Global existence for the higher-order Camassa–Holm shallow water equation

Nonlinear Analysis 74 (2011) 2468–2474

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Global existence for the higher-order Camassa–Holm shallow water equation Lixin Tian ∗ , Pin Zhang, Limeng Xia Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China

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Article history: Received 25 January 2010 Accepted 4 December 2010

In this paper, we investigate the global existence of the higher-order Camassa–Holm equation in the case of k = 2. We prove the local well-posedness of this equation and find a conservation law. Then a global existence result is obtained. © 2011 Published by Elsevier Ltd

Keywords: Higher-order Camassa–Holm Local well-posedness Conservation law Global existence

1. Introduction The nonlinear dispersive wave equation ut + 2ωux − uxxt + 3uux = 2ux uxx + uuxxx ,

(1.1)

was derived by Camassa and Holm [1] as a model for the unidirectional propagation of shallow water waves over a flat bottom. Here u(t , x) stands for the fluid velocity at time t in the spatial x direction (or equivalently the height of the free surface of water above a float bottom), ω is a constant related to the critical shallow water wave speed. Eq. (1.1) is the wellknown Camassa–Holm equation. It has a bi-Hamiltonian structure [2,3] and is completely integrable [1,4]. Moreover, it has many conservation laws (see [5]):

∫

udx,

E1 = R

∫

(u2 + u2x )dx,

E2 =

∫

(u3 + uu2x )dx.

E3 =

R

R

And it also has solitary wave solutions [6–8] to the form ce−|x−ct | , c ∈ R, which is called peakon because they have a discontinuous first derivative at the wave peak. In addition to smooth solutions, the author in [6] obtained that there are a multitude of travelling waves with singularities: cuspons, stumpons and composite waves. The well-posedness of the Camassa–Holm equation has been studied extensively [9–16]. Using a regularization technique, Li and Olver in [13] established the local well-posedness in the Sobolev space H s with any s > 32 for Eq. (1.1). In the condition of the first derivative of initial value belongs to L∞ (R); they also obtained an existence theorem for (1.1) in H s (R), 1 < s ≤ 32 . Similar results for the local well-posedness of (1.1) were obtained by Rodriguez-Blanco [17] by applying the Kato theory [18] for the quasilinear equations. The global well-posedness in case s > 23 to the Camassa–Holm equation  was also established, provided that |u0 | dx < +∞ and (1 − ∂x2 )u0 does not change sign. Furthermore, the global existence and blow-up for this equation have also been well studied in [10,19,12–16]. Here blow-up means that the slope of the solution becomes unbounded while the solution itself stays bounded. The first studies on the Cauchy problem for the CH

∗

Corresponding author. E-mail address: [email protected] (L. Tian).

0362-546X/$ – see front matter © 2011 Published by Elsevier Ltd doi:10.1016/j.na.2010.12.002

L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

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equation as well as the first results on the global existence and wave breaking go back to Refs. [10,12]. Particularly, Constantin and Escher considered the problem of the development of singularities for classical solutions to a new periodic shallow water equation (see [19]). They established the precise blow-up rate and blow-up set for the Camassa–Holm equation. In recent years, many researchers have been researched on the Camassa–Holm equation. They extend the studies to the generalized CH equation, higher-order CH equations and so on. Lixin Tian, Chunyu Shen and Danping Ding gave the optimal control of the viscous Camassa–Holm equation under the boundary condition and proved the existence and uniqueness of optimal solution to the viscous Camassa–Holm equation in a short interval (see [20]). In [21], Robert McLachlan and Xingyou Zhang studied the well-posedness and dynamics of a modified Camassa–Holm equation on the unit circles and obtained the result that it had some significant difference from that of the Camassa–Holm equation. Using geometrical methods, higherorder CH equations have been treated in [22] by Constantin and Kolev. The well-posedness of higher-order Camassa–Holm equations were considered in [23]. The authors established the existence of their global weak solutions and proved a ‘‘weak equals strong’’ uniqueness result. They also present some invariant spaces under the action of the equation. Moreover, more and more authors have interested in the two-component Camassa–Holm equation (see [24,25]). They proved the local wellposedness, global existence and blow-up phenomena for the Camassa–Holm equation, which extended the research of the Camassa–Holm equation to more extensive fields. The formulation of the higher-order Camassa–Holm equation which was recently derived by Coclite, Holden and Karlsen in [23] is 1 Bk (u, u) := A− k Ck (u) − uux ,

