Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 62, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.62 GLOBAL CLASSICAL SOLUTIONS TO EQUATORIAL SHALLOW-WATER EQUATIONS YUE FANG, KAIQIANG LI, XIN XU Abstract. In this article, we study the equatorial shallow-water equations with slip boundary condition in a bounded domain. By exploring the dissipa- tive structures of the system, we obtaining a priori estimates of the solution for small initial data. Then the existence of classical global solutions and exponential stability results are given. 1. Introduction In geophysical fluid dynamics, the rotating shallow water system is a widely used 2D approximation of the 3D geophysical hydrodynamic equations such as the Boussinesq equations and Euler equations [13]. In mid-latitudes, the rotational Coriolis terms are bounded away from zero and the rotation frequency is usually regarded as constant because the variations of the Coriolis force due to the curvature of the Earth can be neglected in many cases. However, the constant rotation frequency assumption is no longer reasonable in the equatorial region [11] since the tangential projection of the Coriolis force from rotation vanishes identically. In this article, we consider the equatorial shallow-water equations [6] Ht + U · ∇H +H∇ · U = 0, Ut + U · ∇U +∇H + yU⊥ = −U, (1.1) in which H = H(t, x, y) is the height, U = (u, v)(t, x, y) is the velocity, x the longitude, y the distance to the equator, and U⊥ = (−v, u)>. System (1.1) is supplemented with the initial and boundary conditions H ∣∣ t=0 = H0, U ∣∣ t=0 = U0, U · n ∣∣ ∂Ω = 0,∫ Ω H0 dx = H > 0, (1.2) where Ω ⊂ R2 is a bounded domain with smooth boundary ∂Ω, n is the unit outward normal vector on the boundary of Ω. 2020 Mathematics Subject Classification. 35Q35, 76N10, 76U05, 76M45. Key words and phrases. Equatorial shallow-water equations; global classical solution; exponential stability. ©2023. This work is licensed under a CC BY 4.0 license. Submitted April 20, 2023. Published September 21, 2023. 1 2 Y. FANG, K. LI, X. XU EJDE-2023/62 There have been a number of studies on the rotating shallow water system by physicists and mathematicians for its physical importance and mathematical chal- lenges. The rotating shallow water equations with constant rotation frequency can be modeled by Ht + U · ∇H +H∇ · U = 0, Ut + U · ∇U +∇H + fU⊥ = S(U), (1.3) where f is a constant and S represents the dissipation term like viscosity, damping, etc. Wang and Xu [17] proved the existence of a local large solution, as well as the existence of a global small solution, to the viscous shallow water system by using Littlewood-Paley theory. By the energy method of Matsumura and Nishida [12], and Sundbye obtained the global well-posedness of the system with small initial data in a bounded domain [15] and in the whole space [16], respectively. If the initial data close to a constant equilibrium state away from the vacuum, the existence of a global solution in Besov spaces was shown by Hao et al. [9]. For inviscid rotating shallow water system, Cheng and Xie [4] established a global classical solution under the zero relative vorticity condition by using the dispersive effect of the system. Subsequently, Qu [3] also proved the formation of singularities when the solution crosses certain thresholds. Moreover, the chemotaxis-shallow water system