EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR 2D SHALLOW WATER EQUATIONS WITH ITS DEGENERATE VISCOSITY

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1 EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR 2D SHALLOW WATER EQUATIONS WITH ITS DEGENERATE VISCOSITY ALEXIS F. VASSEUR AND CHENG YU Abstract. In this paper, we prove the existence of global weak solutions for 2D shallow water equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [27] for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the shallow water equations, for any γ > 1, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions in [22]. 1. Introduction The existence of global weak solutions of 2D shallow water equations with degenerate viscosity has been a long standing open problem. The objective of this current paper is to establish the existence of global weak solutions to the following 2D compressible Navier-Stokes equations: ρ t + div(ρu) = (ρu) t + div(ρu u) + P div(ρdu) =, (1.1) with initial data ρ t= = ρ (x), ρu t= = m (x). (1.2) where P = ρ γ, γ > 1, denotes the pressure, ρ is the density of fluid, u stands for the velocity of fluid, Du = 1 2 [ u + T u] is the strain tensor. For the sake of simplicity we will consider the case of bounded domain with periodic boundary conditions, namely = T 2. In the case γ = 2, this corresponds to the shallow water equations, where ρ(t, x) stands for the height of the water at position x, and time t, and u(t, x) is the 2D velocity at the same position, and same time. In this case, the physical viscosity was formally derived as in (1.1) (see Gent [13]). In this context, the global existence of weak solutions to equations (1.1) is proposed as an open problem by Lions in [22]. A careful derivation of the shallow water equations with the following viscosity term 2div(ρDu) + 2 (ρdivu) can be found in the recent work by Marche [23]. Bresch-Noble [6, 7] provided the mathematical derivation of viscous shallow-water equations with the above viscosity. However, this viscosity cannot be covered by the BD entropy. Date: January 27, Mathematics Subject Classification. 35Q35, 76N1. Key words and phrases. Global weak solutions, compressible Navier-Stokes equations, shallow water equations, vacuum, degenerate viscosity. 1

2 2 ALEXIS F. VASSEUR AND CHENG YU Compared with the incompressible flows, dealing with the vacuum is a very challenging problem in the study of the compressible flows. Kazhikhov and Shelukhin [2] established the first existence result on the compressible Navier-Stokes equations in one dimensional space. Due to the difficulty from the vacuum, the initial density should be bounded away from zero in their work. It has been extended by Serre [29] and Hoff [16] for the discontinuous initial data, and by Mellet-Vasseur [28] in the case of density dependent viscosity coefficient. For the multidimensional case, Matsumura and Nishida [24, 25, 26] first established the global existence with the small initial data, and later by Hoff [17, 18, 19] for discontinuous initial data. To remove the difficulty from the vacuum, Lions in [22] introduced the concept of renormalized solutions to establish the global existence of weak solutions for γ > 9 5 concerning large initial data that may vanish, and then Feireisl- Novotný-Petzeltová [11] and Feireisl [12] extended the existence results to γ > 3 2, and even to Navier-Stokes-Fourier system. In all above works, the viscosity coefficients were assumed to be fixed positive numbers. This is important to control the gradient of the velocity. in the context of solutions close to an equilibrium, a breakthrough was obtained by Danchin [8, 9]. However, the regularity and the uniqueness of the weak solutions for large data remain largely open for the compressible Navier-Stokes equations, even as in two dimensional space, see Vaigant-Kazhikhov [3] (see also Germain [14], and Haspot [15], where criteria for regularity or uniqueness are proposed). The problem becomes even more challenging when the viscosity coefficients depend on the density. Indeed, the Navier-Stokes equations (1.3) is highly degenerated at the vacuum because the velocity cannot even be defined when the density vanishes. It is very difficult to deduce any estimate of the gradient on the velocity field due to the vacuum. This is the essential difference from the compressible Navier-Stokes equations with the non-density dependent viscosity coefficients. The first tool of handling this difficulty is due to Bresch, Desjardins and Lin, see [4], where the authors developed a new mathematical entropy to show the structure of the diffusion terms providing some regularity for the density. The result was later extended for the case with an additional quadratic friction term rρ u u, refer to Bresch-Desjardins [2, 3] and the recent results by Bresch-Desjardins-Zatorska [5] and by Zatorska [32]. Unfortunately, those bounds are not enough to treat the compressible Navier-Stokes equations without additional control on the vacuum, as the introduction of capillarity, friction, or cold pressure. The primary obstacle to prove the compactness of the solutions to (1.3) is the lack of strong convergence for ρu in L 2. We cannot pass to the limit in the term ρu u without the strong convergence of ρu in L 2. This is an other essential difference with the case of non-density dependent viscosity. To solve this problem, a new estimate is established in Mellet-Vasseur [27], providing a L (, T ; L log L()) control on ρ u 2. This new estimate provides the weak stability of smooth solutions of (1.3). The classical way to construct global weak solutions of (1.3) would consist in constructing smooth approximation solutions, verifying the priori estimates, including the Bresch- Desjardins entropy, and the Mellet-Vasseur inequality. However, those extra estimates impose a lot of structure on the approximating system. Up to now, no such approximation scheme has been discovered. In [2, 3], Bresch and Desjardins propose a very nice construction of approximations, controlling both the usual energy and BD entropy. This

3 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 3 allows the construction of weak solutions, when additional terms -as drag terms, or cold pressure, for instance- are added. Note that their result holds true even in dimension 3. However, their construction does not provide the control of the ρu in L (, T ; L log L()). The objective of our current work is to investigate the issue of existence of solutions for shallow water equations (1.1) with large initial data. Building up from the result of Bresch and Desjardins, we establish the logarithmic estimate for the weak solutions similar to [27]. This estimate is obtained for the solutions of Bresch and Desjardins with the additional drag forces. It is showed to be independent on the strength of those drag forces, allowing to pass into the limit when those forces vanish. Since this estimate cannot be derived from the approximation scheme of Bresch and Desjardins, it has to be carefully derived on weak solutions. This is reminiscent to showing the conservation of the energy for weak solutions to incompressible Navier-Stokes equations. This conservation is true for smooth solutions. However, it is a long standing open problem, whether Leray-Hopf weak solutions are also conserving energy. Equation (1.1) can be seen as a particular case of the following Navier-Stokes ρ t + div(ρu) = (ρu) t + div(ρu u) + P div(µ(ρ)du) (λ(ρ)divu) =, (1.3) where the viscosity coefficients µ(ρ) and λ(ρ) depend on the density, and may vanish on the vacuum. When the coefficients verify the following condition: λ(ρ) = 2ρµ (ρ) 2µ(ρ) the system still formally verifies the BD estimates. However, the construction of Bresch and Desjardins in [3] is more subtle in this case. Up to now, construction of weak solutions are known, only verifying a fixed combination of the classical energy and BD entropy (see [5]). Those solutions verify the decrease of this so-called κ-entropy 1, but not the decrease of Energy and BD entropy by themselves. The extension of our result, in this context, for the 2D case, is considered in [31]. It would be of great interest to extend it to the 3D case. The basic energy inequality associated to (1.1) reads as T E(t) + ρ Du 2 dx dt E, (1.4) where and E(t) = E(ρ, u)(t) = E = E(ρ, u)() = ( 1 2 ρ u ) γ 1 ργ dx, ( 1 2 ρ u ) γ 1 ργ dx. Remark that those a priori estimates are not enough to show the stability of the solutions of (1.1), in particular, for the compactness of ρ γ. Fortunately, a particular mathematical 1 Note that κ here is not related to the κ term in (1.6).

