LOCAL EXISTENCE OF CLASSICAL SOLUTIONS TO SHALLOW ...

c 2012 Society for Industrial and Applied Mathematics 

SIAM J. MATH. ANAL. Vol. 44, No. 2, pp. 541–567

LOCAL EXISTENCE OF CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS WITH CAUCHY DATA CONTAINING VACUUM∗ BEN DUAN† , ZHEN LUO‡ , AND YUXI ZHENG§ Abstract. In this paper, we investigate the Cauchy problem for the rotating shallow water equations with physical viscosity. We obtain the local existence of classical solutions without assuming the initial height is small or a small perturbation of some constant status. Moreover, the initial vacuum is allowed and the spatial measure of the set of vacuum can be arbitrarily large. In particular, the initial height can even have compact support; in this case, a blow-up example is given. Key words. viscous compressible rotating shallow water system, Cauchy problem, classical solution, blow-up AMS subject classifications. 35Q35, 35M10, 76N10, 75U05 DOI. 10.1137/100817887

1. Introduction. The nonlinear shallow water equations can be used to describe the horizontal structure of the atmosphere. They simulate the evolution of an incompressible fluid in response to gravitational and rotational accelerations. In general, it is modeled by the three-dimensional (3D) incompressible Navier–Stokes– Coriolis system in a rotating subdomain of R3 together with a nonlinear free moving surface boundary condition for which the stress tension is evolved at the air-fluid interface from above and the Navier boundary condition holds at the bottom. It is also regarded as an important extension of the two-dimensional (2D) compressible Navier– Stokes equations with additional rotating force, and the solutions present many types of motion. Usually, the nonlinear shallow water equations take the form  ht + div(hu) = 0, (1.1) (hu)t + div(hu ⊗ u) + gh∇h + f (hu)⊥ = μΔu + (μ + λ)∇(div u), where x ∈ Ω ⊂ R2 , t ∈ R+ , h(x, t) is the height of the fluid surface, u(x, t) is the horizontal velocity field, g > 0 is the gravity constant, f > 0 is the Coriolis frequency, and μ and λ are the dynamical viscosities satisfying (1.2)

μ > 0,

μ + λ ≥ 0.

Utilizing scaling in (t, x, h), we can assume without loss of generality that g = 1 and ∗ Received by the editors December 13, 2010; accepted for publication (in revised form) October 20, 2011; published electronically March 8, 2012. http://www.siam.org/journals/sima/44-2/81788.html † Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea ([email protected]). This author’s research was supported by National Research Foundation of Korea (NRF) grant 2010-0029638 funded by the Korea government (MEST). ‡ Corresponding author. School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China ([email protected]). This author’s research was partially supported by NSFC 10801111 and 11171223, Fundamental Research Funds for the Central Universities 2010121006, and NSF of Fujian Province of People’s Republic of China 2010J05011. § Department of Mathematical Sciences, 519 Belfer Hall, Yeshiva College, Yeshiva University, New York, NY 10033 ([email protected]). This author’s research was partially supported by NSF DMS 0908207.

541

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BEN DUAN, ZHEN LUO, AND YUXI ZHENG

f = 1. And for sophistication of the model, we extend the term h∇h to ∇hγ , i.e., we shall consider (1.3):  ht + div(hu) = 0, (1.3) (hu)t + div(hu ⊗ u) + ∇hγ + hu⊥ = μΔu + (μ + λ)∇(div u), ˜ be a fixed nonnegative constant. We where γ > 1 is an arbitrary constant. Let h look for solutions (h(x, t), u(x, t)) to the Cauchy problem for (1.1) with the far field behavior (1.4)

u(x, t) → 0,

h(x, t) → ˜h ≥ 0

as |x| → ∞

and initial data (1.5)

(h, u)|t=0 = (h0 ≥ 0, u0 ),

x ∈ R2 ,

satisfying the compatibility condition (1.6)

−μΔu0 − (μ + λ)∇div u0 + ∇h20 + h0 u⊥ 0 = h0 g 1/2

for some g ∈ D1 with h0 g ∈ L2 , where h0 and u0 are functions. First, let us introduce the notation and conventions used throughout this paper. Integral domain may be omitted if it is the entire domain R2 . For 1 < r < ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:     Lr = Lr (R2 ), Dk,r = u ∈ L1loc (R2 )  ∇k u Lr < ∞ , u Dk,r := ∇k u Lr ,    W k,r = Lr ∩ Dk,r , H k = W k,2 , Dk = Dk,2 , D0k = u ∈ Dk ;  u|∂Ω = 0 . Moreover, the material derivative is defined as f˙ := ft + u · ∇f. There is considerable literature studying fluid dynamics. The multidimensional Navier–Stokes system was investigated by Matsumura and Nishida [30, 31, 32], who proved global existence of smooth solutions for data close to a nonvacuum equilibrium, and later by Hoff for discontinuous initial data [21, 22]. Kazhikhov and Vaˇigant [25] obtained global existence of classical solutions in dimension N = 2 for special viscous coefficients with large initial data but for initial density away from zero. See also Desjardins and Lin [15], Mucha and Zaj¸aczkowski [33] and Itoh, Tanaka, and Tani [23]. For the case that the initial density is allowed to vanish, in fundamental work [28], Lions developed an existence theory of global in time weak solutions (see also Feireisl [16]). Better regularity of spatially periodic weak solutions in both two and three spatial dimensions was proved by Desjardins [13, 14] for small time. Later, for initial data near equilibrium, Danchin [11] and Danchin and Desjardins [12] found the optimal global well-posedness of the Cauchy problem in a functional space invariant by the natural scaling of the associated equations. Recently, for dimension N ≥ 3, the local well-posedness of classical solutions containing vacuum were obtained by Cho, Choe, and Kim [1] and Cho and Kim [2, 6]. Such solutions were later shown to exist globally in time by Huang, Li, and Xin [20]. However, the existence of strong or classical solutions to the 2D Cauchy problem is still open. For the shallow water system, there is a great deal of work. The local existence and uniqueness for classical solutions to the Cauchy–Dirichlet problem was studied in [37]

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

543

using Lagrangian coordinates and H¨older space estimates with initial data in C 2+α . Later, assuming the initial data is a small perturbation of a positive constant, Kloeden [24] and Sundbye [35, 36] proved global existence and uniqueness of classical solutions to the Cauchy–Dirichlet problem and also the Cauchy problem, using Sobolev space estimates and the energy method of Matsumura and Nishida [30, 31, 32]. Wang and Xu [38] obtained local solutions for any initial data and global solutions for small ¯ 0 and s > 0. The result was improved by h0 , u0 ∈ H 2+s (R2 ) with h initial data, h0 − ¯ Chen, Miao, and Zhang in [5], who proved the local existence in time for general initial ¯ 0 ∈ B˙ 0 ∩ B˙ 1 data and global existence in time for small initial data where h0 − h 2,1 2,1 0 ¯ ˙ and u0 ∈ B2,1 with the additional condition that h ≥ h0 > 0. A global existence of strong solution in the space of Besov type was obtained recently by Hao, Hisao, and Li [19] for the shallow water equations involving the capillary term h∇Δh with the initial data close to a positive constant equilibrium state. For arbitrarily large initial data, Bresch and Desjardins [7, 8] and Bresch, Desjardings, and Lin [9] proved the global existence of weak solutions for 2D shallow water equations in bounded domain with periodic boundary conditions, where the friction term and the capillary term are involved. Later, Li, Li, and Xin showed the global existence of weak entropy solution for the initial boundary value problem to one-dimensional compressible flows [27]. The same result was obtain by Guo, Jiu, and Xin [17] for the case of multidimensional spherically symmetric weak solutions. We remark that the 2D problem is the critical case for the standard Sobolev embedding theorems in unbounded domain and one cannot bound the Lp -norm of the velocity just in terms of the L2 -norm of the gradient of it. As a consequence, in unbounded domains, a priori estimates in some existence theory in 3D, such as [2], cannot be applied for the 2D case. So far, we only know results in 2D bounded domains [7, 8, 9]. In fact, the existence of strong or classical solutions to the Cauchy problem for (1.3)–(1.5) is open, especially, when the initial height has compact support. In this paper, basic energy estimates are derived on the material derivatives of the velocity, u˙ = ∂t u + (∇ · u)u, in the spirit of D. Hoff’s work in [21]. Actually, one of the main difficulties in this paper is to derive Lp -bounds on the velocity u and the material derivatives of the velocity u: ˙ combining some substantial estimates on a suitably spatial weighted norm of ∇u and ∇u˙ with the Caffarelli–Kohn–Nirenberg inequality yields the Lp -norm of u and u˙ for some constant p = p(μ; λ). In [10], several open problems about shallow water equations are mentioned. Particularly, in section 4, the authors remark on strong solution that “nothing has been done so far, to obtain better results such as local existence of strong solution with initial data including vacuum. Remark that such a situation is important from a physical point of view: the dam break situation”; the dam break problem is the case in which initial height has compact support. In fact, the possible appearance of vacuum is one of the major difficulties when trying to prove existence and strong regularity results to the hyperbolic-parabolic system (1.1), which possesses strong degeneracies near vacuum and singularities in the vacuum region. See [7, 8, 9] for weak solutions containing vacuum and [24, 35, 36, 38, 5] for higher regularity solutions away from vacuum, assuming that the initial height is a small perturbation of a positive constant. In this paper, we investigate the existence of classical solutions for nonnegative initial height, and the set of initial vacuum can be arbitrary large. Therefore, we are able to deal with the problem without assuming the initial height is close to a nonvacuum equilibrium. Furthermore, the initial far fields can be vacuum or nonvacuum. Note that due to the strong nonlinearity of system (1.1), the problem of existence of solutions for large initial data is difficult and previous results for classical solutions