also attracted a lot of attention; see [1, 2, 18, 19] and the references therein. Geophysical equatorial flows are a rich source of new problems in applied math- ematics and partial differential equation theory. Mathematically, the equatorial shallow-water system was first studied by Majda and his collaborators. Dutrifoy and Majda [5] considered the singular limit problem H̃t + U · ∇H̃ + H̃∇ · U + 1 δ ∇ · U = SU , Ut + U · ∇U + 1 δ (∇H̃ + yU⊥) = SH̃ , (1.4) where the small parameter δ represents the Froude number (typical fluid velocity ratio to the gravity wave speed) and the height fluctuations of the fluid. SU and SH̃ are forcing terms. By exploiting the special structure of the system in suitable new variables, they obtained the uniform existence and the convergence of the solutions with unbalanced initial data. In [6], authors proved that, in a suitable limit, solutions of the equatorial shallow water equations would converge to zonal jets. Moreover, with Schochet, they [7] also gave a simpler proof to the above singular limit problem. In this article, we study the global well-posedness to the initial boundary value problem of the equatorial shallow-water equations (1.1)-(1.2). On the one hand, the damping term −U in the momentum equation (1.1) will make the solution of the system dissipative. On the other hand, by the non-penetrating boundary condition (1.2), it is clear that ∫ Ω H(x, t) dx = ∫ Ω H0(x) dx = H. Thus, it is natural to expect that as time goes to infinity the solution of the system will converge to its equilibrium state (H/|Ω|,0) provided that the initial perturbation around this equilibrium state is small, which is the most important part in this article. Before stating our main results, we give some notation. Throughout this paper, C will denote a generic constant which is independent of time. The norms in L2(Ω) EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 3 and in Hs(Ω) are denoted by ‖u‖ ≡ ‖u‖L2(Ω) = (∫ Ω |u|2 dx )1/2 , ‖u‖s ≡ ‖u‖Hs(Ω) = ( ∑ |α|≤s ∫ Ω |Dαu|2 dx )1/2 , where α = (α1, α2) is any multi-index with order |α| = α1 + α2 and Dα = ∂α1 x ∂α2 y . For any vector valued function F = (f1, f2) : Ω→ R2, ‖F‖2s ≡ ‖f1‖2s + ‖f2‖2s. The energy space under consideration is: X3([0, T ],Ω) ≡ { F : Ω× [0, T ]→ R : ∂ltF ∈ L∞([0, T ];H3−l(Ω)), l = 0, 1, 2, 3 } , equipped with norm |||F |||3,T ≡ sup 0≤t≤T |||F (·, t)||| ≡ sup 0≤t≤T [ 3∑ l=0 ‖F (·, t)‖23−l ]1/2 . Our main result reads as follows. Theorem 1.1. Suppose that the initial data satisfies the compatibility condition of order 2, i.e., ∂ltU(0) · n ∣∣ ∂Ω = 0, 0 ≤ l ≤ 2, where ∂ltU(0) is the l-th time derivative at t = 0 for any solution of (1.1)-(1.2), as calculated from (1.1) to yield an expression in terms of H0 and U0. Then there exists a constant ε > 0 such that if ‖(H0 −H/|Ω|, U0)‖3 ≤ ε, the initial-boundary value problem (1.1)-(1.2) admits a unique global solution (H,U) ∈ X3([0,∞),Ω). Moreover, there exist positive constants C and η, which are independent of t, such that ‖(H −H/|Ω|)(·, t)‖3 + ‖U(·, t)‖3 ≤ C‖(H0 −H/|Ω|, U0)‖3 exp{−ηt}. (1.5) We note that the main result in this paper still holds when the physical domain Ω is replaced by T× [0, L] for any L > 0. The proof of Theorem 1.1 is mainly based on the existence theorem and the a priori estimates of a local solution. Since the existence of the local solution can be obtained