4 4 ALEXIS F. VASSEUR AND CHENG YU structure was found in [2, 4], which yields the bound of ρ γ 2 in L 2 (, T ; L 2 ()). More precisely, we have the following Bresch-Desjardins entropy ( ) 1 T 2 ρ u + ln ρ 2 + ργ dx + ρ γ 2 2 dx dt γ 1 T + ρ u T u 2 dx dt ( ρ u 2 + ) ρ 2 + ργ dx. γ 1 Thus, the initial data should be given in such way that ρ L γ (), ρ, ρ L 2 (), m L 1 (), m = if ρ =, m 2 ρ L 1 (). Remark 1.1. The initial condition ρ L 2 () is from the Bresch-Desjardins entropy. (1.5) In particular, we need to recall the following existence result by Bresch-Desjardins [2]. Proposition 1.1. For any κ, there exists a global weak solution to the following system ρ t + div(ρu) =, (ρu) t + div(ρu u) + ρ γ div(ρdu) = r u r 1 ρ u 2 u + κρ ρ, (1.6) with the initial data (1.2) and satisfying (1.5). In particular, we have the following regularity ρ L (, T ; L γ ()), ρu L (, T ; L 2 ()), κ ρ L (, T ; L 2 ()), ρ L (, T ; L 2 ()), ρ γ 2 L 2 (, T ; L 2 ()), ρdu L 2 (, T ; L 2 ()), r u L 2 (, T ; L 2 ()), κ 2 ρ L 2 (, T ; L 2 ()). r ρ 1 4 u L 4 (, T ; L 4 ()), Remark 1.2. The weak solutions (ρ, u) constructed in Proposition 1.1 satisfy the energy inequality and Bresch-Desjardins entropy. Remark 1.3. The existence result of [2] contained the case with κ =, which can be obtained as the limit when κ > goes to in (1.6), by standard compactness analysis. Remark 1.4. The weak formulation reads as T ρu ψ dx t=t ρuψ t dx dt T T = r t= ρ γ divψ dx dt T T uψ dx dt r 1 T ρdu ψ dx dt ρu u : ψ dx dt T ρ u 2 uψ dx dt κ ρdiv(ψρ) dx dt. (1.7)

5 for any test function ψ. GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 5 Our first main result reads as follows: Theorem 1.1. For any δ (, 2), there exists a constant C depending only on δ, such that the following holds true. For any r >, r 1 >, and κ =, there exists a weak solution (ρ, u) verifying all the properties of Proposition 1.1, and satisfying the following Mellet-Vasseur type inequality for every T >, and almost every t < T : ρ(t, x)(1 + u(t, x) 2 ) ln(1 + u(t, x) 2 ) dx + C ρ (1 + u 2 ) ln(1 + u 2 ) dx + C T ( (ρ 2γ 1 δ 2 2 ) 2 δ ) 2 δ 2 ( ( ρ u ργ γ 1 + ρ 2 ) dx ) δ ρ(2 + ln(1 + u 2 )) 2 2 δ dx dt, Remark 1.5. The right hand side constant C of the above inequality can be bounded depending only on δ, ( ) 1 2 ρ u 2 + ργ γ 1 dx, ρ dx, and ρ 2 dx. In particular, it does not depend on r and r 1. This theorem will yield the strong convergence of ρu in space L 2 (, T ; ) when r, r 1 converge to. It will be the key tool of obtaining the existence of weak solutions, as in Mellet-Vasseur [27]. We define the weak solution (ρ, u) to the initial value problem (1.1) in the following sense: for any t [, T ], (1.2) holds in D (), (1.4) holds for almost every t [, T ], (1.1) holds in D ((, T ) )) and the following is satisfied ρ, ρ L ([, T ]; L γ ()), ρ(1 + u 2 ) ln(1 + u 2 ) L (, T ; L 1 ()), ρ γ 2 L 2 (, T ; L 2 ()), ρ L (, T ; L 2 ()), ρu L (, T ; L 2 ()), ρ u L 2 (, T ; L 2 ()). Remark 1.6. The regularity ρ L (, T ; L 2 ()) and ρ γ 2 L 2 (, T ; L 2 ()) are from the Bresch-Desjardins entropy. As a sequence of Theorem 1.1, our second main result reads as follows: Theorem 1.2. Let (ρ, m ) satisfy (1.5) and ρ (1 + u 2 ) ln(1 + u 2 ) dx <. Then, for any γ > 1 and any T >, there exists a weak solution of (1.1)-(1.2) on (, T ). We cannot obtained directly the estimate of Theorem 1.1 from (1.6) with κ =, because we do not have enough regularity on the solutions. But, the estimate is not true for the solutions of (1.6) for κ >. The idea is to obtain a control on ρ(t, x)ϕ n (φ(ρ)u(t, x)) dx