544

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

only consider the case of small initial data; see [24, 35, 36]. Our existence theory of classical solutions in this paper has no restriction on smallness of initial data. We remark that for a 2D Navier–Stokes system in bounded domain, results obtained by Desjardins in [13] showed that the maximal norm of the density controls the breakdown of weak solution, even if vacuum forms in the fluid. That is, vacuum does not yield additional singularities in this case. In this paper, for the Cauchy problem of rotating shallow water equations with spherically symmetry initial data, we obtain that the local smooth solution (h, u) ∈ C 1 ([0; T ]; H s )(s > 2) has to blow up in finite time if the initial height has compact support. Therefore, the solutions in our main result, Theorem 3.9, give a kind of suitable class of solutions which are expected to exist globally in time in the case  h = 0. As above, the main technical difficulties in this papers are to deal with the initial height allowing vacuum, with no restriction on the smallness of initial data or pertur¯ 0 ||, and to gain high regularity estimates to show the bation in the sense of ||h0 − h solution is classical. We obtain the local existence of classical solutions in this paper; ˜ = 0. moreover, we give a blow-up example when the far field state is h The structure of the paper is as follows. In the next section, we recall some useful well-known lemmas. In section 3 we prove the local existence of classical solutions to the Cauchy problem for all sizes of the initial data, with or without initial vacuum. In section 4, assuming the initial height has compact support, we construct a radially symmetric solution and we prove that this solution has to blow up in finite time. 2. Preliminaries. First, the following Caffarelli–Kohn–Nirenberg inequality will play a key role in obtaining the Lp estimate of u when the constant far field state  h = 0. Lemma 2.1 (see [3, 4]). For α ∈ (0, 2), the following estimates hold for all u ∈ C0∞ (R2 ):    α2 (2.1) |x|α−2 |u|2 dx ≤ |x|α |∇u|2 dx, u 2L4/α ≤ C(α) |x|α |∇u|2 dx. 4 The following Lions–Aubin lemma will be used later. Lemma 2.2 (see [34]). Let X0 , X, and X1 be three Banach spaces with X0 ⊂ X ⊂ X1 . Suppose that X0 is compactly embedded in X and that X is continuously embedded in X1 . (i) Let G be bounded in Lp (0, T ; X0 ), where 1 ≤ p < ∞, and let ∂G/∂t be bounded in L1 (0, T ; X1). Then G is relatively compact in Lp (0, T ; X). (ii) Let F be bounded in L∞ (0, T ; X0 ) and ∂F/∂t be bounded in Lr (0, T ; X1), where r > 1. Then F is relatively compact in C([0, T ]; X). The following Schauder fixed point theorem will be needed to obtain the local existence of strong solutions in a bounded smooth domain. Lemma 2.3 (see [18]). Let be a compact convex set in a Banach space B and let T be a continuous mapping of into itself. Then T has a fixed point, that is, T x = x for some x ∈ . Next, the well-known Gagliardo–Nirenberg inequality will be used frequently (see [26]). Lemma 2.4 (Gagliardo–Nirenberg). For p ≥ 2, q ∈ (1, ∞), and r ∈ (2, ∞), there exists some generic constant C > 0 which may depend on q, r such that for f ∈ H 1 (R2 ) and g ∈ Lq (R2 ) ∩ D1,r (R2 ), we have (2.2)

2/p

(p−2)/p

f Lp ≤ C f L2 ∇f L2

,

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

(2.3)

q(r−2)/(2r+q(r−2))

g C (R2 ) ≤ C g Lq

2r/(2r+q(r−2))

∇g Lr

545

.

It is quite well known that for bounded smooth domain Ω ⊂ R2 , the linear hyperbolic problem  ht + div(hv) = 0 in Ω × (0, T ), (2.4) h(x, 0) = h0 in Ω, where v is a known vector field in Ω × (0, T ) such that (2.5)

v ∈ C([0, T ]; H01 ∩ H 3 ) ∩ L2 (0, T ; H 4 ),

has a unique strong solution h for any regular initial data h0 because the vector field v is sufficiently smooth (see [6]). So we state the result without proof. Lemma 2.5. Assume that (2.5) holds and that h0 ≥ δ on [0, T ] × Ω for some δ > 0,  h ≥ 0, h0 −  h ∈ H 3 (Ω). Then (i) there exists a unique solution h to the problem (2.4) such that (2.6)

h− h ∈ C([0, T ]; H 3 ),

ht ∈ C([0, T ]; H 2 );

(ii) the solution h satisfies the estimate   t h(t) −  h H 3 ≤ ( h0 −  h H 3 +  h) exp C v(s) D1 ∩D4 ds 0

for some universal constant C; (iii) the solution h is represented by the formula (2.7)

 t  h(x, t) = h0 (U (0; x, t)) exp − div v(U (s; x, t), s)ds , 0

where U ∈ C([0, T ]; [0, T ] × Ω) is the solution to the initial value problem  ∂ ∂s U (s; x, t) = v(s; U (s; x, t)), 0 ≤ s ≤ T, (2.8) ¯ U (t; x, t) = x, 0 ≤ s ≤ T, x ∈ Ω. Next, we consider the linear parabolic problem ⎧ γ ⊥ ⎪ ⎨hut + hv · ∇u − μΔu − (μ + λ)∇(div u) + ∇h + hu = 0 (2.9) u = 0 ⎪ ⎩ u(x, 0) = u0

in (0, T ) × Ω, on (0, T ) × ∂Ω, in Ω,

where h is a known scalar field in (0, T ) × Ω such that (2.10)

h, hγ ∈ C([0, T ]; H 3 ),

ht , (hγ )t ∈ C([0, T ]; H 2 ),

h ≥ δ on [0, T ] × Ω,

for some constant δ > 0. Recall that L := −μΔ − (μ + λ)∇div is a strongly elliptic operator (see [1], for instance). Then applying a standard method such as a semidiscrete Galerkin method or the method of continuity, we can prove the following

546

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

existence and regularity results on solutions to the linear parabolic problem (2.9). See also [6] for a similar result to Navier–Stokes equations. Lemma 2.6. (i) Assume that u0 ∈ H01 , (2.5), and (2.10) hold. Then there exists a unique strong solution u to the problem (2.9) such that u ∈ C([0, T ]; H01 ) ∩ L2 (0, T ; H 2 ), ut ∈ L2 (0, T ; L2 ). (ii) If in addition u0 ∈ H01 ∩ H 2 and vt ∈ L∞ (0, T ; L2 ), then the solution u satisfies u ∈ L∞ (0, T ; H 2 ), ut ∈ L2 (0, T ; H01), utt ∈ L2 (0, T ; H −1 ). (iii) Finally, if u0 ∈ H01 ∩ H 3 , ut (0) = h(0)−1 (−∇h2 (0) − Lu0 ) − u⊥ (0) − v(0) · ∇u(0) ∈ H01 , and vt ∈ L∞ (0, T ; H01), then the solution u satisfies u ∈ L∞ (0, T ; H 3 ),

ut ∈ L2 (0, T ; H 2 ),

utt ∈ L2 (0, T ; L2).