by the classical local well-posedness theory, we will focus on the a priori estimates of the solution in the majority of this article. The energy method is used to derive these key estimates. Several difficulties need to be overcome during obtaining the energy estimate to the system. The first one is that we can not obtain the normal derivative estimate of the solution because of the presence of the boundary. Here it seems that the missed estimates can only be compensated by estimating the time derivative of the solution and the vorticity ω = ∇×U . However, as we can see later, this will lead to a new difficulty when we try to estimate the vorticity for the non-constant rotation frequency. When taking y-derivatives to the vorticity equation, we will encounter with some trouble terms which need to be controlled properly. We overcome this difficulty by designing some delicate semi-norms to capture the full dissipation mechanism of the system. Finally, since the system is partial dissipative, we need to use the nonlinear interaction of the 4 Y. FANG, K. LI, X. XU EJDE-2023/62 system to derive the dissipative estimate of h, which can be achieved through a Kawashima-type energy estimate [10]. Using the special structure of (1.1) together with induction on the number of spatial derivatives, the estimate of total energy is reduced to those for the vorticity and temporal derivatives. Actually, the method used here is simpler than the classical energy estimate. The plan of the rest of this article is as follows. In Section 2, we reformulate the original system to get a quasi-linear symmetric hyperbolic system, the local existence result and some basic facts which will be used in this paper are given. In Section 3, we prove Theorem 1.1 by energy estimates. 2. Reformulation of the problem To symmetrize the equations, we give a reformulation to the initial-boundary value problem (1.1)-(1.2) in this section. Without loss of generality, we assume H/|Ω| = 1. First, multiplying (1.1)1 by 1/H, we have 1 H Ht + 1 H U · ∇H +∇ · U = 0, Ut + U · ∇U +∇H + yU⊥ = −U. Since equilibrium height is conjectured to be H/|Ω| = 1, we set H = 1 + h and get the desired symmetric system 1 1 + h (ht + U · ∇h) +∇ · U = 0, Ut + U · ∇U +∇h+ yU⊥ = −U, (2.1) with the initial and boundary conditions h ∣∣ t=0 = h0, U ∣∣ t=0 = U0, U · n ∣∣ ∂Ω = 0, (2.2) where h0 = H0 − 1. Now, for the initial boundary value problem (2.1)-(2.2), we can use the same idea as in [14] to establish the existence of classical local solutions. Lemma 2.1. If (h0, U0) ∈ H3(Ω) and satisfies the compatibility condition ∂ltU(0) · n ∣∣ ∂Ω = 0, 0 ≤ l ≤ 2, then there exists a unique local solution (h, U) of problem (2.1)-(2.2) in C1(Ω× [0, T ]) ∩X3([0, T ],Ω) for some finite T > 0. Moreover, there exist positive constants ε0 and C0(T ) such that if ‖h(·, 0)‖3 + ‖U(·, 0)‖3 ≤ ε0, then ‖h‖3,T + ‖U‖3,T ≤ C0(‖h(·, 0)‖3 + ‖U(·, 0)‖3). Now, we give some lemmas to be used later. The first one is an inequality of Sobolev type whose proof can be found in many textbooks [8]. Lemma 2.2. Let Ω be any bounded domain in R2 with smooth boundary. Then (i) ‖f‖L∞(Ω) ≤ C‖f‖H2(Ω), (ii) ‖f‖Lp(Ω) ≤ C‖f‖H1(Ω) for 2 ≤ p <∞, EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 5 for some constant C > 0 depending only on Ω. To give the control of the velocity in terms of ∇·U and the vorticity ω, we need the following