6 6 ALEXIS F. VASSEUR AND CHENG YU at the level κ >, for a ϕ n, suitable bounded approximation of (1 + u 2 ) ln(1 + u 2 ), and a suitable cut-off function φ of ρ, controlling both the large and small ρ. The first step (see section 2) consists in showing that we can control (uniformly with respect to κ) this quantity, for any weak solutions of (1.6) with κ >. This has to be done in several steps, taking into account the minimal regularity of the solutions, the weak control of the solutions close to the vacuum, and the extra capillarity higher order terms. In the limit κ goes to zero, the cut-off function φ has to converge to one in a special rate associated to κ (see section 3 and 4). This provides, for any weak limit of (1.6) obtained by limit κ converges to, a (uniform in n, r, and r 1 ) bound, to: ρ(t, x)ϕ n (φ(ρ)u(t, x)) dx. Note that the bound is not uniform in n, for κ fixed. However, it becomes uniform in n at the limit κ converges to. In section 5, we pass into the limit n goes to infinity, obtaining a uniform bound with respect to r and r 1 of ρ(t, x)(1 + u(t, x) 2 ) ln(1 + u(t, x) 2 ) dx. Section 6 is devoted to the limit r 1 and r converges to. The uniform estimate above provides the strong convergence of ρu needed to obtain the existence of global weak solutions to (1.1) with large initial data. 2. Approximation of the Mellet-Vasseur type inequality In this section, we construct an approximation of the Mellet-Vasseur type inequality for any weak solutions at the following level of approximation system ρ t + div(ρu) =, (ρu) t + div(ρu u) + ρ γ div(ρdu) = r u r 1 ρ u 2 u + κρ ρ, (2.1) with the initial data (1.5), verifying in addition that ρ 1 m for m > and ρ u L (). This restriction on the initial data will be useful later to get the strong convergence of ρu when t converges to. This restriction will be cancel at the very end, (see section 6). First, we recall the energy inequality and the Bresch-Desjardins entropy on any weak solutions of (2.1). By Proposition 1.1 and Remark 1.2, we have the following usual energy inequality associated to equation (2.1), T T T E(t) + ρ Du 2 dx dt + r u 2 dx dt + r 1 ρ u 4 dx dt E, (2.2) where and E = E(t) = ( 1 2 ρ u ) γ 1 ργ + κ ρ 2 dx, ( 1 2 ρ u γ 1 ργ + κ ρ 2 ) dx < +.

7 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 7 This yields the following estimates ρu L (,T ;L 2 ()) E <, ρ L (,T ;L γ ()) E <, κ ρ L (,T ;L 2 ()) E <, ρdu L 2 (,T ;L 2 ()) E <, r u L 2 (,T ;L 2 ()) E <, 4 r 1 ρu L 4 (,T ;L 4 ()) E <. (2.3) Meanwhile, we have the Bresch-Desjardins entropy associated to equation (2.1) as follows: [ 1 2 ρ u + (ln ρ) γ 1 ργ + κ T 2 ρ 2 ] dx + ρ γ 2 2 dx dt T T + κ 2 ρ 2 dx dt + ρ u T u 2 dx dt r log ρ dx [ 1 2 ρ u + (ln ρ ) γ 1 ργ ] dx r log ρ dx + C, where C is bounded by the initial energy. Notice that ρ is bounded uniformly in L (, T ; L γ ()), we have r log + ρ dx C, where log + ρ = log max(ρ, 1). In fact, this gives us log + ρ ρ for all ρ, log + ρ dx ρ dx < +. (2.4) It remains to require that the initial data as r log ρ dx C. This yields the following bounds on the density ρ: and ρ L (,T ;L 2 ()) C <, (2.5) κ 2 ρ L 2 (,T ;L 2 ()) C <, (2.6) ρ γ 2 L 2 (,T ;L 2 ()) C <, (2.7) ρ u L 2 (,T ;L 2 ()) C <, (2.8) where C is bounded by the initial data, uniformly on r, r 1 and κ. In fact, (2.5) yields ρ L (, T ; L p ()), for any 1 p < (2.9) in two dimensional space. Bresch-Desjardins [2, 3] constructed the weak solutions to the system (1.6) for any κ by the natural energy estimates and the Bresch-Desjardins entropy. The term r u turns out to be essential to show the strong convergence of ρu in L 2 (, T ; L 2 ()).

8 8 ALEXIS F. VASSEUR AND CHENG YU Unfortunately, it is not enough to ensure the strong convergence of ρu in L 2 (, T ; L 2 ()) when r and r 1 vanish. We define two C, nonnegative cut-off functions φ m and φ K as follows. φ m (ρ) = 1 for any ρ > 1 m, φ m(ρ) = for any ρ < 1 2m, (2.1) where m > is any real number, and φ m 2m; and φ K (ρ) C (R) is a nonnegative function such tat φ K (ρ) = 1 for any ρ < K, φ K (ρ) = for any ρ > 2K, (2.11) where K > is any real number, and φ K 2 K. We define v = φ(ρ)u, and φ(ρ) = φ m (ρ)φ K (ρ). The following Lemma will be very useful to construct the approximation of the Mellet-Vasseur type inequality. Lemma 2.1. For any fixed κ >, we have ρ L 4 (,T ;) C, v L 2 (,T ;) C, where the constants C depend on κ >, r 1, K and m; and where r < 4 3 and s < 2. ρ t L 4 (, T ; L r ()) + L 2 (, T ; L s ()), Remark 2.1. These estimates are very crucial to construct a suitable test function later. This is the main obstacle to extend the result to 3D. The lack of estimate of ρ in L 4 (, T ; ) fail to control v L 2 (, T ; ). Proof. By (2.5), (2.6), and Gagliardo-Nirenberg inequality, we deduce For v, we have and hence ρ L 4 (,T ;) C. v = (φ(ρ)u) = (φ (ρ) ρ)u + φ(ρ) u, (φ (ρ) ρ)u + φ(ρ) u L 2 (,T ;) C ρ 1 4 u ρ L 2 (,T ;) + C ρ u L 2 (,T ;) C ρ 1 4 u L 4 (,T ;) ρ L 4 (,T ;) + C ρ u L 2 (,T ;), where we used the definition of the function φ(ρ). Indeed, there exists C > such that φ (ρ) + φ(ρ) C for any ρ >. For ρ t, we have ρ 1 4 ρ t = ρ u ρdivu ρ 1 2 = ρ ρ 1 4 uρ 1 4 ρ ρdivu = S1 + S 2.