3. Local existence of classical solutions. In this section, for simplicity, we assume γ = 2 and all results are valid for other cases. We look for the local classical solutions, (h(x, t), u(x, t)), to the Cauchy problem for (1.3) with the far field behavior (1.4) and initial data (1.5). 3.1. Uniform a priori estimates for the linearized problem. In this subsection, we derive some uniform local (in time) a priori estimates for strong solutions (h, u) to the linearized problem (2.4), (2.9), which are stated as in Lemmas 2.5 and 2.6. The estimates we obtained here are independent of the lower bound δ of h0 and size of the domain Ω. The main difficulty is how to obtain the Lp norm for u itself and the material derivatives of the velocity u˙ for some p ≥ 2. The case of  h > 0 is easier because the following lemma holds. ˜ > 0 such that the Lemma 3.1. If  h > 0, then there exists some constant C(h) ˜ ∈ L2 , v ∈ D1 , h1/2 v ∈ L2 : following estimate holds for h − h  2 2 2 2 ˜ (3.1) v L2 ≤ C h|v| dx + h − h L2 ∇v L2 . Proof. Equation (3.1) follows directly from Lemma 2.4 and the following simple fact:    ˜ − h)|v|2 dx ˜ |v|2 dx = h|v|2 dx + (h h  ˜ − h L2 v L2 ∇v L2 . ≤ h|v|2 dx + C h In the rest of this subsection, we assume  h = 0 and deduce some local in time a priori estimates for this case. Set 1/2

(3.2)

c0 := 1 + h0 u0 2L2 + (h0 , h20 ) 2H 3 + u0 2D1 ∩D3 1/2

1/2

+ (h20 , |∇u0 |, h0 |g|, h0 u0 )(1 + |x|α/2 ) 2L2 + g 2D1 ,

where (3.3)

α := μ/(4(2μ + λ)) ∈ (0, 1/8].

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

547

We define  (3.4)

Φ(v, t) := 1 + sup

0≤s≤t

  t ∇v 2H 1 ∩W 1,4/α ds. ∇v 2H 1 + |x|α/2 ∇v 2L2 + 0

From now on, several positive generic constants depending on μ, λ are denoted by C and we always assume that 0 ≤ t ≤ T ≤ 1. We start with the following energy estimate for (h, u) and the preliminary L2 bounds for ∇u and h1/2 u, ˙ in the spirit of Hoff’s work [21]. Lemma 3.2. Let (h, u) be a smooth solution of (2.4), (2.9). Then  (3.5)

sup

0≤s≤t

  h|u|2 + h4 dx +

 t 0

  |∇u|2 dxdt ≤ Cc0 exp Φ(v, t)t1/2

and (3.6)

sup

0≤s≤t

∇u 2L2

 t + 0

h|u| ˙ 2 dxdt ≤ Cc20 exp {Cc0 t exp {CΦ(v, t)}} ,

where c0 as in (3.2) and Φ(v, t) in (3.4). Proof. Note that (2.4)1 implies that h2 satisfies h2t + v · ∇h2 + 2h2 div v = 0.

(3.7) It is easy to check that

  (h, h2 ) 2L2 t ≤ C ∇v L∞ (h, h2 ) 2L2 ,

(3.8)

which together with (2.7) yields that sup ( (h, h2 ) 2L2 + (h, h2 ) 2L∞ )

0≤s≤t

≤ Cc0 exp





t

∇v ds   0 ≤ Cc0 exp Φ(v, t)t1/2 .

(3.9)

L∞

Now, the energy estimate gives (3.10)

1 2



   μ|∇u|2 + (λ + μ)(div u)2 dx ≤ Cδ h2 2L2 + δ ∇u 2L2 , h|u|2 dx + t

which together with (3.9) yields (3.5). Next, multiplying (2.9) by u, ˙ then integrating the resulting equality over Ω, leads to   2 ˙ h|u| ˙ dx = (−u˙ · ∇h2 + μ u · u˙ + (λ + μ)∇div u · u˙ − hu⊥ · u)dx (3.11) :=

4  i=1

Mi .

548

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

Using (2.4) and integrating by parts give  M1 = − u˙ · ∇h2 dx  = ((div u)t h2 − (v · ∇u) · ∇h2 )dx   (3.12)   2 2 = div uh dx + h div udiv v + h2 ∂i vj ∂j ui dx t    1  ≤ div uh2 dx + Cc02 exp CΦ(v, t)t1/2 ∇u 2L2 + ∇v 2L2 , t

due to (3.9). Integration by parts implies  M2 = μ u · udx ˙

  μ 2 = − ∇u L2 t − μ ∂i uj ∂i (vk ∂k uj )dx 2  μ ≤ − ∇u 2L2 t + C ∇v L∞ ∇u 2L2 . 2

(3.13)

Similarly,   λ+μ  div u 2L2 t − (λ + μ) divudiv(v · ∇u)dx 2  λ+μ  ≤− div u 2L2 t + C ∇v L∞ ∇u 2L2 2

M3 = − (3.14)

and

 M4 = −

(3.15)

hu⊥ · udx ˙ ≤δ



h|u| ˙ 2 dx + Cδ



h|u|2 dx.

Combining (3.5), (3.8), and (3.11)–(3.15) with Gronwall’s inequality gives (3.6). The following two lemmas mainly deal with the estimates on the spatial weighted norm of ∇u and ∇u, ˙ which are needed to overcome the difficulties from a possibly large set of initial vacuum and failure of Sobolev embedding theorems in critical space R2 . Lemma 3.3. Let (h, u) be as in Lemma 3.2. Then (3.16) B  (t) +



h|u| ˙ 2 |x|α dx

≤ C (1 + ∇v L∞ + v L∞ ) |x|α/2 ∇u 2L2 + 2μα4 |x|α/2 ∇u ˙ 2L2 + Cβc20 t exp {CΦ(v, t)} , where

λ+μ μ 2 2 2 4 2 B(t) := |x| |∇u| + |divu| − h div u + βh + h|u| dx 2 2    μ ≥ |x|α |∇u|2 + h4 + h|u|2 dx 4 

α



for suitably large β(μ, λ) > 0.

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

549

Proof. Multiplying (2.9) by u|x| ˙ α and then integrating the resulting equality over Ω leads to (3.17)

2



α

h|u| ˙ |x| dx = :=

|x|α (−u˙ · ∇h2 + μ u · u˙ + (λ + μ)∇div u · u˙ − hu⊥ · u)dx ˙

4 

Mi .

i=1

Using (2.4)1 and integrating by parts yield  M1 = − |x|α u˙ · ∇h2 dx  = ((|x|α div u)t h2 + αx · u|x| ˙ α−2 h2 − |x|α (v · ∇u) · ∇h2 )dx   α 2 = |x| div uh dx + α h2 |x|α−2 x · udx ˙ t    + h2 |x|α div udiv v + ∂i vj ∂j ui dx  − α |x|α−2 h2 x · (vdiv u − v · ∇u) dx    α 2 ≤ |x| h div udx + μα4 |x|α |∇u| ˙ 2 dx + C h4 |x|α dx t

(3.18)

2

+ C h L∞ |x|

α/2

∇v 2L2 + C h2 L∞ |x|α/2 ∇u 2L2 ,

where in the last two inequalities we have used (2.1). Integration by parts and using (2.1) imply M2 = μ

(3.19)



|x|α u · udx ˙

  μ α/2 2 |x| ∇u L2 − μ |x|α ∂i uj ∂i (vk ∂k uj )dx =− 2 t  − αμ ∂i uj u˙ j |x|α−2 xi dx  μ |x|α/2 ∇u 2L2 + C ( ∇v L∞ + v L∞ ) ∇u|x|α/2 2L2 ≤− 2 t + Cα ∇u|x|α/2 L2 u|x| ˙ α/2−1 L2 + C v L∞ ∇u 2L2  μ |x|α/2 ∇u 2L2 + C (1 + ∇v L∞ + v L∞ ) |x|α/2 ∇u 2L2 ≤− 2 t μ 4 α/2 2 + α |x| ∇u ˙ L2 + C v L∞ ∇u 2L2 , 2

and similarly, (3.20)  λ+μ  M3 ≤ − |x|α/2 div u 2L2 + C (1 + ∇v L∞ + v L∞ ) |x|α/2 ∇u 2L2 2 t μ 4 α/2 2 + α |x| ∇u ˙ L2 + C v L∞ ∇u 2L2 . 2

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BEN DUAN, ZHEN LUO, AND YUXI ZHENG

Next, M4 (3.21)



hu⊥ · u|x| ˙ α dx

=−

 √ ˙ 2 |x|α dx. ≤ Cδ || h|u||x|1/2 ||2L2 + δ h|u|

Now, multiplying (2.9) by u|x|α then integrating the resulting equation over Ω and using (2.4) yield  1 2 α h|u| |x| dx 2 t    α 2 α−2 |u| h|x| = v · x + h2 ∇u|x|α + αh2 u · x|x|α−2 dx 2    (3.22) − μ|∇u|2 |x|α + αμ|x|α−2 u(∇u)x dx    − (λ + μ)|div u|2 |x|α + α(λ + μ)|x|α−2 (div u)u · x dx  ≤ h4 |x|α dx + C( h L∞ v L∞ + 1) |∇u||x|α/2 2L2 . Multiplying (3.7) by h2 |x|α , we obtain that    α 4 α−1 4 h |v|dx + C |x|α h4 |div v|dx |x| h dx ≤ C |x| t

