved the local solutions to system (1) for any initial data, and global solutions for small initial data in the Sobolev space H s (s > 2) as well as the initial height bounded away from zero. Chen, Miao and Zhang [5] further extended the result of [13] with low regularity by using the weighted Besov space. More precisely, they obtained the following result: then there exist a global unique solution (h −h 0 , u) of (1) such that h ≥ 1 2h 0 and h −h 0 ∈ C(R + ;Ḃ 0 2,1 ∩Ḃ 1 2,1 ), (h −h 0 ) L ∈ L 1 (R + ;Ḃ 2 2,1 ), (h −h 0 ) H ∈ L 1 (R + ;Ḃ 1 2,1 ), u ∈ C(R + ;Ḃ 0 2,1 ) ∩ L 1 (R + ;Ḃ 2 2,1 ).
where f L def = j<0∆ j f and f H def = j≥0∆ j f .
The aim of this paper is to construct the global solutions of the shallow wave equations with initial data in L p -type Besov spaces and large energy. Our key observation is that the energy of the solutions about the post-perturbation equations for the incompressible parts are finite although the energy of the perturbation equations are infinite. The proof rely highly on the perturbation theory and weighted-Chemin-Lerner technique. Denote ν = 1, the projector by P = I − Q := I − ∇∆ −1 div and the heat flows by e t∆ . We are now in the position to state the main result of the present paper: . If there exist two positive constants c 0 and C 0 such that then the system (1) has a unique global solution (h −h 0 , u) such that is small enough, the condition (2) is satisfied automatically. On the other hand, due to the embedding relation can be arbitrarily large. Thus, our theorem extends the previous results obtained by Wang et al. in [13] and Chen et al. in [5]. Moreover, the method used here is quiet different from that of [5,13].
The remaining part of the paper is organized as follows. In Section 2 we give some preliminaries which will be used in the sequel. Sections 3 is devoted to the proof of Theorem 1.2.
2. Preliminary. In this section, we recall some basic facts on Littlewood-Paley theory (see [1]). Throughout this paper, C stands for a generic positive constant which may vary from line to line. Let S(R 2 ) be the Schwartz class of rapidly decreasing functions. Choose two nonnegative radial functions χ, ϕ ∈ S(R 2 ) supported respectively in B = {ξ ∈ R 2 ; |ξ| ≤ 4 3 For h = F −1 ϕ and h = F −1 χ, we define the frequency localization operator as follows:∆ Denote by S h (R 2 ) the space of tempered distributions u such that lim j→−∞Ṡ j u = 0 in S (R 2 ).
By telescoping the series, we thus have the following Littlewood-Paley decomposition We would like to mention that the Littlewood-Paley decomposition has a nice property of quasi-orthogonality: where the positive constant C is independent of f and j.
Now we recall the definition of homogeneous Besov spaces.
Definition 2.2. Let s ∈ R, 1 ≤ p, r ≤ ∞, the homogenous Besov spaceḂ s p,r is defined byḂ The homogenous Besov spaces obey the inclusion relations stated in the following lemma. (1) In addition to the general time-space such as L ρ T (0, T ;Ḃ s p,r ), we introduce a useful mixed time-space homogeneous Besov space L ρ T (Ḃ s p,r ), which was introduced by Chemin and Lerner in [4].
Using the Minkowski inequality, it is easy to verify that For the convenience of readers, we recall the following form of weighted Chemin-Lerner type norm: The Bony's decomposition is a very effective method to estimate the nonlinear terms in fluid motion equations. Here, we recall the decomposition in the homogeneous context: wherė Next, we give some multilinear estimates in the Besov spaces which will be used in this paper.
Using Lemma 2.1 and the property of quasi-orthogonality, we get that Similarly, using the fact s 2 ≤ 2 p , we also obtain Now we turn to estimate the last term. Denote 1 r def = 1 p + 1 2 ≤ 1. Applying Lemma 2.1, Hölder's inequality, Young's inequality and the fact s 1 + s 2 > 0, one has This completes the proof of the lemma. ).
Proof. We first write uv as follows .
For the last termṘ(u, v), we get This completes the proof of this lemma.
There exists a constant C > 0 depending only on s such that for all j ∈ Z and k = 1, 2, we have where (c j ) j∈Z denotes a sequence such that c j 1 ≤ 1.
Let us recall the parabolic regularity estimate for the heat equation to end this section.
Then there holds ).