Ak (u) :=

k − (−1)j ∂x2j u,

(1.2)

j =0

Ck (u) = −uAk (∂x u) + Ak (u∂x u) − 2∂x uAk (u) where k is a positive integer. In cases k = 0 and k = 1, Eq. (1.2) becomes the inviscid Burgers equation ut + 3uux = 0 and the Camassa–Holm equation ut − uxxt + 3uux = 2ux uxx + uuxxx . In this paper we only consider the global existence of the case k = 2 of Eq. (1.2). That is mt + 2ux m + umx = 0,

t > 0, x ∈ R

(1.3)

where m = u − uxx + uxxxx , u(t , x) describes the horizontal velocity of the fluid. It also can be rewritten as ut − ∂t (∂x2 u) + ∂t (∂x4 u) + 3uux − 2ux uxx − uuxxx + 2ux ∂x4 u + u∂x5 u = 0.

(1.4)

For m = u − uxx , Eq. (1.3) is the Camassa–Holm equation (1.1). When k > 2, the local well-posedness of Eq. (1.2) is hard to prove and the global existence is complicated to estimate. We will continue to discuss this case in the later studies. The structure of this paper is organized as follows. In Section 2, by applying the Kato theory, the local well-posedness of the higher-order Camassa–Holm equation is studied. In Section 3, we get a conservation law for the higher-order Camassa–Holm equation. Thus a norm estimate is established. By applying this norm estimate, the global existence result is obtained in Section 4. 2. Local well-posedness and main lemmas In this section, we will first use the Kato theory to establish the local well-posedness for Eq. (1.3) (or Eq. (1.4)), then we will give the main lemmas. The Cauchy problem of Eq. (1.3) is



mt + 2ux m + umx = 0, t > 0, x ∈ R, m(0, x) = (1 − ∂x2 + ∂x4 )u0 (x), x ∈ R.

(2.1)

Note that if √

1

√

(θ −2 − θ 2 )−1 (θ −1 e−|bx|−|ax| −1 − θ e−|bx|+|ax| −1 ), 2 √ π where θ = b + a −1 = e 12 i , x, a, b ∈ R. Then (1 − ∂x2 + ∂x4 )−1 f = p ∗ f for all f ∈ L2 (R) and p ∗ m = u. Using this identity, p(x) =

Eq. (2.1) takes the form

  

 ut + uux = −∂x p ∗ u(0, x) = u0 (x),

u2 +

x ∈ R.

1 2

u2x −

1 2



u2xx − 3∂x (∂x u∂x2 u) ,

t > 0, x ∈ R,

(2.2)

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L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

From [23], we know that p(x) has the Fourier transform pˆ (x) given by pˆ (x) =

1 1 + ξ2 + ξ4

,

ξ ∈ R.

We also have p(x) ≥ 0,

‖p(x)‖w3,1 (R) , ‖p(x)‖w3,∞ (R) ≤ C0

(2.3)

for some constant C0 > 0. If we denote A(u) = u∂x ,

f (u) = −∂x (1 − ∂ + ∂ ) 2 x

4 −1 x



2

u +

1 2

u2x

−

1 2

u2xx



− 3∂x (∂x u∂ ) , 2 xu

then (2.2) has the form

∂u + A(u)u = f (u), t ≥ 0, x ∈ R, ∂t u(0, x) = u0 (x), x ∈ R.