≤ C h2 L∞ 1/2

≤ Cc0

(3.23) Then

 sup

0≤s≤t



1/2 |x|α/2 ∇v L2 |x|α h4 dx 

exp{CΦ(v, t)}

1/2 |x|α h4 dx .

|x|α h4 dx ≤ Cc0 t exp{CΦ(v, t)},

which together with (3.23) gives  α 4 |x| h dx ≤ Cc20 t exp {CΦ(v, t)} . (3.24) t

Combining (3.17)–(3.24) implies (3.16). Lemma 3.4. Let (h, u) be as in Lemma 3.2. Then the following estimate holds for q = 4/α : (3.25)     ˙ 2 + h|u|2 dx sup ∇(h, h2 ) 2L2 ∩Lq + ∇2 u 2L2 + (1 + |x|α ) |∇u|2 + h4 + h|u| 0≤s≤t

 t

2



t

˙ dxdt + ∇2 u 2Lq dt (1 + |x| )|∇u| 0 0   ≤ Cc30 exp Cc0 exp {CΦ(v, t)} t(q−2)/(2(q−1)) . +

α

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

∂ ∂xk

551

Proof. First, let vk be the kth component of a vector v and denote the operator by ∂k , k = 1, 2. Operating ∂t + ∂k (vk ·) on each term of (2.9)1 gives hu˙ t + hv · ∇u˙

(3.26)

  = μΔu˙ + (μ + λ)∇div u˙ − μ ∂k (∂i vk ∂i u) − ∂i (∂k vk ∂i u) + ∂i (∂i vk ∂k u)   − h(u) ˙ ⊥ + (μ + λ) ∇(div vdiv u) − ∇(∂j vk ∂k uj ) − div(∇v⊥ div u) + div(h2 ∇v⊥ ) + ∇(h2 div v).

Multiplying (3.26) by u˙ and integrating over Ω, we have by (3.6) and (3.5)   2 ˙ dx + (λ + μ) |div u| ˙ 2 dx h|u| ˙ dx + μ |∇u| t   2 2 ≤ C |∇v| |∇u| dx + h4 |∇v|2 dx  (q−2)/(q−1) q/(q−1)  ∇u 2L2 + h2 2L2 ≤ C ∇v L2 ∇2 v Lq   (q−2)/(q−1) q/(q−1) ∇2 v Lq , ≤ Cc20 exp Cc0 t1/2 exp {CΦ(v, t)} ∇v L2



2

which yields  sup

(3.27)

0≤s≤t

2

 t

h|u| ˙ dx + μ 0

2

|∇u| ˙ dxdt + (λ + μ)

 t 0

|divu| ˙ 2 dxdt

  ≤ Cc20 exp Cc0 exp {CΦ(v, t)} t(q−2)/(2(q−1)) ,

where q = 4/α. Similarly, multiplying (3.26) by u|x| ˙ α and integrating over Ω lead to     1 α 2 α 2 ˙ dx + μ |x| |∇u| ˙ dx + (λ + μ) |x|α |divu| ˙ 2 dx |x| h|u| 2 t    2 α−1 α−1 ≤ α h|v||u| ˙ |x| dx + μα |x| |∇u|| ˙ u|dx ˙ + (λ + μ)α |x|α−1 |divu|| ˙ u|dx ˙     α α−1 +C |∇u||∇v| + h2 |∇v| |∇u||x| ˙ + α|u||x| ˙ dx   ˙ 2 dx + α3 (2μ + λ) |x|α |∇u| ˙ 2 dx ≤ CΦ1/2 (v, t) |x|α h|u|   + δ |x|α |∇u| ˙ 2 dx + Cδ ∇v 2L∞ |x|α (|∇u|2 + h4 )dx. By (3.3), we have 

  |x|α h|u| ˙ 2 dx + μ |x|α |∇u| ˙ 2 dx + (λ + μ) |x|α |divu| ˙ 2 dx t 1/2 ≤ CΦ (v, t) |x|α h|u| ˙ 2 dx + C ∇v 2L∞ B(t).

552

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

Therefore, we get for q = 4/α,

(3.28)

   μ α 2 |x|α |∇u| ˙ dx + ˙ 2 dx B(t) + |x| h|u| 2 t    ≤ CΦ1/2 (v, t) |x|α h|u| ˙ 2 dx + Cc20 exp CΦ(v, t)t1/2 Φ(v, t)   + C 1 + ∇v L∞ + v L∞ + ∇v 2L∞ B(t)    (q−2)/(q−1) q/(q−1) ≤ C Φ(v, t) + ∇v L2 ∇2 v Lq ˙ 2 dx B(t) + |x|α h|u|   + Cc20 exp CΦ(v, t)t1/2 Φ(v, t).

Gronwall’s inequality thus gives (3.29)

sup

0≤s≤t

    B(t) + |x|α h|u| ˙ 2 dx ≤ Cc20 exp CΦ(v, t)t(q−2)/(2(q−1)) ,

which together with (3.28) yields  t (3.30) 0

  ˙ 2 dxdt ≤ Cc20 exp CΦ(v, t)t(q−2)/(2(q−1)) . |x|α |∇u|

Moreover, by virtue of (3.7), we have for p ≥ 2, ( ∇h2 Lp ) ≤ C ∇v L∞ ∇h2 Lp + C h2 L∞ ∇2 v Lp , which together with (3.9) yields that for p ∈ [2, q], sup ∇h2 Lp

0≤s≤t

(3.31)

 t   t 2 2 2 ∞ p ∞ p ≤ C exp C ∇v L ds ∇h0 L + h L ∇ v L ds 0  0  ≤ Cc0 exp CΦ(v, t)t1/2 .

Similarly, we deduce from (1.3)1 and (3.9) that   sup ∇h Lp ≤ Cc0 exp CΦ(v, t)t1/2 .

0≤s≤t

Noticing that u satisfies 

μΔu + (μ + λ)∇(div u) = hu˙ + ∇h2 + hu⊥ u=0

in Ω, on ∂Ω,

we thus derive from the standard Lp estimate for elliptic equations (see [18]) that for p ∈ (1, ∞), (3.32)

  ˙ Lp + ∇h2 Lp + hu⊥ Lp , ∇2 u Lp ≤ C hu

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

553

which together with (3.29)–(3.31), (3.9), and (2.1) yields that 

t

0

∇2 u 2Lq dt  t   hu ˙ 2Lq + ∇h2 2Lq + hu⊥ 2Lq dt ≤C 0  t  2 α 2 2 2 2 ˙ + |∇u| )dx + ∇h Lq dt ≤C h L∞ (1 + |x|) (|∇u| 0   ≤ Cc30 exp CΦ2 (v, t)t(q−2)/(2(q−1)) .

Then, it follows from (3.32), (3.5), (3.27), (3.31), and (3.9) that (3.33)

  sup ∇2 u 2L2 ≤ Cc30 exp Cc0 exp {CΦ(v, t)} t(q−2)/(2(q−1)) .

0≤s≤t

Remark 3.1. The estimates in Lemmas 3.3 and 3.4 are substantial when investigating the Cauchy problem in two dimensions. It is quite different for 3D problems, in which the L6 -norm of u is bounded by the L2 -norm of ∇u directly from the standard Sobolev inequality (see [2]). Also, the estimates hold when the initial height vanishes at far field. In particular, the initial height can even have compact support. Finally, Lemma 3.5 will close our arguments on the uniform a priori estimates for  h = 0. Lemma 3.5. Assume that (h, u) is as in Lemma 3.2. Then there exists a time T ∗ ∈ (0, 1] depending only on c0 , μ, λ, α such that Φ(u, T ∗ ) ≤ M, provided Φ(v, T ∗ ) ≤ M with some given M = M (μ, c0 ) > 1. Proof. Lemmas 3.2–3.4 imply that for q = 4/α,   Φ(u, t) ≤ Cc30 exp Cc0 exp {CΦ(v, t)} t(q−2)/(2(q−1)) , which yields that Φ(u, T ∗ ) ≤ M by choosing M = Cc30 eCc0 and (3.34)

  T ∗ = min e−2CM(q−1)/(q−2) , 1 .