9 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 9 Thanks to (2.3), (2.5) and (2.9), we have S 1 L 4 (, T ; L r ()) for 1 r < 4 3. By (2.3) and (2.9), we have Thus, we have S 2 L 2 (, T ; L s ()) for 1 s < 2. ρ t L 4 (, T ; L r ()) + L 2 (, T ; L s ()). We introduce a new C (R 2 ), nonnegative cut-off function ϕ n which is given by { (1 + x 2 )(1 + ln(1 + x 2 )) if x < n, ϕ n (x) = (1 + 8n 2 ) ln(1 + 4n 2 ) if x 2n, where n > are large, and (2.12) ϕ n(x) + ϕ n(x) C n for any x n. The first step of constructing the approximation of the Mellet-Vasseur type inequality is the following lemma: Lemma 2.2. For any weak solutions to (2.1) constructed in Proposition 1.1, and any ψ(t) D( 1, + ), we have T T T ψ t ρϕ n (v) dx dt ψ(t)ϕ n(v)f dx dt + ψ(t)s : (ϕ n(v)) dx dt (2.13) = ρ ϕ n (v )ψ() dx, where S = ρφ(ρ)(du + κ ρi), where I is an identical matrix. and F = ρ 2 uφ (ρ)divu + 2ρ γ 2 ρ γ 2 φ(ρ) + ρ φ(ρ)du + r uφ(ρ) + r 1 ρ u 2 uφ(ρ) + κρ φ(ρ) ρ κφ(ρ) ρ ρ, (2.14) In this proof, κ, m and K are fixed. So the dependence of the constants appearing in this proof will not be specified. Multiplying φ(ρ) on both sides of the second equation of (2.1), we have (ρv) t ρuφ (ρ)ρ t + div(ρu v) ρu u φ(ρ) + 2ρ γ 2 ρ γ 2 φ(ρ) div(φ(ρ)ρdu) + ρ φ(ρ)du + r uφ(ρ) + r 1 ρ u 2 uφ(ρ) κ (ρφ(ρ) ρ) + κρ φ(ρ) ρ κφ(ρ) ρ ρ =. Remark 2.2. Both ρ and ρ t are functions, so the above equality are justified by regularizing ρ and passing into the limit.

10 1 ALEXIS F. VASSEUR AND CHENG YU We can rewrite the above equation as follows where S and F are as in (2.14), and here we used (ρv) t + div(ρu v) divs + F =, (2.15) ρuφ (ρ)ρ t + ρu uφ (ρ) ρ = ρuφ (ρ)(ρ t + ρ u) = ρ 2 uφ (ρ)divu. We should remark that, thanks to (2.3)-(2.8), F L 4 3 (,T ;L 1 ()) C, S L 2 (,T ;) C, since ρφ(ρ) and ρφ(ρ) bounded. Those bounds depend on K and κ. We first introducing a test function ψ(t) D(, + ). Essentially this function vanishes for t close t =. We will later extend the result for ψ(t) D( 1, + ). We define a new function Φ = ψ(t)ϕ n(v), where f(t, x) = f η k (t, x), k is a small enough number. Note that, since ψ(t) is compactly supported in (, ). Φ is well defined on (, ) for k small enough. We use it to test (2.15) to have T ψ(t)ϕ n(v)[(ρv) t + div(ρu v) divs + F ] dx dt =, which in turn gives us T ψ(t)ϕ n(v)[(ρv) t + div(ρu v) divs + F ] dx dt =. (2.16) The first term in (2.16) can be calculated as follows T ψ(t)ϕ n(v)(ρv) t dx dt where T = = = T T T ψ(t)ϕ n(v)(ρv) t dx dt + ψ(t)ϕ n(v)[(ρv) t (ρv) t ] dx dt ψ(t)ϕ n(v)(ρ t v + ρv t ) dx dt + R 1 T ψ(t)ρ t ϕ n(v)v dx dt + ψ(t)ρϕ n (v) t dx dt + R 1, R 1 = T ψ(t)ϕ n(v)[(ρv) t (ρv) t ] dx dt. (2.17) Thanks to the first equation in (2.1), we can rewrite the second term in (2.16) as follows T ψ(t)ϕ n(v)div(ρu v) dx dt T T (2.18) = ψ(t)ρ t ϕ n (v) dx dt ψ(t)ρ t ϕ n(v)v + R 2,

11 and GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 11 R 2 = T By (2.16)-(2.18), we have T ψ(t)(ρϕ n (v)) t dx dt + R 1 + R 2 ψ(t)ϕ n(v)[div(ρu v) div(ρu v)]. T ψ(t)ϕ n(v)divs dx dt T + ψ(t)ϕ n(v)f =. Notice that v converges to v almost everywhere and ρϕ n (v)ψ t ρϕ n (v)ψ t So, up to a subsequence, We have T (ρϕ n (v))ψ t dx dt T in L 1 (, T ; ). (2.19) (ρϕ n (v))ψ t dx dt as k. (2.2) Since ϕ n(v) converges to ϕ n(v) almost everywhere, and is uniformly bounded in L (, T ; ), we have T T ψ(t)ϕ n(v)f ψ(t)ϕ n(v)f as k. (2.21) Noticing that we have v v v L 2 (, T ; ), strongly in L 2 (, T ; ). Since S converges to S strongly in L 2 (, T ; ), and ϕ n(v) converges to ϕ n(v) almost everywhere and uniformly bounded in L (, T ; ), we get T T ψ(t)ϕ n(v)divs dx dt = ψ(t)s : (ϕ n(v)) dx dt, (2.22) which converges to T ψ(t)s : (ϕ n(v)) dx dt. (2.23) To handle R 1 and R 2, we use the following lemma due to Lions, see [21]. Lemma 2.3. Let f W 1,p (R N ), g L q (R N ) with 1 p, q, and 1 p + 1 q 1. Then, we have div(fg) w ε div(f(g w ε )) L r (R N ) C f W 1,p (R N ) g L q (R N ) for some C independent of ε, f and g, r is determined by 1 r = 1 p + 1 q. In addition, as ε if r <. div(fg) w ε div(f(g w ε )) in L r (R N ) This lemma includes the following statement.

12 12 ALEXIS F. VASSEUR AND CHENG YU Lemma 2.4. Let f t L p (, T ), g L q (, T ) with 1 p, q, and 1 p + 1 q we have 1. Then, (fg) t w ε (f(g w ε )) t L r (,T ) C f t L p (,T ) g L q (,T ) for some C independent of ε, f and g, r is determined by 1 r = 1 p + 1 q. In addition, as ε if r <. (fg) t w ε (f(g w ε )) t in L r (, T ) With Lemma 2.3 and Lemma 2.4 in hand, we are ready to handle the terms R 1 and R 2. For κ >, by Lemma 2.1 and Poincare inequality, we have v L 2 (, T ; L p ()) for any 1 p <. We also have, by Lemma 2.1, ρ t L 4 (, T ; L r ()) + L 2 (, T ; L s ()), where 1 r < 4 3 and 1 s < 2. Thus, applying Lemma 2.4, T R 1 ψ(t)ϕ n(v)[(ρv) t (ρv) t ] dx dt C(ψ) T Similarly, applying Lemma 2.3, we conclude ϕ n(v)[(ρv) t (ρv) t ] dx dt as k. (2.24) By (2.2)-(2.25), we have T ψ t ρϕ n (v) dx dt R 2 as k. (2.25) T + T ψ(t)ϕ n(v)f dx dt ψ(t)s : (ϕ n(v)) dx dt =, (2.26) for any test function ψ D(, ). Now, we need to consider the test function ψ(t) D( 1, ). For this, we need the continuity of ρ(t) and ( ρu)(t) in the strong topology at t =. In fact, thanks to (2.3), we have ρ t L 2 (, T ; H 1 ()), ρ L 2 (, T ; H 1 ()). This gives us ρ C([, T ]; L 2 ()), thanks to Theorem 3 on page 287, see [1]. Meanwhile, we have ρ L (, T ; L p ()) for any 1 p <, and hence ρ C([, T ]; L p ()) for any 1 p <. (2.27)

