Vanishing viscosity limit of the rotating shallow water equations with far field vacuum

. In this paper, we consider the Cauchy problem of the rotating shallow water equations, which has height-dependent viscosities, arbitrarily large initial data and far ﬁeld vacuum. Firstly, we establish the existence of the unique local regular solution, whose life span is uniformly positive as the viscosity coeﬃcients vanish. Secondly, we prove the well-posedness of the regular solution for the inviscid ﬂow. Finally, we show the convergence rate of the regular solution from the viscous ﬂow to the inviscid ﬂow in L ∞ ([0 ,T ]; H s (cid:48) ) for any s (cid:48) ∈ [2 , 3) with a rate of (cid:15) 1 − s (cid:48) 3 .

1. Introduction. In this paper, we consider the Cauchy problem of the rotating shallow water equations, which have height-dependent viscosities, arbitrarily large initial data and far field vacuum. The shallow water equations simulate the evolution of an incompressible fluid in regard to gravitational and rotational accelerations. It is also considered as an important extension of the two-dimensional compressible Navier-Stokes equations with additional rotating force, and the solutions present many types of motion. In general, the rotating shallow water equations with the viscous flow have the form    h t + div(hu) = 0, where x = (x 1 , x 2 ) ∈ R 2 , t ≥ 0, h is the height of the fluid surface, u = (u (1) , u (2) ) is the horizontal velocity field, u ⊥ = (−u (2) , u (1) ), g > 0 is the gravity constant, f > 0 is the Coriolis frequency, and T denotes the viscosity stress tensor with the following form T = µ(h)(∇u + (∇u) ) + λ(h)divuI 2 , where I 2 is the 2 × 2 identity matrix, µ(h) = αh is the shear viscosity coefficient, λ(h) = βh, µ(h) + λ(h) is the bulk viscosity coefficient, α and β are both constants satisfying α > 0, α + β ≥ 0.
Without loss of generality, we can assume that g = 1, f = 1 and ∈ (0, 1]. We only consider the local classical solution satisfying the initial data (h, u)| t=0 = (h 0 (x), u 0 (x)), (5) and the far field behavior (h, u) → (0, 0) as |x| → +∞, t > 0. (6) Throughout this paper, we will adopt the following simplified notations for Sobolev spaces: There is a great deal of work studying the shallow water equations. Ton [18] used Lagrangian coordinates and Hölder space estimates to study the local existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem. Kloeden [10] and Sundbye [16,17] used Sobolev space estimates and the energy method to prove the global well-posedness of classical solutions to the Cauchy-Dirichlet problem and also the Cauchy problem when the initial data are a small perturbation. Wang-Xu [19] obtained local solutions and global solutions for small initial data h 0 −h 0 , u 0 ∈ H 2+s (R 2 ) withh 0 , s > 0. Later, Chen-Miao-Zhang [4] improved their results to the initial data h 0 −h 0 ∈Ḃ 0 2,1 ∩Ḃ 1 2,1 , u 0 ∈Ḃ 0 2,1 with h ≥h 0 > 0. Hao-Hsiao-Li [9] established the global existence of strong solutions in the space of Besov type when the initial data are close to a positive constant equilibrium state. For arbitrarily large initial data, the global existence of weak solutions in bounded domain with periodic boundary conditions was obtained by Bresch-Desjardins [1,2] and Bresch-Desjardins-Lin [3], and the global existence of weak entropy solution for the one-dimensional initial boundary value problem was proved by Li-Li-Xin [11]. Later, Guo-Jiu-Xin [8] obtained the similar result in [11] for the multi-dimensional spherically symmetric weak solutions. Duan-Zheng-Luo [7] established a local existence theory of strong solutions for the rotating shallow water equations with constant viscosity coefficients. When viscosity coefficients are height-dependent, Li-Pan-Zhu [12,14,20,21] proposed a new quantity ∇ρ/ρ, which should belong to space L 6 ∩ D 1 ∩ D 2 , to obtain the local existence of classical solutions with far vacuum. Some other results on degenerate viscosities and initial vacuum can be seen in [13,15]. Recently, via introducing a symmetric structure for the quantity ∇ρ/ρ and a "quasi-symmetric hyperbolic"-"elliptic" coupled structure for (ρ, u), Ding-Zhu [6] showed the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for the compressible fluid with far field vacuum. The motivation of this paper is to establish the vanishing viscosity limit of the rotating shallow water equations from the viscous flow to the inviscid flow. This paper is organized as follows. In §2, we introduce some definitions of regular solutions and state our main results. In §3, we reformulate problem (1)-(5)-(6) into two coupled symmetric systems, and prove the existence and uniqueness of the regular solution to this reformulated systems. Based on this result, we can obtain the well-posedness of problem (1)-(5)- (6). In §4, we establish the existence of a unique regular solution for problem (4)-(5)- (6). Finally, we show the convergence rate of the regular solution to the rotating shallow water equations from the viscous flow to the inviscid flow in §5.