3.2. Local existence of strong solutions. In this subsection, based on the uniform estimates in section 3.1 for solutions to the linearized problem, we use the fixed point theory to show that the shallow water equations (1.3)–(1.5) have unique local strong solutions (h, u). Lemma 3.6. Let Ω ⊂ R2 be a bounded smooth domain. Suppose that the initial data (h0 , u0 ) satisfy (3.35)

u0 ∈ D 1 ∩ D 3 ,

˜ h2 − h ˜ 2) ∈ H 3, (h0 − h, 0

554

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

h0 ≥ δ on Ω for some δ > 0 and the compatibility condition (1.6). In addition, we ˜ = 0, and assume that h |x|α/2 ∇u0 ∈ L2 ,

(3.36)

1/2

|x|α/2 h20 ∈ L2 ,

|x|α/2 h0 g ∈ L2

for α as in (3.3). Then for T ∗ as in Lemma 3.5, there exists a unique strong solution (h, u) to the problem (1.3) on Ω × [0, T ∗ ] with (3.37)

(h, u)|t=0 = (h0 , u0 ),

u|∂Ω = 0,

satisfying

(3.38)

⎧ 2 ∞ ∗ 1,p 2 ∞ ∗ p ⎪ ⎨(h, h ) ∈ L (0, T ; W ), (h, h )t ∈ L (0, T ; L ), u ∈ C([0, T ∗ ]; Lq ∩ D1 ∩ D2 ) ∩ L2 (0, T ∗ ; D2,q ), √ ⎪ ⎩ ut ∈ L2 (0, T ∗ ; Lq ∩ D1 ), hut ∈ L∞ (0, T ∗ ; L2 )

for q = 4/α and any p ∈ [2, q]. Moreover, the following estimate holds: 

h|u|2 dx + ∇2 u L2 (Ω) + (h, h2 ) W 1,p (Ω) + (h, h2 )t Lp (Ω) ∗ 0≤t≤T Ω    + sup (1 + |x|α ) |∇u|2 + h4 + h|u| ˙ 2 + h|u|2 dx



sup

(3.39)

0≤t≤T ∗



T∗



+ 0

Ω

Ω

(1 + |x|α )|∇u| ˙ 2 dxdt +



T∗ 0

∇2 u 2Lq (Ω) dt

≤ C(μ, λ, c0 ). Remark 3.2. After combing the uniform estimate (3.39) with Caffarelli–Kohn– Nirenberg inequality and some approximations, we bound the Lq -norm of u and ut in both bounded and unbounded domain in R2 ; see Proposition 3.7. Therefore, the higher regularity of ut and, consequently, the classical solution (h, u) in R2 can be expected; see (3.48) and Remark 3.3. Proof. For T ∗ and M as in Lemma 3.5, set B := L2 (0, T ∗ ; H01 ) and    := v ∈ L∞ (0, T ∗ ; H01 ∩ D2 ) ∩ L2 (0, T ∗ ; D2,q ) vt ∈ B, Φ(v, T ∗ ) ≤ M . It is easy to see from Lemma 2.2 that is a convex and compact subset of the Banach space B. For any v ∈ , by virtue of Lemma 2.5, there exists a unique solution h = h(v) solving (2.4) on Ω × [0, T ∗] and satisfying (2.6). Moreover, according to Lemma 2.6, there exists a unique solution u = T (v, h(v)) solving the problem (2.9) on Ω × [0, T ∗ ]. Here, (h, u) satisfies the a priori estimates stated in Lemmas 3.2–3.5, which yield that T maps to . Next, we will show T is a continuous operator in B.

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

555

By (3.25) and (2.4)1 , we have sup ( h(v) W 1,q + (h(v))t Lq ) ≤ C(μ, c0 ).

(3.40)

0≤t≤T ∗

Let vn ∈ , n = 1, 2, . . . , converge to v in B, that is, in L2 (0, T ∗ ; H01 ) as n → +∞,

vn → v

(3.41)

which together with vn ∈ implies (3.42)

vn  v

w∗ in L∞ (0, T ∗ ; H01 ∩ D2 ) ∩ L2 (0, T ∗ ; D2,q ) as n → +∞.

It thus follows from Lemma 2.2 and (3.40) that up to a subsequence, in C(Ω × [0, T ∗ ]) as nj → +∞.

h(vnj ) → h

(3.43)

Taking the limits in (2.4), where h, v are replaced by h(vnj ) and vnj , respectively, we obtain that h is a weak solution to (2.4), and then, by the uniqueness of the weak solutions due to (3.42) and (3.43), h = h(v), which again together with (3.43) and (3.40) implies h(vn ) → h

(3.44)

in C(Ω × [0, T ∗]) as n → +∞

and (3.45)

h(vn )  h

w∗ in L∞ (0, T ∗ ; W 1,q ) as n → +∞.

Denoting by un = T (vn , h(vn )), by virtue of Lemmas 3.2–3.5 and Lemma 2.2, we get that up to a subsequence, (3.46)

uni  u

w∗ in L∞ (0, T ∗ ; H01 ∩ D2 ) ∩ L2 (0, T ∗; D2,q ) as ni → +∞

and (3.47)

uni → u

in B as ni → +∞.

Letting ni → +∞ in (2.9), where h, u, v are replaced by h(vni ), uni , vni , respectively, we obtain that u is a weak solution to (2.9) by (3.46), (3.45), and (3.42), which again yield the uniqueness of the weak solutions to (2.9), and then u = T (v, h(v)), which implies (3.47) holds for un itself. That means T is a continuous operator in B. By the Schauder fixed point theory, Lemma 2.3, there exists some u ∈ such that T (u, h(u)) = u, which together with Lemmas 3.2–3.4 implies (3.39). Then the uniqueness of u and (3.38) follow from (3.39), standard Sobolev embedding theory, and Lemma 2.1. Now, we can prove the local existence of unique strong solution (h, u) to the Cauchy problem (1.3)–(1.5). ˜ ≥ 0, assume that the initial data (h0 ≥ 0, u0 ) satisfy Proposition 3.7. For h ˜ = 0, in addition to (3.35) and (1.6), we assume that (3.35) and (1.6). Moreover, if h (3.36) holds. Then there exist a small time T ∗ > 0 and a unique strong solution (h, u) to the Cauchy problem (1.3), (1.4), (1.5) such that ⎧ ˜ h2 − h ˜ 2 ) ∈ L∞ (0, T ∗ ; H 3 ), (h − h, ⎪ ⎪ ⎪ ⎪ ⎪ (h, h2 )t ∈ L∞ (0, T ∗ ; H 1 ), (h, h2 )tt ∈ L2 (0, T ∗ ; L2 ) ⎪ ⎪ ⎪ ⎨u ∈ C([0, T ∗ ]; Lq ∩ D1 ∩ D3 ) ∩ L2 (0, T ∗ ; D4 ), √ (3.48) ⎪ ut ∈ L∞ (0, T ∗ ; D1 ) ∩ L2 (0, T ∗; Lq ∩ D2 ), hut ∈ L∞ (0, T ∗ ; L2 ), ⎪ ⎪ √ ⎪ ⎪ ⎪ hutt ∈ L2 (0, T ∗ ; L2 ), ⎪ ⎪ ⎩ 1/2 √ t hutt ∈ L∞ (0, T ∗; L2 ), tutt ∈ L2 (0, T ∗ ; D1 ),

556

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

where q=2

if

 h > 0;

q=

4 α

 h = 0.

if

Remark 3.3. As we will see in the next subsection, the unique strong solution obtained in Proposition 3.7 becomes a classical one, as long as t1/2 u ∈ L∞ (0, T ∗ ; D4 ),

tut ∈ L∞ (0, T ∗ ; Lq ∩ D2 )

for some q ≥ 2. Proof. First, we consider the case  h = 0. Define BR = {x ||x| < R } and −2 ψ R (x) = ψ(x/R), gR (x) = ψ R (x)g(x), hR 0 = h0 + R

for (t, x) ∈ [0, T ∗ ] × R2 , where 0 ≤ ψ ∈ C0∞ (B1 ) is a smooth cut-off function such that ψ = 1 in B1/2 . 1 3 Let uR 0 ∈ H0 (BR ) ∩ H (BR ) be the unique solution to the elliptic boundary value problem R LuR 0 = F0

in BR

and

  uR 0 ∂BR = 0,

where 2 F0R = −∇(hR 0) +

 R ⊥ hR 0 g − h 0 u0 .

2 Extending uR 0 to R by defining zero outside BR , we can show that

uR 0 → u0

in D1 (BR ) as R → ∞.

The proof is similar to that in [6], so we omit it here. By Lemma 3.6, there exists some T ∗ > 0 such that for each R > 1, the initial boundary value problem (1.3), R R R (3.37), with Ω = BR and (h0 , u0 ) = (hR 0 , u0 ) for h0 , u0 defined as above, has a unique strong solution (h, u) satisfying (3.38) and (3.39). We denote such (h, u) by (hR , uR ). Extending (hR , uR ) to R2 by zero outside BR , we have from (3.39) that for p ∈ [2, q],  2 sup ψ R hR W 1,p 0≤t≤T ∗   ≤ C sup (hR )p dx + C sup 0≤t≤T ∗

0≤t≤T ∗

BR

BR

|∇hR |p dx

≤C and 

R R

sup

0≤t≤T ∗

R2



p

|(ψ h )t | dx ≤ C sup

0≤t≤T ∗

BR

p |hR t | dx ≤ C.