Theorem 2.1. Given u0 ∈ H s (R), s > 29 . Then there exists a T > 0 depending on ‖u0 ‖H s , and a unique solution u = u(t , x) satisfying Eq. (2.1) (or Eq. (2.2)) such that u ∈ C ([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)). Moreover, the map u0 ∈ H s → u ∈ C ([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) is continuous. In order to prove this theorem, we only need to verify all conditions in the Kato Theorem. We need the following lemma. Lemma 2.1 (See [10,12,17]). The operator A(u) = u∂x with u ∈ H s , s ≥ 4 belongs to G(H s−1 , 1, β) for some β > 0. Lemma 2.2. Let A(u) = u∂x , u ∈ H s , s ≥ 4 be given. Then A(u) ∈ L(H s , H s−1 ), and ∀u, y, ω ∈ H s , we have

‖(A(u) − A(y))ω‖H s−1 ≤ µ ‖u − y‖H s ‖ω‖H s . Proof. Note that H s−1 is a Banach algebra and that

(A(u) − A(y))ω = (u − y)∂x ω. Then we have

‖(A(u) − A(y))ω‖H s−1 = ‖(u − y)∂x ω‖H s−1 ≤ ‖u − y‖H s−1 ‖∂x ω‖H s−1 ≤ µ ‖u − y‖H s ‖ω‖H s . Taking y = 0 in the above inequality, we obtain that A(u) ∈ L(H s , H s−1 ). This completes the proof of this lemma. 1



Lemma 2.3. Let B(u) = QA(u)Q −1 − A(u), u ∈ H s , s ≥ 4, Q = (1 − ∂x2 + ∂x4 ) 2 . Then B(u) ∈ L(H s−1 ), and for ∀u, y ∈ H s , ω ∈ H s−1 , we have

‖(B(u) − B(y))ω‖H s−1 ≤ c ‖u − y‖H s ‖ω‖H s−1 . Proof. We know that

(B(u) − B(y))ω = Q (u − y)Q −1 ∂x ω − (u − y)∂x ω = (Q (u − y)Q −1 − (u − y))∂x ω = [Q , u − y]Q −1 ∂x ω. Applying the Fourier transformation, we have

  ‖(B(u) − B(y))ω‖H s−1 = [Q , u − y]Q −1 ∂x ωH s−1  s−1  s−1 s−3   ≤ c1 Q 2 [Q , u − y]Q − 2 Q 2 ∂x ω 2  s−1   s−L3   2   − s−2 1  ≤ c1 Q [Q , u − y]Q  2 Q 2 ∂x ω 2 L(L )

≤ c ‖u − y‖H s ‖ω‖H s−1 . 

L

L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

Lemma 2.4. Let f (u) = −∂x (1 − ∂x2 + ∂x4 )−1 (u2 + 21 u2x − 12 u2xx − 3∂x (∂x u∂x2 u)), u ∈ H s , s > (i) ‖f (u) − f (v)‖H s−1 ≤ c ‖u − v‖H s−1 ; (ii) ‖f (u) − f (v)‖H s ≤ c ‖u − v‖H s .

2471 9 , 2

then

Proof. f (u) − f (v) = −∂x p ∗



u2 − v 2 +

1 2

u2x −

1 2

 1 1 2 vx2 − u2xx + vxx − 3∂x (∂x u∂x2 u) + 3∂x (∂x v∂x2 v) . 2

2

Now we only prove (i), since the proved method of (ii) is similar to it.

  ∂x p ∗ (u2 − v 2 )

H s−1

  ≤ C0 (u2 − v 2 )H s−4 ≤ C0 ‖(u + v)‖H s−4 ‖(u − v)‖H s−4 ≤ c ‖(u − v)‖H s−1 .

In a similar way,

  ∂x p ∗ (u2 − v 2 ) s−1 ≤ c ‖(u − v)‖H s−1 . x x  H   ∂x p ∗ (u2 − v 2 ) s−1 ≤ C0 ∂ 2 (u − v) s−4 xx xx x H H ≤ c ‖(u − v)‖H s−1 . ‖∂x p ∗ ∂x (ux uxx − vx vxx )‖H s−1 ≤ C0 ‖∂x (ux uxx − vx vxx )‖H s−4 ≤ C0 ‖ux uxx − ux vxx + (ux vxx − vx vxx )‖H s−3 = C0 ‖ux (uxx − vxx ) + ((ux − vx )vxx )‖H s−3 ≤ C0 ‖u‖H s−2 ‖u − v‖H s−1 + ‖u − v‖H s−2 ‖v‖H s−1 ≤ c ‖(u − v)‖H s−1 . From the above estimates, we proved (i). Here we have used the fact that H s is a Banach algebra for s > the proof of this lemma. 