lemma, see [20]. Lemma 2.3. Let U ∈ Hs(Ω) be a vector-valued function satisfying U · n ∣∣ ∂Ω = 0, where n is the unit outer normal of ∂Ω. Then ‖U‖s ≤ C(‖∇ × U‖s−1 + ‖∇ · U‖s−1 + ‖U‖s−1) (2.3) for s ≥ 1, and the positive constant C depends only on s and Ω. 3. Existence of global solutions and long time behavior In this section, we shall prove the existence and the large time behavior of global solutions to (2.1)-(2.2). To this end, we need to derive a key a priori estimates, which are the main part of this section. For convenience, we set the total energy W (t) ≡ |||h(t)|||2 + |||U(t)|||2 = 3∑ l=0 ( ‖∂lth(t)‖23−l + ‖∂ltU(t)‖23−l ) . (3.1) The main result in this section is as follows. Theorem 3.1. Suppose that the initial data (h0, U0) ∈ H3(Ω) and satisfies the compatibility condition of order 2. If there exists a small enough positive constant ε such that ‖(h0, U0)‖3 ≤ ε, then there is a unique global classical solution of (2.1)- (2.2) such that W (t) ≤ CW (0)e−ηt, (3.2) where C and η are positive constants independent of t. The proof of Theorem 3.1 is based on a detailed energy estimates. To simplify the presentation, we define E(t) ≡ 3∑ l=0 ( ‖∂lth‖2 + ‖∂ltU‖2 ) , V (t) ≡ 2∑ l=0 ‖∂ltω‖22−l. (3.3) By the definition of E(t), V (t) and W (t), we find that the total energy W (t) can be controlled by E(t) and V (t) as long as (h, U) is a sufficiently small solution of (2.1)-(2.2). Lemma 3.2. Let (h, U) be solution of (2.1)-(2.2). Suppose that there is a small constant δ̄ > 0 such that W (t) ≤ δ̄, then W (t) ≤ C(E(t) + V (t)). (3.4) Proof. Rewriting (2.1)2 as ∇h = −(U + Ut + U · ∇U + yU⊥), (3.5) and taking the L2 inner product with ∇h yields ‖∇h‖2 = ∫ Ω −(U + Ut + U · ∇U + yU⊥) · ∇h dx. Combining Lemma 2.2 with the Cauchy-Schwarz inequality, we obtain ‖∇h‖2 ≤ C(‖U‖2 + ‖Ut‖2) + C‖U‖2L∞‖∇U‖2 + ‖yU‖2 ≤ C(‖U‖2 + ‖Ut‖2) + CW (t)3/2 ≤ C(E(t) + V (t)) + CW (t)3/2. (3.6) 6 Y. FANG, K. LI, X. XU EJDE-2023/62 Similarly, by rewriting (2.1)1 as ∇ · U = − 1 1 + h (ht + U · ∇h), (3.7) we have ‖∇ · U‖2 ≤ C(‖ht‖2 +W (t)3/2). (3.8) Using Lemma 2.3 with s = 1 and (3.8), one has ‖U‖21 ≤ C(‖ω‖2 + ‖∇ · U‖2 + ‖U‖2) ≤ C(‖ω‖2 + ‖ht‖2 + ‖U‖2 +W (t)3/2) ≤ C(E(t) + V (t)) + CW (t)3/2. (3.9) Applying ∂lt with 1 ≤ l ≤ 2 to (3.5) and (3.7), taking inner product with ∂lt∇h and ∂lt∇ · U respectively, then using Lemma 2.3 to control ‖∂ltU‖1, we find that 2∑ l=0 ( ‖∂lth(t)‖21 + ‖∂ltU(t)‖21 ) ≤ C(E(t) + V (t)) + CW (t)3/2, (3.10) where we used the smallness of δ̄ and Sobolev inequality. Now, following similar procedure as above and by an induction on the number of spatial derivatives, one has 1∑ l=0 ( ‖∂lth(t)‖22 + ‖∂ltU(t)‖22 ) ≤ C(E(t) + V (t)) + CW (t)3/2, (3.11) ‖h(t)‖23 + ‖U(t)‖23 ≤ C(E(t) + V (t)) + CW (t)3/2. (3.12) Combining estimates (3.10)–(3.12), we have W (t) ≤ C(E(t) + V (t)) + CW (t)3/2. (3.13) Noting that W (t) ≤ δ̄, the proof is complete. � Lemma 3.3. Let (h, U) be a solution of (2.1)-(2.2). Suppose that there is a small constant δ̄ > 0 such that W (t) ≤ δ̄. Then d dt ( ‖h‖2 + ‖ht‖2 + ‖ √ 1 1 + h htt‖2 + ‖ √ 1 1 + h httt‖2 + 3∑ l=0 ‖∂ltU‖2 ) + 2 3∑ l=0 ‖∂ltU‖2 ≤ CW (t)3/2. (3.14) Proof. By calculating h(1 + h) · (2.1)1 + U · (2.1)2, we have 1 2 d dt (h2 + |U |2) + |U |2 = − [ − hU · ∇h+ U · (U · ∇U) +∇ · (hU) +∇ · (h2U) ] . (3.15) Integrating (3.15) over Ω and using integration by parts and the non-penetration boundary condition (2.2), we