13 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 13 On the other hand, we see ess lim sup ρu ρ u 2 dx t ( ess lim sup ( 1 t 2 ρ u 2 + ργ + ess lim sup t κ ess lim sup t γ 1 + κ ρ 2 ) dx ( 1 2 ρ u 2 + ργ ρ u ( ρ u ) ρu) dx + ( ργ γ 1 ργ γ 1 ) ρ ρ 2 dx + 2κ ess lim sup ρ ( ρ ρ) dx. t ( 2 ) γ 1 ) + κ ρ 2 dx (2.28) We have ess lim sup ρ ( ρ ρ) dx =. (2.29) t Indeed, for any ε >, there exists R ε C () such that and ρ R ε L 2 () ε, ρ ( ρ ρ) dx = ( ρ R ε )( ρ ρ ) dx divr ε (ρ ρ ) dx, ρ(t) converges to ρ in L p () when t for any 1 p <. So, for ε > fixed, we have divr ε (ρ ρ ) dx =. But, ρ L (, T ; L 2 ()), then ( ρ R ε )( ρ ρ ) dx ε ( ρ L (,T ;L 2 () + ρ L ()) 2, this is true for any ε >. Thus, we have (2.29). So, using (2.2), and (2.27), we have ess lim sup ρu ρ u 2 dx t 2ess lim sup ρ u ( ρ u ρu) dx t = 2ess lim sup t = B 1 + B 2. ( ρ u ( ρ u ρuφ m (ρ)) dx + ρ u (1 φ m (ρ)) ) ρu dx Because (ρ, u) is a weak solution to (2.1), one deduces that ρu C([, T ]; L 2p p+2 weak ()), for any 1 p <. (2.3)

14 14 ALEXIS F. VASSEUR AND CHENG YU We consider B 1 as follows B 1 = ( 2ess lim sup ρ u ( φ m(ρ) (ρ u ρu)) dx ρ ρ u 2 ( φ m(ρ) t ρ then we have B 1 =, where we used (2.27) and (2.3). Since m m, and ρ 1 m, we have ρ φ m(ρ ) B 2 ρ u L (,T ;) ρu L (,T ;L 2 ())ess lim sup 1 φ m (ρ) L 2 (,T ;) =. t Thus, we have which gives us By (2.27) and (2.31), we get 1 lim τ τ ess lim sup ρu ρ u 2 dx =, t τ Considering (2.26) for the test function, we get T τ ψ τ (t) = ψ(t) T ψ t ρϕ n (v) dx dt ρ ) ) dx, ρu C([, T ]; L 2 ()). (2.31) T + ρϕ n (v) dx dt = for t τ, ψ τ (t) = ψ(τ) t τ ψ τ (t)ϕ n(v)f dx dt ρ ϕ n (v ) dx. for t τ, ψ τ (t)s : (ϕ n(v)) dx dt = ψ(τ) τ Passing into the limit as τ, this gives us T T ψ t ρϕ n (v) dx dt ψ(t)ϕ n(v)f dx dt + T ψ(t)s : (ϕ n(v)) dx dt = 3. Recover the limits as m τ ρ ψ()ϕ n (v ) dx. ρϕ n (v) dx dt. (2.32) In this section, we want to recover the limits in (2.13) as m. Here, we should remark that (ρ, u) is any fixed weak solution to (2.1) verifying Proposition 1.1 with κ >. For any fixed weak solution (ρ, u), we have φ m (ρ) 1 almost everywhere for (t, x), and it is uniform bounded in L (, T ; ), we also have and thus r φ K (ρ)u L 2 (, T ; ), v m = φ m φ K u φ K u almost everywhere for (t, x)

15 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 15 as m. By the Dominated Convergence Theorem, we have as m, and hence, we have for any 1 p <. Meanwhile, for any fixed ρ, we have v m φ K u in L 2 (, T ; ) ϕ n (v m ) ϕ n (φ K u) in L p (, T ; ) φ m(ρ) almost everywhere for (t, x) as m. We calculate φ m(ρ) 2m as 1 2m ρ 1 m, and otherwise, φ m(ρ) =, thus, we have and We can find that T ψ (t)(ρϕ n (v m )) dx dt ρφ m(ρ) 1 for all ρ. ρ ϕ n (v m ) T ψ (t)(ρϕ n (φ K (ρ)u)) dx dt ρ ϕ n (φ K (ρ )u ) as m. To pass into the limits in (2.32) as m, we rely on the following Lemma: Lemma 3.1. If a m L (,T ;) C, a m a a.e. for (t, x) and in L p (, T ; ) for any 1 p <, f L 1 (, T ; ), then we have T φ m (ρ)a m f dx dt and Proof. We have T T af dx dt as m, ρφ m (ρ)a m f dx dt as m. φ m (ρ)a m f af φ m (ρ)f f a m + a m f af = I 1 + I 2. For I 1 : φ m (ρ)f f a.e. for (t, x) and φ m (ρ)f f 2 f a.e. for (t, x), by Lebesgue s Dominated Convergence Theorem, we conclude T φ m (ρ)f f dx dt

16 16 ALEXIS F. VASSEUR AND CHENG YU as m, which in turn yields T φ m (ρ)a m f a m f dx dt a m L (,T ;) T φ m (ρ)f f dx dt as m. Following the same line, we have T a m f af dx dt as m. Thus we have T φ m (ρ)a m f dx dt T af dx dt as m. We now consider T ρφ m(ρ)a m f dx dt. Notice that ρφ m(ρ) C, and ρφ m(ρ) converges to almost everywhere, so ρφ m(ρ)a m f C f, and by Lebesgue s Dominated Convergence Theorem, T ρφ m(ρ)a m f dx dt as m. Calculating T T = = T ψ(t)s m : (ϕ n(v m )) dx dt ψ(t)s m ϕ n(v m ) ( φ m φ K u + φ m φ K u + φ m φ K u) dx dt T φ m (ρ)a m1 f 1 dx dt + ρφ m(ρ)a m2 f 2 dx dt, where a m1 = φ m (ρ)ϕ n(v m ), f 1 = ψ(t)ρφ K (ρ) (Du + κ ρi) (u φ K (ρ) + φ K (ρ) u), and a m2 = ϕ n(v m )φ m (ρ)φ K (ρ)u = ϕ n(v m )v m, f 2 = ψ(t)φ K (ρ) (Du + κ ρi) ρ = ψ(t)φ K (ρ)(κ ρ ρ + ρdu ρ). So applying Lemma 3.1 to (3.1), we have T T ψ(t)s m : (ϕ n(v m )) dx dt ψ(t)s : (ϕ n(φ K (ρ)u)) dx dt as m, where S = φ K (ρ)ρ(du + κ ρi). Letting F m = F m1 + F m2, where F m1 = ρ 2 uφ (ρ)divu + ρ φ(ρ)du + κρ φ(ρ) ρ = ρ ( φ m(ρ)φ K (ρ) + φ m (ρ)φ K(ρ) ) (ρudivu + ρ Du + κ ρ ρ), (3.1)