1 . 2

This completes

In the following, we give the main lemmas. Lemma 2.5. Given u0 ∈ H s (R), s ≥ 4, and let T be the maximal existence time of the solution u = u(t , x) to Eq. (2.1) (or Eq. (2.2)) with the initial u0 . Then the corresponding solution blows up in finite time if and only if lim inf{inf ux (t , x)} = −∞.

t →T x∈R

Proof. The proof of this lemma is similar to [25], so we omit it here.



Introduce the following differential equation



qt = u(t , q), t ∈ [0, T ), x ∈ R, q(0, x) = x, x ∈ R.

(2.4)

By applying classical results in the theory of ordinary differential equation, one can obtain a result on q which is crucial in studying the global existence. Lemma 2.6 (See [24,25]). Let u ∈ C ([0, T ); H s ) ∩ C 1 ([0, T ); H s−1 ), s ≥ 4. Then Eq. (2.4) has a unique solution q ∈ C 1 ([0, T ) × R; R). Moreover, the map q(t , ·) is an increasing diffeomorphism of R with qx (t , x) = exp

t

∫

ux (s, q(s, x)) > 0,

∀(t , x) ∈ [0, T ) × R.

0

3. Conservation law In this section we derive a conservation law and an a priori estimate for the solution to Eq. (2.1). Theorem 3.1. Let u0 ∈ H s (R), s ≥ 4, and let T be the maximal existence time of the solution u = u(t , x) to Eq. (2.1) (or Eq. (2.2)) with the initial u0 . Then we have E (t ) =

∫

(u2 + u2x + u2xx )dx =

∫ R

R

(u20 + u20,x + u20,xx )dx,

∀t ∈ [0, T ).

(3.1)

Moreover, we have

‖u(t , ·)‖2L∞ (R) ≤

1 2

‖u0 ‖2H 2 (R) ,

∀t ∈ [0, T ).

(3.2)

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L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

Proof. Using Eq. (1.4) and integration by parts, we can get dE (t ) dt

∫

(2uut + 2ux uxt + 2uxx uxxt )dx

= R

∫

(2uut − 2uuxxt − 2ux uxxxt )dx

= ∫R =

(2uut − 2uuxxt + 2uuxxxxt )dx

∫R = ∫R =

2u(ut − uxxt + uxxxxt )dx 2u(−3uux + 2ux uxx + uuxxx − 2ux uxxxx − uuxxxxx )dx

R

∫

∫

= −6

uux dx + 4 R

∫

R

∫

u2 uxxx dx − 4

uux uxx dx + 2

∫

u2 uxxxxx dx

uux uxxxx dx − 2

R

R

R

= 0. This implies E (t ) = E (0) for all t ∈ [0, T ). Using this conservation law, we obtain u2 (t , ·) ≤

≤ =

1

∫

2 R ∫ 1 2 R ∫ 1 2

R

(u2 + u2x )dx (u2 + u2x + u2xx )dx (u20 + u20,x + u20,xx )dx

hence, we get 1 ‖u0 ‖2H 2 (R) , ∀t ∈ [0, T ). 2 This completes the proof of this theorem. 

‖u(t , ·)‖2L∞ (R) ≤

4. Global existence In this section, we establish the global existence of solution to Eq. (2.1) (or Eq. (2.2)). Lemma 4.1. Let u0 ∈ H s (R), s ≥ 4, and let T be the maximal existence time of the solution u = u(t , x) to Eq. (2.1) (or Eq. (2.2)) with the initial u0 . Then we have

√

lim inf{ux (t , x)} ≥ −

t →T x∈R

2 2

(1 + ‖ ‖ ) exp u0 2H 2



5 2

C0 ‖

u0 2H 2

‖

 T

.

Proof. By Lemma 2.6, we know that q(t , ·) is an increasing diffeomorphism of R with qx (t , x) = exp

t

∫

ux (s, q(s, x)) > 0,

∀(t , x) ∈ [0, T ) × R.