obtain 1 2 d dt (‖h‖2 +‖U‖2)+‖U‖2 ≤ ‖h‖L2‖U‖L∞‖∇h‖L2 +‖U‖L2‖U‖L∞‖∇U‖L2 . (3.16) Then we deduce that d dt (‖h‖2 + ‖U‖2) + 2‖U‖2 ≤ CW (t)3/2. (3.17) EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 7 Differentiating (1 + h) · (2.1)1 and (2.1)2 with respect to t, multiplying by ht, Ut respectively, we obtain 1 2 d dt (h2 t + |Ut|2) + |Ut|2 = −htUt · ∇h− htU · ∇ht − h2 t∇ · U − hht∇ · Ut − Ut · (Ut · ∇U)− Ut · (U · ∇Ut)−∇ · (htUt). Similarly, 1 2 d dt (‖ht‖2 + ‖Ut‖2) + ‖Ut‖2 ≤ ‖ht‖L4‖Ut‖L4‖∇h‖L2 + ‖ht‖L4‖U‖L4‖∇ht‖L2 + ‖ht‖L4‖ht‖L4‖∇U‖L2 + ‖h‖L4‖ht‖L4‖∇Ut‖L2 + ‖Ut‖L4‖Ut‖L4‖∇U‖L2 + ‖Ut‖L4‖U‖L4‖∇Ut‖L2 . (3.18) Using the Sobolev embedding theorem, one has d dt (‖ht‖2 + ‖Ut‖2) + 2‖Ut‖2 ≤ CW (t)3/2. (3.19) In a similar way, we can obtain the following second order and third order esti- mates d dt (‖ √ 1 1 + h htt‖2 + ‖Utt‖2) + 2‖Utt‖2 ≤ CW (t)3/2, (3.20) d dt (‖ √ 1 1 + h httt‖2 + ‖Uttt‖2) + 2‖Uttt‖2 ≤ CW (t)3/2. (3.21) Thus, combining (3.17), (3.19), (3.20), and (3.21), we obtain (3.14). This completes the proof. � The estimate (3.14) contains the dissipation in velocity due to the friction term −U in the equation. To close our a priori assumption W (t) ≤ δ̄, it is important to derive the dissipation of h, which will be done in the following lemma. Lemma 3.4. Let (h, U) be the solution of (2.1)-(2.2). If there is a small constant δ̄ > 0 such that W (t) ≤ δ̄, then d dt ( 3∑ l=1 ∫ Ω (−∂l−1 t h∂lth) dx ) + 3∑ l=0 ‖∂lth‖2 ≤ CW (t)3/2 + c0 3∑ l=0 ‖∂ltU‖2. (3.22) Proof. Since ∫ Ω h dx = ∫ Ω (H − 1) dx = 0, using Poincaré’s inequality, we have ‖h‖2 ≤ C‖∇h‖2. This together with (3.6) yields ‖h‖2 ≤ C(‖U‖2 + ‖Ut‖2) + CW (t)3/2. (3.23) A direct calculation of ∂t[(1 + h)(2.1)1]− (1 + h)∇ · (2.1)2 gives htt + (U · ∇h)t + ht(∇ · U)− (1 + h)∇ · (U · ∇U +∇h+ U + yU⊥) = 0. (3.24) 8 Y. FANG, K. LI, X. XU EJDE-2023/62 Multiplying by h, we obtain htth+ (U · ∇h)th+ ht(∇ · U)h− (1 + h)∇ · (U · ∇U +∇h+ U + yU⊥)h = (hht)t − h2 t + (U · ∇h)th+ hht(∇ · U) + (1 + h)h∇ · Ut = (hht)t − h2 t + (Ut · ∇h)h+ (U · ∇ht)h+ hht(∇ · U) +∇ · [(h2 + h)Ut] − Ut · ∇(h2 + h) = (hht)t − h2 t + (Ut · ∇h)h+∇ · (hhtU)− hht(∇ · U)− ht(U · ∇h) + hht(∇ · U) +∇ · [(h2 + h)Ut]− Ut · ∇(h2 + h) = (hht)t − h2 t + (Ut · ∇h)h− ht(U · ∇h)− 2hUt · ∇h− Ut · ∇h +∇ · [(h2 + h)Ut + hhtU ] = 0. (3.25) Integrating over Ω and using Cauchy-Schwarz inequality, we have − d dt ∫ Ω hht dx + ‖ht‖2 ≤ C ( W (t)3/2 + ‖∇h‖2 + ‖Ut‖2 ) . (3.26) Combining this with (3.6), we have − d dt ∫ Ω hht dx + ‖ht‖2 ≤ C ( W (t)3/2 + ‖U‖2 + ‖Ut‖2 ) , (3.27) Similarly, we have − d dt ∫ Ω hthtt dx + ‖htt‖2 ≤ C ( W (t)3/2 + ‖Ut‖2 + ‖Utt‖2 ) , (3.28) − d dt ∫ Ω htthttt dx + ‖httt‖2 ≤ C ( W (t)3/2 + ‖Utt‖2 + ‖Uttt‖2 ) . (3.29) Collecting (3.23), (3.27)-(3.29), the proof is complete. � Let c1 ≡ max{2, c0}, and E1(t) ≡ c1 ( ‖h‖2 + ‖ht‖2 + ‖ √ 1 1 + h htt‖2 + ‖ √ 1 1 + h httt‖2 + 3∑ l=0 ‖∂ltU‖2) − 3∑ l=1 ∫ Ω (−∂l−1 t h∂lth) dx. (3.30) Then, from the estimates in Lemma 3.3 and Lemma 3.4, we have the following result. Lemma 3.5. There exist constants C2, C > 0 such that d dt E1(t) + C2E(t) ≤ CW (t)3/2. (3.31) Proof. Taking c1 × (3.14) + (3.22) yields d dt E1(t) + c0 3∑ l=0 ‖∂ltU‖2 + 3∑ l=0 ‖∂lth‖2 ≤ CW (t)3/2. (3.32) Letting C2 = min{c0, 1}, inequality (3.31) follows directly from (3.32). � EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 9 Lemma 3.6. Let (h, U) be the solution of (2.1)-(2.2). Suppose that there is