17 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 17 where and where φ K (ρ)(ρudivu + ρ Du + κ ρ ρ) L 1 (, T ; ), ρφ K(ρ)(ρudivu + ρ Du + κ ρ ρ) L 1 (, T ; ), F m2 = φ m (ρ)φ K (ρ)(2ρ γ 2 ρ γ 2 + r u + r 1 ρ u 2 u κ ρ ρ), ) φ K (ρ) (2ρ γ γ 2 ρ 2 + r u + r 1 ρ u 2 u κ ρ ρ L 1 (, T ; ). Applying Lemma 3.1, we have T ψ(t)ϕ n(v m )F m dx dt where T ψ(t)ϕ n(φ K (ρ)u)f dx dt, F = ρ 2 uφ K(ρ)divu + 2ρ γ 2 ρ γ 2 φk (ρ) + ρ φ K (ρ)du + r uφ K (ρ) + r 1 ρ u 2 uφ K (ρ) + κρ φ K (ρ) ρ κφ K (ρ) ρ ρ. Letting m in (2.32), we have T ψ (t)(ρϕ n (φ K (ρ)u)) dx dt T + T ψ(t)s : (ϕ n(φ K (ρ)u)) dx dt = which in turn gives us the following lemma: ψ(t)ϕ n(φ K (ρ)u)f dx dt ψ()ρ ϕ n (φ K (ρ )u ) dx, Lemma 3.2. For any weak solutions to (2.1) verifying in Proposition 1.1, we have T T ψ (t)(ρϕ n (φ K (ρ)u)) dx dt ψ(t)ϕ n(φ K (ρ)u)f dx dt T + where S = φ K (ρ)ρ(du + κ ρi), and ψ(t)s : (ϕ n(φ K (ρ)u)) dx dt = ψ()ρ ϕ n(φ K (ρ )u ) dx, F = ρ 2 uφ K(ρ)divu + 2ρ γ 2 ρ γ 2 φk (ρ) + ρ φ K (ρ)du + r uφ K (ρ) + r 1 ρ u 2 uφ K (ρ) + κρ φ K (ρ) ρ κφ K (ρ) ρ ρ. where I is an identical matrix. (3.2) 4. Recover the limits as κ and K. The objective of this section is to recover the limits in (3.2) as κ and K. In this section, we assume that K = κ 3 4, thus K when κ. First, we address the following lemma.

18 18 ALEXIS F. VASSEUR AND CHENG YU Lemma 4.1. Let κ and K, we have ρ γ 2 κ ρ γ 2 strongly in L 2 (, T ; ), (4.1) and ρ γ 2 κ ρ γ 2 weakly in L 2 (, T ; H 1 ()), ρ κ ϕ n (φ K (ρ κ )u κ ) ρϕ n (u) strongly in L 1 (, T ; ), ρ γ 2 κ ϕ n(φ K (ρ κ )u κ ) ρ γ 2 ϕ n (u) strongly in L 2 (, T ; ). Proof. We calculate (ρ γ 2 ) t as follows (ρ γ 2 κ ) t = u κ ρ γ 2 κ + γ 2 ρ γ 2 κ divu κ = γ 2 div(ρ γ 2 κ u κ ) + (1 γ 2 )u κ ρ γ κ, which in turn yields (ρ γ 2 κ ) t is uniformly bounded in L (, T ; W 1,1 ()). This, together with ρ γ 2 κ L 2 (,T ;) C, gives us (4.1) by Aubin-Lions Lemma. And thus we have ρ γ 2 κ ρ γ 2 weakly in L 2 (, T ; H 1 ()). Meanwhile, we have (ρ κ ) t is uniformly bounded in L 2 (, T ; W 1,2 ()), and ρ κ L 4 (,T ;L s ()) C s, for 1 s < 4 3. Applying Aubin-Lions Lemma, one obtains ρ κ ρ strongly in L 4 (, T ; L s ()). When κ, we have ρ κ u κ ρu strongly in L 2 (, T ; ), (see [2]). Thus, up to a subsequence, for almost every (t, x) such that ρ(t, x), we have u κ (t, x) = ρκ u κ ρκ u(t, x), and φ K (ρ κ )u κ u(t, x), as κ. For almost every (t, x) such that ρ(t, x) =, ρ κ ϕ n (φ K (ρ κ )u κ ) C n ρ κ (t, x) = ρϕ n (u) (4.2) as κ. Hence, ρ κ ϕ n (φ K (ρ κ )u κ ) converges to ρϕ n (u) almost everywhere, and so ρ κ ϕ n (φ K (ρ κ )u κ ) converges to ρϕ n (u) in L 1 (, T ; ). By the uniqueness of the limit, the convergence holds for the whole sequence. Similarly, we have ρ γ 2 κ ϕ n(φ K (ρ κ )u κ ) ρ γ 2 ϕ n (u) strongly in L 2 (, T ; ). Lemma 4.2. For any κ >, we have κ 1 4 ρκ L 4 (,T ;) C, where C is uniform with respect to κ.