0

Then we have inf ux (t , q(t , x)) = inf ux (t , x),

x∈R

∀t ∈ [0, T ).

x∈R

(4.1)

Set M (t , x) = ux (t , q(t , x)). By Eq. (2.4), we have

∂M = (utx + uuxx )(t , q(t , x)). ∂t Differentiating the first equation in (2.2) with respect to x, we have utx + u2x + uuxx + ∂x2 p ∗



u2 +

1 2

u2x −

1 2



u2xx − 3∂x (∂x u∂x2 u)

So Mt = −u2x − ∂x2 p ∗ u2 + 12 u2x − 12 u2xx − 3∂x (∂x u∂x2 u) .





= 0.

L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

2473

We first consider

  1 1 ∂x2 p ∗ u2 + u2x − u2xx − 3∂x (∂x u∂x2 u) 2 2   ∫ ∫ 1 1 = ∂x2 p u2 + u2x − u2xx (t , ξ )dξ − 3 ∂x2 p(∂x (∂x u∂x2 u))(t , ξ )dξ 2 2 R R   ∫ ∫ 1 1 = ∂x2 p u2 + u2x − u2xx (t , ξ )dξ + 3 ∂x3 p(∂x u∂x2 u)(t , ξ )dξ 2

R

2

R

       1 ≤ ∂x2 pL∞ u2 + u2x + u2xx L1 + 3 ∂x3 pL∞ · u2x + u2xx L1 2

≤ C 0 ‖ u‖ =

5 2

C0 ‖

2 H2

3

+ C 0 ‖ u‖ 2

u0 2H 2

‖

2 H2

.

Let f (t , x) = −∂x2 p ∗ (u2 + 12 u2x − 12 u2xx − 3∂x (∂x u∂x2 u)). Then

|f (t , x)| ≤

5 2

C0 ‖u0 ‖2H 2 ,

∀(t , x) ∈ [0, T ) × R.

(4.2)

Thus Mt = −u2x + f (t , x),

(t , x) ∈ [0, T ) × R.

(4.3)

Consider the following function

w(t , x) = 1 + M 2 (t , x),

(t , x) ∈ [0, T ) × R.

(4.4)

By the Sobolev imbedding theorem, we have 0 < w(0, x) = 1 + M 2 (0, x) = 1 + u2x (0, q(0, x))

= 1 + u20,x (x) ≤ 1 + ‖u0 ‖2H 2 .

(4.5)

Differentiating (4.4) with respect to t and using (4.2) and (4.3), we obtain

∂w = 2MMt = 2M (−M 2 + f (t , x)) ∂t ≤ (1 + M 2 ) |f (t , x)| 5

≤ w(t , x) C0 ‖u0 ‖2H 2 .

(4.6)

2

By Gronwall’s inequality and using (4.5) and (4.6), we get

w(t , x) ≤ w(0, x) exp



5 2

C0 ‖

≤ (1 + ‖ ‖ ) exp u0 2H 2

u0 2H 2



5 2



‖

t

C0 ‖

u0 2H 2

‖

 T

,

for all (t , x) ∈ [0, T ) × R. On the other hand,

w(t , x) ≥

1 2

+ M 2 (t , x) ≥

√

2 |M (t , x)| ,

∀(t , x) ∈ [0, T ) × R.

Thus 1 M (t , x) ≥ − √ w(t , x) 2   1 5 ≥ − √ (1 + ‖u0 ‖2H 2 ) exp C0 ‖u0 ‖2H 2 T 2 2 for all (t , x) ∈ [0, T ) × R.

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L. Tian et al. / Nonlinear Analysis 74 (2011) 2468–2474

Then by (4.1) and the above inequalities, we have inf ux (t , q(t , x)) = inf ux (t , x)

x∈R

x∈R

1

≥ − √ (1 + ‖u0 ‖2H 2 ) exp 2

This completes the proof of this lemma.



5 2

C0 ‖u0 ‖2H 2 T



.



By this lemma and Lemma 2.5, we can get the following theorem. Theorem 4.1. Let u0 ∈ H s (R), s > initial u0 exists globally in time.

9 . 2

Then the corresponding strong solution u = u(t , x) to Eq. (2.1) (or Eq. (2.2)) with the

Acknowledgements This research was supported by the National Nature Science Foundation of China (No. 71073073, 91010011) and Nature Science Foundation of Jiangsu (No. BK2010329) and Outstanding Personnel Program in Six Fields of Jiangsu (No. 6-A-029). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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