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

557

Similarly, (3.39) gives sup ∇(ψ R uR ) 2H 1 (R2 ) + sup ψ R uR 2Lq (R2 ) 0≤t≤T ∗   2 R 2 R 2 ≤ C sup |∇ ψ | |u | dx + C sup |∇ψ R |2 |∇uR |2 dx 0≤t≤T ∗ R2 0≤t≤T ∗ R2   R 2 R 2 ψ |∇ u | dx + C sup ψ R |∇uR |2 dx + C sup 0≤t≤T ∗ R2 0≤t≤T ∗ R2  |∇ψ R |2 |uR |2 dx + C sup uR 2Lq (BR ) + C sup 0≤t≤T ∗ R2 0≤t≤T ∗   −2 R 2 ≤ CR sup |u | dx + C sup |∇uR |2 dx 0≤t≤T ∗ BR 0≤t≤T ∗ BR  |∇2 uR |2 dx + C sup uR 2Lq (BR ) + C sup

0≤t≤T ∗

0≤t≤T ∗

0≤t≤T ∗

BR

≤ CR−2+2(q−2)/q sup uR 2Lq (BR ) + C sup uR 2Lq (BR ) + C 0≤t≤T ∗ 0≤t≤T ∗  |x|α |∇uR |2 dx + C ≤ C sup 0≤t≤T ∗

BR

≤C and



T∗

(ψ R uR )t 2Lq (R2 ) dt

0

 ≤C  ≤C

T∗ 0

T∗

2 uR t Lq (BR ) dt

 2  ˙R  u 

Lq (BR )

0



T∗

dt + C 0

uR · ∇uR 2Lq (BR ) dt

≤ C, which together with (3.39) yields  T∗ ∇(ψ R uR )t 2L2 (R2 ) dt 0

 ≤

T∗

0

≤ CR

2 |∇ψ R ||uR t | L2 (BR ) dt +

−2



T



∗

0

2 uR t L2 (BR ) dt



T∗

0 T∗

+ 0

2 ψ R ∇uR t L2 (BR ) dt

 R 2 ∇u˙  2 L (B

R)

dt

∇(uR · ∇uR ) 2L2 (BR ) dt

+ 0

≤ CR−4/q  +C ≤ C.

T∗



0



T∗

T∗

0

2 uR t Lq (BR ) dt + C

 0

T∗

 R 4 ∇u  4

L (BR )

dt

uR 2L∞ ∇2 uR 2L2 (BR ) dt + C

Therefore, by Lemma 2.2 and by a diagonalization procedure, we can choose a sub-

558

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

sequence of (hR , uR ), still denoted by (hR , uR ), such that as R → ∞ (3.50)

w∗ in L∞ (0, T ∗ ; W 1,q (R2 )) ∩ W 1,∞ (0, T ∗ ; Lp (R2 )), ψ R hR → h in C(B¯n × [0, T ∗ ]),

(3.51)

ψ R uR  u

w∗ in L∞ (0, T ∗ ; D1 (R2 ) ∩ D2 (R2 )),

(3.52)

ψ R uR  u

w in H 1 (0, T ∗ ; Lq (R2 ) ∩ D1 (R2 )),

(3.49)

ψ R hR  h

and (3.53)

ψ R uR → u in C([0, T ∗ ]; W 1,q (Bn ))

for any positive integer n. Now, for any function φ ∈ C0∞ (R2 × [0, T ∗ )), we take (ψ R )r φ with r ≥ 4 as a test function in (1.3), (3.37). Then letting R → ∞, it is easy to check that (3.49)–(3.53) yield that (h, u) is a unique strong solution of (1.3), (1.4), h > 0, we can bound (1.5) on R2 × [0, T ∗ ] satisfying (3.38) with Ω = R2 . In the case  u L∞ (0,T ∗ ;L2 (R2 )) and (ut , u) ˙ L2 (0,T ∗ ;L2 (R2 )) according to Lemma 3.1. Then, it is easy to get the unique strong solution as in [2]. In order to show (3.48), we can follow the steps in section 3 of [20] and obtain the higher order estimates of the solutions ˙ L2 (0,T ∗ ;Lq (R2 )) are bounded, where q = 2 if  h>0 since u L∞ (0,T ∗ ;Lq (R2 )) and (ut , u)  and q = 4/α if h = 0. The proof of Proposition 3.7 is completed. 3.3. Regularity analysis. In this subsection, we will show the necessary higher regularity for the classical solution and then prove our main result. We remark here that for arbitrary data, which may include vacuum states, the following lemma is substantial for the regularity of solution. Here, we estimate the spatial weighted norm of suitable quantities to overcome the difficulties coming from the space dimension and the appearance of vacuum. Moreover, an elaborate spatial weight is adopted in order to close the estimates when using the energy estimate method. Lemma 3.8. If  h = 0, the following estimate holds:   (3.54) sup t ut 2L8/α ∩D2 + ∇4 u 2L2 ≤ C. 0≤t≤T ∗

Proof. Setting v = u in (3.26) and multiplying the resulting equation by |x|α/2 (u) ˙ t, we obtain after integrating by parts that     1 d α/2 2 α/2 2 α/2 2 |x| h|(u) ˙ t | dx + ˙ dx + (μ + λ) |x| |div u| ˙ dx μ |x| |∇u| 2 dt   α/2 = − h(u · ∇u)|x| ˙ (u) ˙ t dx − h|x|α/2 (u) ˙ ⊥ · (u) ˙ t dx   μα α(μ + λ) − ∂i u˙ k (u˙ k )t |x|α/2−2 xi dx − div u(( ˙ u) ˙ t · x)|x|α/2−2 dx 2 2   k α/2 ˙ t )dx − μ (∂k uk ∂i u)∂i (|x|α/2 (u) ˙ t )dx + μ (∂i u ∂i u)∂k (|x| (u)  + μ (∂i uk ∂k u)∂i (|x|α/2 (u) ˙ t )dx   − (μ + λ) (div u)2 div(|x|α/2 (u) ˙ t )dx − (∂j uk ∂k uj )div(|x|α/2 (u) ˙ t )dx   ˙ t )dx − (∇uk div u) · ∂k (|x|α/2 (u)   2 k α/2 (3.55) − (h ∇u ) · ∂k (|x| (u) ˙ t )dx − h2 div udiv(|x|α/2 (u) ˙ t )dx.

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

559

We use (2.1), (3.39), and (3.48) to estimate each terms on the right-hand side of (3.55) as follows:      α/2 α/2 ⊥  h(u · ∇u)|x| ˙ (u) ˙ t dx + h|x| (u) ˙ · (u) ˙ t dx    α/4 ≤ C h1/2 (u) ˙ t |x|α/4 L2 u L∞ |∇u||x| ˙ L2 + h1/2 u|x| ˙ α/4 L2 1 1/2 α/4 2 h (u) ˙ t |x|α/4 2L2 + C |∇u||x| ˙ L2 + C h1/2 u|x| ˙ α/2 2L2 2 + C h1/2 u ˙ 2L2 1 α/2 2 ≤ h1/2 (u) ˙ t |x|α/4 2L2 + C |∇u||x| ˙ L2 2 + C( ∇u ˙ 2L2 + ∇u 2L2 + h1/2 ut 2L2 ); ≤

(3.56)

next, −

(3.57)

μα 2



∂i u˙ k (u˙ k )t |x|α/2−2 xi dx   μα d k k α/2−2 i (α−2)/2 x dx + C |∇(u) ˙ t ||u||x| ˙ dx ≤− ∂i u˙ u˙ |x| 2 dt  μα d ≤− ∂i u˙ k u˙ k |x|α/2−2 xi dx 2 dt + C ∇(utt + ut · ∇u + u · ∇ut ) L2 u|x| ˙ (α−2)/2 L2  μα d ∂i u˙ k u˙ k |x|α/2−2 xi dx ≤− 2 dt α/2 2 L2 + ∇utt 2L2 + ut 2L4/α + ∇2 ut 2L2 + 1). + C( |∇u||x| ˙

Also, we can estimate the terms  μ (∂i uk ∂i u)∂k (|x|α/2 (u) ˙ t )dx  d ˙ ≤μ (∂i uk ∂i u)∂k (|x|α/2 u)dx dt  (3.58) α/2 (α−2)/2 ˙ + |u||x| ˙ )dx + C |∇ut ||∇u|(|∇u||x|  d (∂i uk ∂i u)∂k (|x|α/2 u)dx ˙ + C ∇u|x| ˙ α/2 2L2 + C ≤μ dt and  −

(3.59)