19 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 19 Proof. Applying Gagliardo-Nirenberg inequality, we have κ 1 4 ρκ L 4 () C ρ κ 1 2 L 2 () κ ρ κ 1 2 L 2 (), which in turn gives us T κ ρ κ 4 dx dt C ρ κ 2 L (,T ;L 2 ()) κ ρ κ 2 L 2 (,T ;). With above two lemmas in hand, we are ready to recover the limits in (3.2) as κ and K. We have the following lemma. Lemma 4.3. Let K = κ 3 4, and κ, we have T ψ (t) ρϕ n (u) dx dt C T n + + C ( 1 2 ρ u 2 + ργ γ 1 + ρ 2 ) dx + ψ() ψ(t) ρ γ ϕ n(u) dx dt ρ ϕ n (u ) dx (4.3) Proof. Here, we use (ρ κ, u κ ) to denote the weak solutions to (2.1) verifying Proposition 1.1 with κ >. By Lemma 4.1, we can handle the first term in (3.2), that is, T T ψ (t)(ρ κ ϕ n (φ K (ρ κ )u κ )) dx dt ψ (t)(ρϕ n (u)) dx dt (4.4) and ψ() ρ ϕ n(φ K (ρ )u ) dx ψ() ρ ϕ n(u ) dx (4.5) as κ and K = κ 3 4. By Lemma 4.1, we have T ψ(t)ϕ n(φ K (ρ κ )u κ )2ρ γ 2 κ ρ γ 2 κ φ K (ρ κ ) dx dt as κ. Note that T T ψ(t)ϕ n(u)2ρ γ 2 ρ γ 2 dx dt (4.6) ψ(t)ϕ n(φ K (ρ κ )u κ )(r u κ + r 1 ρ κ u κ 2 u κ ) dx dt, (4.7) so this term can be ignored. We treat the other terms in F one by one, T ψ(t)ϕ n(φ K (ρ κ )u κ )ρ 2 κu κ φ K(ρ κ )divu κ dx dt C(n, ψ) ρ 1 4 κ u κ L 4 (,T ;) ρ κ divu κ L 2 (,T ;) φ K(ρ κ ) ρ 5 4 κ L 4 (,T ;) C(n, ψ)κ 3 4 (4.8)

20 2 ALEXIS F. VASSEUR AND CHENG YU as κ, where we used Sobolev inequality, Lemma 4.2, and φ K (ρ κ) 2 K ; T C(n, ψ) ψ(t)ϕ n (φ K (ρ κ )u κ )ρ κ φ K (ρ κ )Du κ dx dt 1 Kκ 1 4 C(n, ψ) κ ( ) κ 1 4 ρκ L 4 (,T ;) ρ κ Du κ L 2 (,T ;) ρ κ L 4 (,T ;) (4.9) as κ ; T κ ψ(t)ϕ n(φ K (ρ κ )u κ )ρ κ φ K (ρ κ ) ρ κ dx dt ( ) 2C(n, ψ)κ 1 4 κ 1 4 ρκ L 4 (,T ;) κ ρ κ L 2 (,T ;) (4.1) 2C(n, ψ)κ 1 4 as κ, where we used ρ κ φ K (ρ κ) 1. Finally T κ ψ (t)ϕ n (φ K (ρ κ )u κ )φ K (ρ) ρ ρ dx dt ( ) 2C(n, ψ)κ 1 4 κ 1 4 ρκ L 4 (,T ;) κ ρ κ L 2 (,T ;) (4.11) 2C(n, ψ)κ 1 4 as κ. For the term S κ = φ K (ρ κ )ρ κ (Du κ + κ ρ κ I) = S 1 + S 2, we calculate as follows T T = = + T T ψ(t)s 1 : (ϕ n(φ K (ρ κ )u κ )) dx dt = A 1 + A 2. ψ(t)φ K (ρ κ )ρ κ Du κ : (ϕ nφ K (ρ κ )u κ )) dx dt ψ(t)[ u κ ϕ n(φ K (ρ κ )u κ )ρ κ ]Du κ φ K (ρ κ ) 2 dx dt ψ(t)ρ κ ϕ n(φ K (ρ κ )u κ ) φ K (ρ κ ) Du κ (u κ φ K (ρ κ )) dx dt (4.12)

21 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 21 For A 1, we have A 1 = T C T n + ψ(t)ϕ n(φ K (ρ κ )u κ )ρ κ Du κ 2 φ K (ρ κ ) 2 dx dt ψ(t) ( 1 + ln(1 + u κ 2 ) ) ρ κ Du κ 2 1 uκ nφ K (ρ κ ) 2 dx dt T 2u i + ψ(t)ρ κu j κ κ 1 + u κ 2 ju k κd i u j κ1 uκ nφ K (ρ κ ) 2 dx dt C T n + ψ(t) ( 1 + ln(1 + u κ 2 ) ) ρ κ Du κ 2 1 uκ nφ K (ρ κ ) 2 dx dt C T C T n C ψ(t)ρ κ u κ 2 1 uκ nφ K (ρ κ ) 2 dx dt ρ κ u κ 2 dx dt, (4.13) where we used (2.12) and ϕ n(u) + ϕ n(u) C n for u n. For A 2, thanks to ϕ n(φ K u κ )(φ K u κ ) C n, we can control it as follows A 2 C(n, ψ) ρ κ Du κ L 2 (,T ;) ρ κ L 4 (,T ;)(κ 1 4 ρκ L 4 (,T ;)) φ K L (,T ;) C Kκ 1 4 = C κ (4.14) κ 1 4 as κ. We need to treat the term related to S 2, T κ = κ + κ T T ψ(t)s 2 : (ϕ n(φ K (ρ κ )u κ )) dx dt = B 1 + B 2, ψ(t) u κ ϕ n(φ K (ρ κ )u κ ) : ρ κ φ K (ρ κ ) 2 ρ κ dx dt ψ(t)u κ φ K (ρ κ )ϕ n(φ K (ρ κ )u κ ) ρ κ ρ κ φ K(ρ κ ) ρ κ dx dt (4.15) we control B 1 as follows B 1 C(n, ψ) ρ κ u κ L 2 (,T ;) κ ρ κ L 2 (,T ;) Kκ Cκ 1 8 (4.16)

22 22 ALEXIS F. VASSEUR AND CHENG YU as κ, where we used φ 2 K ρκ L (,T ;) K. For B 2, we have ( ) B 2 C(n, ψ) κ 1 4 ρκ L 4 (,T ;) ρ κ L 4 (,T ;) κ 1 4 κ ρ κ L 2 (,T ;) K C κ 1 4 K = Cκ (4.17) as κ. With (4.4)-(4.17), letting κ in (3.2), dropping the positive terms on the left side, we have the following inequality T ψ (t) ρϕ n (u) dx dt C T n + + C T ψ(t) ρ γ ϕ n(u) dx dt + ψ() ρ u 2 dx dt, ρ ϕ n (u ) dx which in turn gives us Lemma Recover the limits as n. In this section, we aim at recovering the Mellet-Vasseur type inequality for the weak solutions at the approximation level of compressible Navier-Stokes equations by letting n. In particular, we prove Theorem 1.1 by recovering the limit from Lemma 4.3. In this section, (ρ, u) are the fixed weak solutions. By (2.12), we have ϕ n (u) = (1 + u 2 ) ln(1 + u 2 ) for u n, ϕ n(u) + ϕ n(u) C n for u n. (5.1) Our task is to bound the right term of (4.3), T ψ(t) ρ γ ϕ n(u) dx dt T ψ(t)ϕ n(u) uρ γ T 1 u n dx dt + C ψ(t) 2u iu k 1 + u 2 iu k ρ γ 1 u n dx dt T + C ψ(t)(1 + ln(1 + u 2 ))(divu)ρ γ 1 u n dx dt C T n ρ u L 2 (,T ;) ρ γ 1 2 L 2 (,T ;) + C ρ u 2 1 u n dx dt T + C ρ 2γ 1 T dx dt + C (1 + ln(1 + u 2 ))(divu)ρ γ 1 u n dx dt. (5.2)