2.
Definition and main results. In this section, we introduce some definitions of regular solutions and state our main results.
Firstly, we denote some notations. For any 2 × 2 matrices M = (m ij ) 2×2 and A i (i = 1, 2), vectors U = (U 1 , U 2 ) and V = (V 1 , V 2 ), we set We now introduce some definitions of regular solutions. Under the assumption that h > 0, (1) 2 can be rewritten into where the so-called Lamé operator L and operator Q are given by It is well-known that the quantity ∇h/h is very important for the analysis on velocity. In order to solve the essential obstacle, Li-Pan-Zhu [12] first found that the quantity ψ = ∇h/h = 2∇ √ h/ √ h satisfies the following equation By (9) and (1), we can obtain two coupled symmetric systems: the symmetric hyperbolic system for the quantity ψ and the symmetric hyperbolic-parabolic coupled system for the quantities ( √ h, u), see (16). Thus, we can introduce a proper class of solutions called regular solutions to problem (1)-(5)-(6).
The condition (B1) and (B2) imply that the regular solutions do not contain vacuum in any local point, but have vacuum at infinity. When the density approaches to vacuum, the velocity is determined by the condition (E1) and (D2), which is a physically reasonable way.
Before considering the limit of the regular solution from the viscous flow to the inviscid flow, we first prove the following uniform local-in-time well-posedness of problem (1)-(5)-(6) independent of .
then there exists a time T * > 0 independent of , such that there is a unique regular Based on the uniform existence in Theorem 2.3, we can establish the following existence of the regular solution for problem (4)-(5)-(6).
Moreover, we can show the following vanishing viscosity convergence of the regular solution form the viscous flow to the inviscid flow.
satisfy (10), then when → 0, h , u converges to (h, u) in the following sense for any constant s ∈ [0, 3). Moreover, we also have where C > 0 is a constant depending only on the quantities T * , α, β, h 0 , u 0 .
3. Well-posedness of regular solutions for the viscous flow. In this section, we first reformulate the rotating shallow water equations with the viscous flow into two coupled symmetric systems, and prove the existence and uniqueness of the regular solution to this reformulated systems, whose life span is uniformly positive as the viscosity coefficients go to zero. Based on the well-posedness of this reformulated systems, we establish the well-posedness of regular solutions for the viscous flow (Theorem 2.3).
we can rewrite system (1) into the following nonlinear symmetric form: where In addition, (ψ, φ, u) satisfies the following initial data and the following far field behavior: In order to prove Theorem 2.3, we first establish the existence and some uniform estimates of regular solutions for the reformulated problem (16)- (18): then there exists a time T * > 0 independent of , and a unique classical solution (ψ, φ, u) to problem (16)- (18) satisfying Moreover, there exists a constant E 0 depending only on the quantities T * , α, β, h 0 , u 0 such that 3.2. Linearization. By the linearization for problem (16)-(18), we can obtain the following linearized problem We now prove the global existence of classical solutions to the linear problem (22). Lemma 3.2. Suppose that the initial data (ψ 0 , U 0 ) satisfy (19). Then for any T > 0, there exists a unique classical solution (ψ, U ) to problem (22) satisfying The proof of this lemma could be obtained by the standard hyperbolic and parabolic theory, see [5]. Here we omit its details.
3.3. The a priori estimates independent of . In this subsection, we will establish a priori estimates for (ψ, U ), and in particular, some of the estimates are independent of .
We first chose a constant T > 0 and a positive constant c 0 large enough such that and sup 0≤t≤T * for some fixed constants T * ∈ (0, T ) and c 1 satisfying 1 ≤ c 0 ≤ c 1 . Here, T * and c 1 depend only on the quantities T * , α, β, c 0 and the initial data h 0 , u 0 , and are determined later, see (39).
In the following, we will prove a series of uniform local (in time) estimates given in Lemmas 3.3-3.4. Hereinafter, we denote a generic positive constant by C ≥ 1, which is only dependent of the quantities T * , α, β, and may be different from line to line.
From [6], we know that the function ψ satisfies the following estimates. for Next, we prove a priori estimates for U , some of which are independent of .

ROTATING SHALLOW WATER EQUATIONS 319
Step 2. We consider the estimates for U t . From (22) 2 , we have Therefore, the proof of this lemma is completed.
Therefore, if we define the constants c 1 and T * by then we deduce that sup 0≤t≤T * ||ψ|| 2 3.4. Proof of Theorem 3.1. We will use the classical iteration scheme and the existence results in §3.2-3.3 to prove Theorem 3.1. As shown in §3.3, we chose the constants c 0 , c 1 and suppose that is the solution of the following Cauchy problem: Therefore, there exists a time T * * ∈ (0, T * ] such that (φ 0 , u 0 ) satisfies

ZHIGANG WANG
Our proof is divided into two steps.
We can observe that the time T * > 0 is independent of .
3.5. Proof of Theorem 2.3. From Theorem 3.1, we can obtain a unique classical solution (ψ, φ, u) to problem (16)- (18) satisfying (20), which means that By (16), it is easy to conclude that (h, u) satisfies the equation (1) in the sense of distribution. where Since h 0 > 0, by we can also conclude Therefore, problem (1) where T * and C is independent of . Thanks to the uniform estimates (54), we can derive that as → 0 there exists a subsequence of solutions ( √ h , u ) converges to a limit ( √ h, u) in weak or weak * snese.
By virtue of Lemma 2.4 in [6], for any R > 0 we can also obtain a subsequence of (h , u ) satisfy ( where B R is a ball with radius R centered at origin. Using the lower semi-continuity of norms, (55) and (56), we can derive that (h, u) satisfies Moreover, we can also conclude that U = ( √ h, u) satisfies the following system in the sense of distribution: Using (57) and (58), it is easy to see that Thus, we have Similarly as the proof of Theorem 2.3, we can conclude (h, u) is a regular solution of problem (4)-(5)-(6).
Thus, it is easy to derive that U = U − U satisfies U (t, x)| t=0 = (0, 0). (64) Now, we will prove some necessary estimates for U in order to establish the vanishing viscosity limit from U to U when → 0.
Firstly, we give the estimate of U in L 2 space.