˙ t )dx (h2 ∇uk ) · ∂k (|x|α/2 (u)  d (h2 ∇uk ) · ∂k (|x|α/2 u)dx ˙ ≤− dt  α/2 (α−2)/2 + C (|(h2 )t ||∇u| + h2 |∇ut |)(|∇u||x| ˙ + |u||x| ˙ )dx  d ≤− ˙ + C ∇u|x| ˙ α/2 2L2 + C. (h2 ∇uk ) · ∂k (|x|α/2 u)dx dt

560

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

It follows that in the same way as (3.57)–(3.59), we can estimate the other terms on the right-hand side of (3.55). Therefore, we arrive at (3.60)     1 1 d α/2 2 α/2 2 α/2 2 |x| h|u˙ t | dx + ˙ dx + (μ + λ) |x| |divu| ˙ dx μ |x| |∇u| 2 2 dt d α/2 2 ≤ Ψ(t) + C( ∇u ˙ 2L2 + |∇u||x| ˙ L2 + ∇utt 2L2 + ut 2L4/α ) dt + C ∇2 ut 2L2 + C, where

  μα α(μ + λ) ∂i u˙ k u˙ k |x|α/2−2 xi dx − divu( ˙ u˙ · x)|x|α/2−2 dx 2 2   k α/2 + μ (∂i u ∂i u)∂k (|x| u)dx ˙ − μ (∂k uk ∂i u)∂i (|x|α/2 u)dx ˙  ˙ + μ (∂i uk ∂k u)∂i (|x|α/2 u)dx   ˙ − (∂j uk ∂k uj )div(|x|α/2 u)dx ˙ − (μ + λ) (div u)2 div(|x|α/2 u)dx   − (∇uk div u) · ∂k (|x|α/2 u)dx ˙   − (h2 ∇uk ) · ∂k (|x|α/2 u)dx ˙ − h2 div udiv(|x|α/2 u)dx. ˙

Ψ(t) := −

It is easy to show that |Ψ(t)| ≤ μα2 |x|α/4 ∇u ˙ 2L2 + (μ + λ)α2 |x|α/4 ∇u ˙ 2L2     (α−2)/2 dx +C |∇u|2 + |∇u| |x|α/2 |∇u| ˙ + |u||x| ˙ (3.61)

≤ (2μ + λ)α2 |x|α/4 ∇u ˙ 2L2 + C ∇u|x|α/4 L2 |x|α/4 |∇u ˙ L2 ≤ 2(2μ + λ)α2 |x|α/4 ∇u ˙ 2L2 + C ∇u|x|α/2 2L2 + C ∇u 2L2 μ ≤ |x|α/4 ∇u ˙ 2L2 + C. 8

Multiplying (3.60) by t and integrating the resulting equation over (0, T ∗ ), we deduce from (3.61), (3.48), and (3.39) that  T∗   α/2 2 ˙ dx + t |x|α/2 h|u˙ t |2 dxdt t|x| |∇u| sup 0≤t≤T ∗

(3.62)

0



≤C

T∗

0



α/2 2 ( ∇u ˙ 2L2 + |∇u||x| ˙ L2 + t ∇utt 2L2 + ut 2L4/α )dt

T∗

+C 0

( ∇2 ut 2L2 + 1)dt

≤ C. It follows from (3.62) and (2.1) that sup t u ˙ 2L8/α ≤ C sup t |x|α/4 ∇u ˙ 2L2 ≤ C,

0≤t≤T ∗

0≤t≤T ∗

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

561

which yields that (3.63)

  sup t ut 2L8/α ≤ C sup t u ˙ 2L8/α + u · ∇u 2L8/α ≤ C.

0≤t≤T ∗

0≤t≤T ∗

Next, the standard L2 -estimate for the elliptic system, (3.39) and (3.48) yield that for p ∈ [q, ∞) ∇2 ut L2 ≤ C μΔut + (μ + λ)∇div ut L2 = hutt + ht ut + ht u · ∇u + hut · ∇u + hu · ∇ut + ∇(h2 )t + (hu⊥ )t L2 (3.64)

≤ C ( hutt L2 + ( ht L2p/(p−2) + h L2p/(p−2) )( ut Lp + u Lp ) + ht L4 u L∞ ∇u L4 + ut Lp ∇u L2p/(p−2)  + u L∞ ∇ut L2 + ∇(h2 )t L2 ≤ C h1/2 utt L2 + C ut Lp + C.

The standard L2 -estimate for the elliptic system again yields that for p ∈ [q, ∞) ∇4 u L2 ≤ C ∇2 (μΔu + (μ + λ)∇div u) L2 (3.65)

≤ C ∇2 (hu) ˙ L2 + C ∇3 h2 L2 + C hu⊥ L2 ≤ C + C ∇u H 2 + C ut Lp + C ∇ut H 1 + C ∇3 h2 L2 ,

where one has used the following simple facts:

(3.66)

∇2 (hut ) L2   ≤ C ∇2 h L2 ( ut Lp + ∇ut H 1 ) + ∇h L4 ∇ut L4 + ∇2 ut L2 ≤ C ut Lp + C ∇ut H 1

and

(3.67)

∇2 (hu · ∇u) L2   ≤ C ∇2 (hu) L2 ∇u H 2 + ∇(hu) L4 ∇2 u L4 + ∇3 u L2   ≤ C 1 + ∇2 h L2 u L∞ + ∇h L4 ∇u L4 + ∇2 u L2 ∇u H 2 ≤ C ∇u H 2

due to (3.39), (3.48), and (2.3). Choosing p = 8/α in both (3.64) and (3.65), together with (3.48), yields that (3.68)

sup t ∇4 u 2L2 ≤ C.

0≤t≤T ∗

Thus, (3.54) follows from (3.63) and (3.68) directly. We finish the proof of Lemma 3.8. We are now in a position to prove our main result, Theorem 3.9. ˜ ≥ 0 and Ω = R2 , assume that the initial data (h0 ≥ 0, u0 ) Theorem 3.9. For h satisfy the condition in Proposition 3.7. Then there exist a small time T ∗ > 0 and a unique solution (h, u) to the Cauchy problem (1.3), (1.4), (1.5) on R2 × [0, T ∗ ] such

562

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

that for any 0 < τ < T ∗ , ⎧ ˜ h2 − (h) ˜ 2 ) ∈ C([0, T ∗ ]; H 3 ), (h − h, ⎪ ⎪ ⎪ ⎪ ⎪ u ∈ C([0, T ∗ ]; Lq ∩ D1 ∩ D3 ) ∩ L2 (0, T ∗ ; D4 ) ∩ L∞ (τ, T ∗ ; D4 ), ⎪ ⎪ ⎪ ⎨u ∈ L∞ (0, T ∗ ; D1 ) ∩ L2 (0, T ∗; Lq ∩ D2 ) t ⎪ u ∈ L∞ (τ, T ∗ ; Lq1 ∩ D2 ) ∩ H 1 (τ, T ∗ ; D1 ), ⎪ ⎪ √t √ ⎪ ⎪ ⎪ hut ∈ L∞ (0, T ∗ ; L2 ), hutt ∈ L2 (0, T ∗ ; L2 ), ⎪ ⎪ √ ⎩ 1/2 ∞ ∗ 2 hutt ∈ L (0, T ; L ), t where

 q = q1 = 2 q = 4/α, q1 = 8/α

if if

 h > 0,  h = 0.

Remark 3.4. The unique solution obtained in Theorem 3.9 becomes a classical one for positive time. Remark 3.5. The local classical solution obtained in Theorem 3.9 exists for arbitrary large initial data and the initial height need not be positive and may vanish in open sets. In particular, the initial height can even have compact support. ˜ = 0, Theorem 3.9 is an easy consequence of Proposition 3.7 and Proof. If h Lemma 3.8. It remains to prove that (h, u) becomes a classical solution for positive time, that is, for any 0 < τ ≤ T ∗ ,   (3.69) ut , ∇2 u, ∇h, ht ∈ C [τ, T ∗ ] × R2 . It follows from (3.48) and (3.54) that for any 0 < τ ≤ T ∗ , sup ∇2 u H 2 ≤ C(τ ),

τ ≤t≤T ∗

which, together with (3.48), (∇2 u)t ∈ L2 (R2 × (0, T ∗ )), implies that for p > 2,   (3.70) ∇2 u ∈ C([τ, T ∗ ]; H 1 ∩ W 1,p ) → C [τ, T ∗ ] × R2 . By virtue of (3.54) and (3.48),  (3.71)

sup ( ut L8/α + ∇ut H 1 ) +

τ ≤t≤T ∗

τ

T∗

(∇ut )t 2L2 dt ≤ C(τ ),

which implies for p > 2, ∇ut ∈ C([τ, T ∗ ]; L2 ∩ Lp ).