23 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 23 The last term on above inequality can be controlled as follows T (1 + ln(1 + u 2 ))(divu)ρ γ 1 u n dx dt T C (1 + ln(1 + u 2 ))ρ Du 2 1 u n dx dt and C T T ( C 2 + C T (1 + ln(1 + u 2 ))ρ 2γ 1 1 u n dx dt (ρ 2γ 1 δ 2 2 ) 2 δ ) 2 δ 2 (1 + ln(1 + u 2 ))ρ 2γ 1 1 u n dx dt, ( ρ(1 + ln(1 + u 2 ))) 2 δ 1 u n dx By (5.1)-(5.4), we have T ψ (t) ρϕ n (u) dx dt ρ ϕ n (u ) dx + C ( 1 2 ρ u 2 + ργ γ 1 + ρ 2 ) dx + C T n ρ u L 2 (,T ;) ρ γ 1 2 L 2 (,T ;) + C 1 T ( + C 2 (ρ 2γ 1 δ 2 2 ) 2 δ ) 2 δ 2 Letting n, we have T ψ (t) ρ(1 + u 2 ) ln(1 + u 2 ) dx dt ψ() ρ (1 + u 2 ) ln(1 + u 2 ) dx + C T ( + C 2 (ρ 2γ 1 δ 2 2 ) 2 δ ) 2 δ 2 ( which gives our first main result Theorem 1.1. ) δ ρ u 2 1 u n dx dt + C n ( ) δ ρ(1 + ln(1 + u 2 ))) 2 2 δ 1 u n dx dt. (5.3) 2 dt. (5.4) ( 1 2 ρ u 2 + ργ γ 1 + ρ 2 ) dx ) δ ρ(2 + ln(1 + u 2 )) 2 2 δ dx dt, 6. Recover the weak solutions The objective of this section is to apply Theorem 1.1 to prove Theorem 1.2. In particular, we aim at establishing the existence of global weak solutions to (1.1)-(1.2) by letting r and r 1. Let r = r = r 1, we use (ρ r, u r ) to denote the weak solutions to (2.1) verifying Proposition 1.1 with κ =. Here, we remark that the initial data should satisfy the following conditions, more precisely, ρ r ρ strongly in L γ (), ρ r u r ρ u strongly in L 2 ()

24 24 ALEXIS F. VASSEUR AND CHENG YU as r, and ρ is bounded in L 1 L γ (), ρ a.e. in, ρ u 2 = m 2 /ρ is bounded in L 1 (), ρ is bounded in L 2 (), 1 ρ (1 + u 2 ) ln(1 + u 2 ) dx C <. 2 By (2.2), (2.4), one obtains the following estimates, and by Theorem 1.1, we have sup t [,T ] ρ r u r L (,T ;L 2 ()) C; ρ r L (,T ;L 1 L γ ()) C; ρ r u r L 2 (,T ;L 2 ()) C; ρ r L (,T ;L 2 ()) C; ρ γ/2 r L 2 (,T ;L 2 ()) C, (6.1) (6.2) ρ r u r 2 ln(1 + u r 2 ) dx C. (6.3) It is necessary to remark that all above estimates on (6.2) and (6.3) are uniformly on r. In particular, this C only depends on the initial conditions (6.1). Thus, we can make use of all estimates to recover the weak solutions by letting r. Meanwhile, we have the following estimates from (2.2), T r u r 2 dx dt C, T (6.4) rρ r u r 4 dx dt C. To establish the existence of global weak solutions, we should pass into the limits as r. Following the same line as in [27], we can show the convergence of the density and the pressure, prove the strong convergence of ρ r u r in space L 2 loc (, T ; ), the convergence of the diffusion terms. We remark that Theorem 1.1 is the key tool to show the strong convergence of ρ r u r. Here, we list all related convergence from [27]. In particular, ρr ρ almost everywhere and strongly in L 2 loc (, T ; )), (6.5) ρ r ρ in C (, T ; L 3 2 loc ()); the convergence of pressure the convergence of the momentum and ρ r u r ρ γ r ρ γ strongly in L 1 loc (, T ; ); (6.6) ρ r u r ρu strongly in L 2 (, T ; L p loc ()) for p [1, 3/2); ρr u r ρu strongly in L 2 loc (, T ; ); (6.7)

25 GLOBAL SOLUTIONS TO THE SHALLOW WATER EQUATIONS 25 and the convergence of the diffusion terms ρ r u r ρ u in D, ρ r T u r ρ T u in D. (6.8) It remains to prove that terms ru r and rρ r u r 2 u r tend to zero as r. Let ψ be any test function, then we estimate the term ru r T T ru r ψ dx dt r r 2 ur ψ dx dt ( T ( ) 1 ) r 1 2 r u r 2 2 dx ψ L 2 () dt (6.9) 1 T 2 r 1 2 r u r 2 dx dt + 1 T 2 r 1 2 ψ 2 dx dt as r, due to (6.4). We also estimate rρ r u r 2 u r as follows T rρ r u r 2 T u r ψ dx dt r 1 4 ψ L (,T ;) T r ρ 4 r r ρ 4 r u r 3 ψ dx dt ( rρ r u r 4 dx) 3 4 ( ρ r 3 dx) 1 12 dt (6.1) as r. The global weak solutions to (2.1) verifying Proposition 1.1 with κ = is in the following sense, that is, (ρ r, u r ) satisfy the following weak formulation T T ρ r u r ψ dx t=t ρ r u r ψ t dx dt ρ r u r u r : ψ dx dt T = r T t= ρ γ r divψ dx dt u r ψ dx dt r T T ρdu r ψ dx dt ρ r u r 2 u r ψ dx dt, (6.11) where ψ is any test function. Letting r in the weak formulation (6.11), and applying (6.5)-(6.1), one obtains that T T ρu ψ dx t=t t= ρuψ t dx dt ρu u : ψ dx dt T T (6.12) ρ γ divψ dx dt ρdu ψ dx dt =. Thus we proved Theorem 1.2.

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