a small constant δ̄ > 0 such that W (t) ≤ δ̄, then d dt V1(t) + C5V1(t) ≤ CW (t)3/2 + C6E(t), (3.33) where C5, C6, C > 0 are constants and V1 is defined as (3.66). Proof. Applying the curl to (2.1)2 yields ωt + ω = −U · ∇ω − ω(∇ · U)− y∇ · U − v. (3.34) Multiplying the above equation by ω, we obtain ωωt + ω2 = −ωU · ∇ω − ω2(∇ · U)− yω∇ · U − ωv. Integrating the resulting equation and using the boundary condition, we obtain 1 2 d dt ‖ω‖2 + ‖ω‖2 ≤ C3W (t)3/2 − ∫ Ω yω∇ · U dx− ∫ Ω ωv dx. (3.35) Dealing with the last two terms on the right-hand side of the above inequality,∫ Ω yω∇ · U dx = ∫ Ω yω[− 1 1 + h (ht + U · ∇h)] dx, where we used the equality ∇ · U = − 1 1 + h (ht + U · ∇h). Combining Hölder’s inequality, Lemma 2.2 and Cauchy-Schwarz inequality, we ob- tain ∫ Ω yω∇ · U dx ≤ C3 ( W (t)3/2 + E(t) ) + 1 8 ‖ω‖2, (3.36)∫ Ω ωv dx ≤ C3E(t) + 1 8 ‖ω‖2. (3.37) Putting the above two estimates into (3.35), we obtain d dt ‖ω‖2 + 1 2 ‖ω‖2 ≤ C4 ( W (t)3/2 + E(t) ) , (3.38) where C4 > 1. First order estimate. Differentiating (3.34) with respect to x and multiplying the resulting equation by ωx, we obtain ωxωtx + ω2 x = −ωxUx · ∇ω − ωxU · ∇ωx − ω2 x(∇ · U) − ωxω(∇ · Ux)− yωx∇ · Ux − ωxvx. Integrating the above equation over Ω and using Hölder’s inequality and Lemma 2.2, we obtain 1 2 d dt ‖ωx‖2 + ‖ωx‖2 ≤ C3W (t)3/2 − ∫ Ω yωx∇ · Ux dx− ∫ Ω ωxvx dx. (3.39) Similarly, we can estimate the last two terms on the right-hand side of the above inequality as follows:∫ Ω yωx∇ · Ux dx = ∫ Ω yωx [ 1 (1 + h)2 hx(ht + U · ∇h) ] dx 10 Y. FANG, K. LI, X. XU EJDE-2023/62 − ∫ Ω yωx 1 1 + h (htx + Ux · ∇h+ U · ∇hx) dx. Using Hölder’s inequality and Lemma 2.2, we have∫ Ω yωx∇ · Ux dx ≤ C3W (t)3/2 + c3 ∫ Ω ωxhtx dx. (3.40) Based on (2.1)2, the Hölder’s inequality, Lemma 2.2 and Cauchy-Schwarz inequality, we obtain c3 ∫ Ω ωxhtx dx = c3 ∫ Ω ωx(−u+ yv − ut − U · ∇u)t dx = c3 ∫ Ω ωx(−ut + yvt − utt − Ut · ∇u− U · ∇ut) dx ≤ c3 ( ‖ωx‖‖ut‖+ ‖ωx‖‖vt‖+ ‖ωx‖‖utt‖ ) + C3W (t)3/2 ≤ 1 8 ‖ωx‖2 + C3‖ut‖2 + 1 8 ‖ωx‖2 + C3‖vt‖2 + 1 8 ‖ωx‖2 + C3‖utt‖2 + C3W (t)3/2 ≤ C3 ( W (t)3/2 + E(t) ) + 3 8 ‖ωx‖2. (3.41) From (3.9), it is clear that∫ Ω ωxvx dx ≤ ‖ωx‖‖vx‖ ≤ ‖ωx‖‖U‖1 ≤ C3 ( W (t)3/2 + E(t) + ‖ω‖2 ) + 1 8 ‖ωx‖2, (3.42) which together with (3.39)-(3.41) gives d dt ‖ωx‖2 + 1 2 ‖ωx‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ω‖2 ) . (3.43) Differentiating (3.34) with respect to y and multiplying the resulting equation by ωy, we obtain ωyωty + ω2 y = −ωyUy · ∇ω − ωyU · ∇ωy − ω2 y(∇ · U) − ωyω(∇ · Uy)− ωy∇ · U − yωy∇ · Uy − ωyvy. Integrating the above equation over Ω, we have 1 2 d dt ‖ωy‖2 + ‖ωy‖2 ≤ C3W (t)3/2 − ∫ Ω ωy∇ · U dx− ∫ Ω yωy∇ · Uy dx− ∫ Ω ωyvy dx. (3.44) Now we estimate the last three terms on the right-hand side of the above inequality as follows:∫ Ω yωy∇ · Uy dx = ∫ Ω yωy [ 1 (1 + h)2 hy(ht + U · ∇h) ] dx − ∫ Ω yωy 1 1 + h (hty + Uy · ∇h+ U · ∇hy) dx. Using Lemma 2.2, we obtain∫ Ω yωy∇ · Uy dx ≤ C3W (t)3/2 + c3 ∫ Ω ωyhty dx. (3.45) EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 11 By (2.1)2, one has c3 ∫ Ω ωyhty dx = c3 ∫ Ω ωy(−v − yu− vt − U · ∇v)t dx ≤ C3 ( W (t)3/2 + E(t) ) + 3 8 ‖ωy‖2 (3.46) and ∫ Ω ωyvy dx ≤ ‖ωy‖‖vy‖ ≤ ‖ωy‖‖U‖1 ≤ C3 ( W (t)3/2 + E(t) + ‖ω‖2 ) + 1 8 ‖ωy‖2. (3.47) Similar derivations show that∫ Ω ωy∇ · U dx ≤ ‖ωy‖‖∇ · U‖ ≤ ‖ωy‖‖U‖1 ≤ C3 ( W (t)3/2 + E(t) + ‖ω‖2 ) + 1 8 ‖ωy‖2. (3.48) Combining (3.44)-(3.48), we obtain d