(3.72)

It follows from (3.71) and (3.72) directly that   (3.73) ut ∈ C [τ, T ∗ ] × R2 . We deduce from (3.48) that ∇h ∈ L∞ (0, T ∗ ; H 2 ),

(∇h)t ∈ L∞ (0, T ∗ ; L2 ),

which yields that for p > 2, (3.74)

  ∇h ∈ C([0, T ∗ ]; H 1 ∩ W 1,p ) → C [0, T ∗ ] × R2 .

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

563

According to (1.3)1 , (3.70), and (3.74), we have   ht ∈ C [0, T ∗ ] × R2 , ˜ > 0, after bounding which together with (3.70), (3.73), and (3.74) gives (3.69). If h u L2 (R2 ) and u ˙ L2 (R2 ) by Lemma 3.1, we can obtain Theorem 3.9 as above. The proof of Theorem 3.9 is completed. 4. Blow-up example. In this section, we investigate the blow-up behavior of 2D smooth radially symmetric solutions to shallow water equations (1.3) when the initial height has compact support. If the initial data (h0 , u0 ) are radially symmetric, i.e., (4.1)

h0 (x) = h0 (r),

x u0 (x) = v0 (r) , r

r = |x|,

satisfying conditions (3.35), (1.6), and (3.36), the shallow water equations has unique local radially symmetric solutions (h, u), (4.2)

h(x, t) = h(r, t),

x u(x, t) = v(r, t) . r

Now, following the idea in [39], we can state the blow-up result on smooth solutions when the initial data (h0 , u0 ) are radially symmetric and the initial height is compactly supported. Theorem 4.1. For  h = 0, suppose that (h0 ≡ 0, u0 ) satisfy (4.1) and that the nonnegative initial height h0 has compact support. Then any radially symmetric solution (h, u) ∈ C 1 ([0, T ]; H s )(s > 2) to the Cauchy problem (1.3) with initial data (h0 , u0 ) has to blow up in finite time, that is, T must be finite. Proof. Since the support of the initial radially symmetric height h0 is compact, we can assume that    supph0 = BR0 (0) = x ∈ R2  |x| ≤ R0 for some R0 > 0. We denote by X(t; x0 ) the particle path starting at x0 when t = 0, i.e., d X(t, x0 ) = u(X(t; x0 ), t), dt

X(t = 0; x0 ) = x0 .

Set (4.3)

Sp (t) := {(x, t) |x = X(t; x0 ) ∀x0 ∈ BR0 (0) } .

It follows from the continuity equation that the height is transported along particle paths, so that (4.4)

suppx h(x, t) ⊂ Sp (t).

Consequently, h = 0 on {t} × (R2 \ Sp (t)) and one has from the momentum equation and (4.2) that  v (2μ + λ)∂r vr + = 0 on {t} × (R2 \ Sp (t)), r

564

BEN DUAN, ZHEN LUO, AND YUXI ZHENG

since Δu(x, t) = ∇div u(x, t) = ∂r (vr + vr ) xr . Therefore, v(r, t) = θ(t)r +

β(t) r

on {t} × (R2 \ Sp (t)).

Now, because u ∈ C 1 ([0, T ]; H s )(s > 2), we have u(x, t) ≡ 0 in x ∈ R2 \ Sp (t), and (4.5)

Sp (t) = BR0 (0),

∀0 < t < T.

Following [39], we denote 

 2 (1 + t)2 hγ dx |x − (1 + t)u|2 hdx + γ −1   = x2 hdx − 2(1 + t) hu · xdx   2 + (1 + t)2 hγ dx, t ∈ [0, T ]. hu2 + γ−1

Iγ (t) = (4.6)

Then d Iγ (t) = dt (4.7)

   2 2 γ h dx (x ht − 2hu · x)dx − 2(1 + t) (hu)t · x − hu − γ−1    2 (hγ )t dx, + (1 + t)2 (hu2 )t + γ−1 

2

= I1 + I2 + I3 ,

t ∈ [0, T ].

Direct calculation by using (4.2), (4.4), and u ∈ C 1 ([0, T ]; H s )(s > 2) yields that I1 = 0, 



− div(hu ⊗ u) · x − ∇hγ · x − hu⊥ · x + μΔu · x   2 hγ dx + (μ + λ)(∇div u) · x dx + 2(1 + t) hu2 + γ−1     +∞   |x|2 v 2 γ + vr rdr ∂r = −2(1 + t) 2− h dx + (2μ + λ) γ−1 r r 0     v 2(γ − 2) γ = −2(1 + t) + vr dx h − 2(2μ + λ) γ−1 r

I2 = −2(1 + t)

and I3 = (1 + t)2



2 [−∇hγ · u + μΔu · u + (μ + λ)(∇div u) · u] dx  2 + (1 + t)2 [−div(hγ u) − ((hγ ) h − hγ )div u] dx γ−1  +∞ v  x  x 2 + vr · v rdr (2μ + λ)∂r = 2(1 + t) r r r 0    2 v = −2(1 + t)2 (2μ + λ) + vr dx. r

LOCAL CLASSICAL SOLUTIONS TO SHALLOW WATER EQUATIONS

565

Therefore, Iγ (t)

(4.8)



     2  v v 2 + vr dx − (1 + t) + vr dx = 2(2μ + λ) 2(1 + t) r r  4(1 + t)(2 − γ) + hγ dx γ−1   4(1 + t)(2 − γ) γ ≤ h dx + 2(2μ + λ) dx γ−1 Sp (t)  4(1 + t)(2 − γ) = hγ dx + 2(2μ + λ)|BR0 (0)|. γ−1

If γ ≥ 2, (4.8) gives Iγ (t) ≤ 2(2μ + λ)|BR0 (0)|, which yields that Iγ (t) ≤ Iγ (0) + 2(2μ + λ)|BR0 (0)|t. Thus, we have  (γ − 1)Iγ (0) (4.9) hγ dx ≤ (1 + t)−2 + (2μ + λ)|BR0 (0)|(γ − 1)(1 + t)−1 2 due to (4.6). On the other hand, if 1 < γ < 2, let α = 2 − 2(γ − 1). It follows from (4.8) that   (1 + t)−α Iγ (t) ≤ 2(2μ + λ)|BR0 (0)|(1 + t)2γ−4 ,

(4.10) which gives

(1 + t)−α Iγ (t) ≤ Iγ (0) + 2(2μ + λ)|BR0 (0)|F (t),

(4.11) where

F (t) =

⎧ 2γ−3 ⎪ ⎪ ⎨ (1+t)

if

2γ − 3 = 0;

⎪ ⎪ ⎩ ln(1 + t)

if

2γ − 3 = 0.

2γ−3

This yields that (4.12)  (γ − 1)Iγ (0) (1 + t)−2(γ−1) + (γ − 1)(2μ + λ)|BR0 (0)|F (t)(1 + t)−2(γ−1) . hγ dx ≤ 2 Finally, from the continuity equation, (4.5), (4.4), and H¨ older’s inequality, we have  0
2) solution cannot exist globally. Therefore, for rotation shallow water equations, the solutions in our main result, Theorem 3.9, give a possible class, in which the globally in time solutions are ˜ = 0. expected in the case h Acknowledgments. The authors would like to thank Chunjing Xie for helpful discussions and the referees for valuable comments and suggestions. REFERENCES [1] Y. Cho, H.J. Choe, and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), pp. 243–275. [2] H.J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), pp. 504–523. [3] L.A. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), pp. 259–275. [4] F. Catrina and Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), pp. 229–258. [5] Q. Chen, C. Miao, and Z. Zhang, Well-posedness for the viscous shallow water equations in critical spaces, SIAM J. Math. Anal., 40 (2008), pp. 443–474. [6] Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math, 120 (2006), pp. 91–129. [7] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), pp. 211–223. [8] D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl. (9), 86 (2006), pp. 362–368. [9] D. Bresch, B. Desjardins, and C.K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), pp. 843–868. [10] D. Bresch, B. Desjardins, and G. M´ etivier, Recent mathematical results and open problems about shallow water equations, Adv. Math. Fluid Mech., Birkh¨ auser, Basel, 2006, pp. 15– 31. [11] R. Danchin, Global existence in critical spaces for flows of compressible viscous and heatconductive gases, Arch. Ration. Mech. Anal., 160 (2001), pp. 1–39. [12] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 18 (2001), pp. 97–133. [13] B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. Partial Differential Equations, 22 (1997), pp. 977–1008. [14] B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluid, Arch. Ration. Mech. Anal., 137 (1997), pp. 135–158. [15] B.Desjardins and C.K. Lin, A survey of the compressible Navier-Stokes equations, Taiwanese J. Math., 3 (1999), pp. 123–137. [16] E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, UK, 2004. [17] Z.H. Guo, Q.S. Jiu, and Z.P. Xin, Radially symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), pp. 1402–1427.

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