In order to deal with composition functions in the Besov spaces, we also need the following proposition: Lemma 2.11. (see [1]) For α < 0 < β, let G be a smooth function defined on the open interval (α, β) so that G(0) = 0. Assume that f : R → I ⊂ (α, β) for an interval I. Then we have the estimate Let T * be the largest time so that there holds (5). Hence to prove Theorem 1.2, we only need to prove that T * = ∞.
Without loss of generality, we assume thath 0 = 1. Replacing h by h + 1 in (1), we reformulate the system (1) as To prove T * = ∞, we need to produce some a priori estimates for the incompressible part and the compressible part of the solutions.
Throughout we make the assumption that sup t∈R+, x∈R 2 which will enable us to use freely the composition estimate stated in Proposition 2.11. Note that asḂ 2 p p,1 → L ∞ , condition (7) will be ensured by the fact that the constructed solution has small norm.

3.1.
Estimates for incompressible part of the solutions. Applying project operator P to the second equation in (6), we can get the incompressible part of the shallow wave equations as follows: Assuming Pu F is the solution of the following Cauchy problem of heat equation: Denote Pū = Pu − Pu F . Subtracting (9) from (8) gives where PM = − P(Pu F · ∇Pū) − P(Pū · ∇Pu F ) − P(Pu F · ∇Qu) − P(Qū · ∇Pū) − P(Pū · ∇Pū) − P(Pū · ∇Qu) − P(Qu · ∇Pū) − P(Qu · ∇Qu) Throughout this paper, we denote with λ > 0 and similar notations for Pū λ , Qu λ . Applying Lemma 2.10 to the Cauchy problem (9) gives So h λ , Pū λ and Qu λ are well defined. Now, we begin to present the estimates for the incompressible part of the solutions.
Multiplying by exp − λ t 0 f (τ )dτ on both hand side of the first equation in (10) and applying∆ j , we obtain Taking the L 2 inner product of the resultant equation with∆ j Pū λ gives Integrating the above inequality over [0, t] and summing up the resultant inequality with respect to j, we arrive at where we use the Minkowski inequality and the fact Pū| t=0 = 0. To hand the first two terms in (PM ) λ , we start with the Lemma 2.6 and interpolation inequality that Similarly, we get t 0 P(Pu F · ∇Qu) λ Ḃ0 2,1 Applying Lemma 2.6, Lemma 2.11 and the embedding relationḂ In addition, it's easy to see that Inserting estimates (14)-(18) into (13) and choosing λ ≥ 4(C + 1), we can obtain the estimate of the incompressible part of the solutions 3.2. Estimates for compressible part of the solutions. In this section, we mainly study the estimates for compressible part of the solutions. We first get by applying operator Q to the second equation of (6) that ∂ t h + u · ∇h + div Qu = −hdiv u, with Applying∆ j to the first equation in (20) gives Taking L 2 inner product of∆ j h with (21) and using integrating by parts, we have Applying∆ j to the second equation in (20) gives We can get by using a similar derivation of (22) that 1 2 Applying the gradient ∇ on (21), we have Taking L 2 inner product of∆ j ∇h with the previous equation implies Testing (23) by ∇∆ j h and (25) by∆ j Qu, we arrive at By summing up (22), (24), (26) and (27), we get 1 2 It is readily seen that for all j ∈ Z and for a constant c. Therefore, we deduce from (28) that Multiplying by exp − λ t 0 f (τ )dτ on both hand side of the above equation yields From (24), we get This together with (31) yields Note that the embedding relationḂ 2 p p,1 (R 2 ) → L ∞ (R 2 ) with p ≥ 2, Lemmas 2.3, 2.6 and the definition of the weighted-Chemin-Lerner-norm, we get Similarly, by Lemma 2.8, we have Applying Lemma 2.9 and the embedding relationḂ Using Lemma 2.6, we easily obtain t 0 (hdiv Qu λ , ∇(hdiv Qu λ ) Ḃ0 To get the estimates for the compressible part of the solutions, we have to deal with the last term about QG in (32). In fact, from u = Pu+Qu = (Pu F +Pū)+Qu, one can deduce that Following the same line, one has By virtue of Lemma 2.6 and the interpolation inequality, we arrive at Similarly, By summing up (37)-(41), we obtain (17) and (42), one can get Combining with (19) and (44), we finally conclude that In what follows, we will prove that T * * = T * under the assumptions of (2).
If not, assuming that T * * < T * , for any t ≤ T * * , we get from (45) that As a consequence, for t ≤ T * * , where we used the estimate (11). If the smallness condition (2) is satisfied, then we deduce from (48) that c 0 2 for t ≤ T * * , which contradicts with (46). Whence we conclude that T * * = ∞ and the conclusion of Theorem 1.2 follows.