dt ‖ωy‖2 + 1 2 ‖ωy‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ω‖2 ) . (3.49) Therefore, d dt ‖∂ω‖2 + 1 2 ‖∂ω‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ω‖2 ) , (3.50) where ∂ denotes spatial derivatives ∂x and ∂y. Finally, differentiating (3.34) with respect to t and multiplying the resulting equation by ωt, we obtain ωtωtt + ω2 t = −ωtUt · ∇ω − ωtU · ∇ωt − ω2 t (∇ · U) − ωtω(∇ · Ut)− yωt∇ · Ut − ωtvt. Integrating the above equation over Ω gives 1 2 d dt ‖ωt‖2 + ‖ωt‖2 ≤ C3W (t)3/2 − ∫ Ω yωt∇ · Ut dx− ∫ Ω ωtvt dx. (3.51) Since ∫ Ω yωt∇ · Ut dx = ∫ Ω yωt [ 1 (1 + h)2 ht(ht + U · ∇h) ] dx − ∫ Ω yωt 1 1 + h (htt + Ut · ∇h+ U · ∇ht) dx, we have ∫ Ω yωt∇ · Ut dx ≤ C3W (t)3/2 + c3 ∫ Ω ωthtt dx ≤ C3W (t)3/2 + c3‖ωt‖‖htt‖ ≤ C3 ( W (t)3/2 + E(t) ) + 1 8 ‖ωt‖2. (3.52) By Cauchy-Schwarz inequality, we have∫ Ω ωtvt dx ≤ ‖ωt‖‖vt‖ ≤ ‖ωt‖‖Ut‖ ≤ C3E(t) + 1 8 ‖ωt‖2, (3.53) 12 Y. FANG, K. LI, X. XU EJDE-2023/62 which together with (3.51)-(3.52) implies d dt ‖ωt‖2 + 1 2 ‖ωt‖2 ≤ C4 ( W (t)3/2 + E(t) ) . (3.54) Second order estimate. Firstly, differentiating (3.34) with respect to t for second order and multiplying the resulting equation by ωtt, we obtain ωttωttt + ω2 tt = −ωttUtt · ∇ω − 2ωttUt · ∇ωt − ωttU · ∇ωtt − ω2 tt(∇ · U) − 2ωttωt∇ · Ut − ωttω(∇ · Utt)− yωtt∇ · Utt − ωttvtt. Integrating over Ω gives 1 2 d dt ‖ωtt‖2 + ‖ωtt‖2 ≤ C3W (t)3/2 − ∫ Ω ωttU · ∇ωtt dx− ∫ Ω yωtt∇ · Utt dx− ∫ Ω ωttvtt dx. (3.55) Since ∫ Ω ωttU · ∇ωtt dx = ∫ Ω ∇ · (ω 2 tt 2 U)− ω2 tt 2 ∇ · U dx = − ∫ Ω ω2 tt 2 ∇ · U dx ≤ C3W (t)3/2, (3.56) and∫ Ω yωtt∇ · Utt dx = ∫ Ω yωtt [htt(1 + h)2 − 2h2 t (1 + h) (1 + h)4 (ht + U · ∇h) ] dx + 2 ∫ Ω yωtt ht (1 + h)2 (htt + Ut · ∇h+ U · ∇ht) dx − ∫ Ω yωtt 1 1 + h (httt + Utt · ∇h+ 2Ut · ∇ht + U · ∇htt) dx, we obtain ∫ Ω yωtt∇ · Utt dx ≤ C3W (t)3/2 − ∫ Ω y 1 + h ωtthttt dx ≤ C3W (t)3/2 + 1 8 ‖ωtt‖2 + C3‖httt‖2 ≤ C3 ( W (t)3/2 + E(t) ) + 1 8 ‖ωtt‖2. (3.57) Since ∫ Ω ωttvtt dx ≤ ‖ωtt‖‖Utt‖ ≤ C3E(t) + 1 8 ‖ωtt‖2, (3.58) from (3.55)-(3.57) we obtain d dt ‖ωtt‖2 + 1 2 ‖ωtt‖2 ≤ C4 ( W (t)3/2 + E(t) ) . (3.59) Secondly, applying ∂2 x to (3.34) and multiplying by ωxx, we obtain ωxxωtxx + ω2 xx = −ωxxUxx · ∇ω − 2ωxxUx · ∇ωx − ωxxU · ∇ωxx − ω2 xx(∇ · U) − 2ωxxωx∇ · Ux − ωxxω(∇ · Uxx)− yωxx∇ · Uxx − ωxxvxx. EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 13 Thus 1 2 d dt ‖ωxx‖2 + ‖ωxx‖2 ≤ C3W (t)3/2 − ∫ Ω ωxxU · ∇ωxx dx − ∫ Ω yωxx∇ · Uxx dx− ∫ Ω ωxxvxx dx. (3.60) In a similar fashion, we find that∫ Ω ωxxU · ∇ωxx dx = ∫ Ω ∇ · (ω 2 xx 2 U)− ω2 xx 2 ∇ · U dx ≤ C3W (t)3/2 (3.61) and ∫ Ω yωxx∇ · Uxx dx = ∫ Ω yωxx [hxx(1 + h)2 − 2h2 x(1 + h) (1 + h)4 (ht + U · ∇h) ] dx + 2 ∫ Ω yωxx hx (1 + h)2 (htx + Ux · ∇h+ U · ∇hx) dx − ∫ Ω yωxx 1 1 + h (htxx + Uxx · ∇h+ 2Ux · ∇hx + U · ∇hxx) dx. Straightforward calculation gives∫ Ω yωxx∇ · Uxx dx ≤ C3W (t)3/2 − ∫ Ω y 1 + h ωxxhtxx dx ≤ C3W (t)3/2 − ∫ Ω y 1 + h ωxx(−utx + yvtx − uttx − Utx · ∇u− Ut · ∇ux − Ux · ∇ut − U · ∇utx) dx ≤ C3W (t)3/2 + 3 8 ‖ωxx‖2 + C3(‖utx‖2 + ‖vtx‖2 + ‖uttx‖2) ≤ C3W (t)3/2 + 3 8 ‖ωxx‖2 + C3(‖Ut‖21 + ‖Ut‖21 + ‖Utt‖21) ≤ C3 ( W (t)3/2 + E(t) + ‖ωt‖2 + ‖ωtt‖2 ) + 3 8 ‖ωxx‖2. (3.62) Similarly,∫ Ω ωxxvxx dx ≤ ‖ωxx‖‖U‖2 ≤ C3 ( W (t)3/2 + E(t) + ‖ω‖2 + ‖ω‖21 ) + 1 8 ‖ωxx‖2. (3.63) Combining this with (3.60)-(3.63), we obtain d dt ‖ωxx‖2 + 1 2 ‖ωxx‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 + ‖ωtt‖2 + ‖ω‖2 + ‖ω‖21 ) , d dt ‖ωyy‖2 + 1 2 ‖ωyy‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 + ‖ωtt‖2 + ‖ω‖2 + ‖ω‖21 ) , d dt ‖ωxy‖2 + 1 2 ‖ωxy‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 + ‖ωtt‖2 + ‖ω‖2 + ‖ω‖21 ) , 14 Y. FANG, K. LI, X. XU EJDE-2023/62 d dt ‖ωtx‖2 + 1 2 ‖ωtx‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 ) , d dt ‖ωty‖2 + 1 2 ‖ωty‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 ) . Therefore, d dt ‖∂2ω‖2 + 1 2 ‖∂2ω‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 + ‖ωtt‖2 + ‖ω‖2 + ‖ω‖21 ) , (3.64) d dt ‖∂ωt‖2 + 1 2 ‖∂ωt‖2 ≤ C4 ( W (t)3/2 + E(t) + ‖ωt‖2 ) . (3.65) Calculating 20C2 4 (3.38) + 8C4[(3.50) + (3.54) + (3.59)] + (3.64) + (3.65), letting C5 ≡ min{C4−1 10C4 , 1 4}, and defining V1(t) ≡ 20C2 4‖ω‖2 + 8C4(‖∂ω‖2 + ‖ωt‖2) + 8C4‖ωtt‖2 + ‖∂2ω‖2 + ‖∂ωt‖2, (3.66) we obtain d dt V1(t) + C5V1(t) ≤ CW (t)3/2 + C6E(t). This completes the proof. � With the above lemmas, we can give the following proof. Proof of Theorem 3.1. Calculating 2C6 C2 (3.31) + (3.33) we have d dt (2C6 C2 E1(t) + V1(t) ) + ( C6E(t) + C5V1(t) ) ≤ CW (t)3/2. (3.67) Set E2(t) ≡ 2C6 C2 E1(t). (3.68) Then according to (3.3), (3.30), (3.68) and the definition of c1, C2 and C6, we can easily see that E(t) and E2(t) are equivalent, i.e., there exist constants c4, c5 > 0 such that c4E2(t) ≤ E(t) ≤ c5E2(t). (3.69) Using (3.67) and (3.69), we have d dt ( E2(t) + V1(t) ) + ( c4C6E2(t) + C5V1(t) ) ≤ CW (t)3/2. (3.70) Let C7 ≡ min{c4C6, C5}. Then d dt ( E2(t) + V1(t) ) + C7 ( E2(t) + V1(t) ) ≤ CW (t)3/2. (3.71) On the other hand, from (3.66) and the definition of C4 we can see that V (t) and V1(t) are also equivalent, i.e., there exist constants c6, c7 > 0 such that c6V1(t) ≤ V (t) ≤ c7V1(t). (3.72) From (3.4), (3.69) and (3.72), we see that W (t) ≤ C1 ( c5E2(t) + c7V1(t) ) . (3.73) Letting C8 ≡ max{c5C1, c7C1}, then we obtain W (t) ≤ C8 ( E2(t) + V1(t) ) . (3.74) EJDE-2023/62 GLOBAL SOLUTIONS TO SHALLOW-WATER EQUATIONS 15 Using the a priori assumption W (t) ≤ δ̄, from (3.71), (3.74) and the smallness of δ̄ we obtain d dt ( E2(t) + V1(t) ) + C7 2 ( E2(t) + V1(t) ) ≤ 0. (3.75) Integrating over [0, t] for any t > 0 gives( E2(t) + V1(t) ) + C7 2 ∫ t 0 ( E2(t′) + V1(t′) ) dt′ ≤ ( E2(0) + V1(0) ) . (3.76) Using ‖(h0, U0)‖3 ≤ ε, we can choose sufficiently small ε such that W (t) ≤ Cε < δ̄. This justifies the a priori assumption. Finally, the exponential decay of E2(t)+V1(t) follows directly from (3.75). Thus W (t) also decays exponentially because of (3.74). This completes the proof. � Acknowledgments. The authors would like to thank the anonymous referees whose comments help us improve the contain of this article. K. Li was supported by Natural Science Foundation of China (Grant No. 12101534) and by the Natural Science Foundation of Shandong Province (Grant No. ZR2021QA052). X. 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Yue Fang School of Mathematical Sciences, Ocean University of China, Qingdao, China Email address: yyfang0929@163.com Kaiqiang Li (corresponding author) School of Mathematics and Information Sciences, Yantai University, Yantai, China Email address: kaiqiangli19@163.com Xin Xu School of Mathematical Sciences, Ocean University of China, Qingdao, China Email address: xx@ouc.edu.cn 1. Introduction 2. Reformulation of the problem 3. Existence of global solutions and long time behavior First order estimate Second order estimate Acknowledgments References