This paper is concerned with time-periodic solution of the weakly dissipative Camassa-Holm equation with a periodic boundary condition. The existence and uniqueness of a time periodic solution is presented.

1. Introduction

The Camassa-Holm equation

(1.1)

modeling the unidirectional propagation of shallow water waves over a flat bottom, where u(t, x) represents the fluid’s free surface above a flat bottom (or equivalently, the fluid velocity at time t ≥ 0 and in the spatial x direction).

Since the equation was derived physically by Camassa and Holm [1, 2], many researchers have paid extensive attention to it. The Camassa-Holm equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [3, 4]. It has a bi-Hamiltonian structure [5, 6] and is completely integrable [1, 2, 7–11]. It is a reexpression of geodesic flow on the diffeomorphism group of the circle [12] and on the Virasoro group [13]. Its solitary waves are peaked [7], and they are orbitally stable and interact like solitons [14–16]. The peakons capture a characteristic of the traveling waves of greatest height-exact traveling solutions of the governing equations for water waves with a peak at their crest [17–19].

The Cauchy problem of the Camassa-Holm equation has been extensively studied. It has been shown that this equation is locally well posed [20–25] for initial data u₀ ∈ H^s(ℝ) with s > 3/2. Moreover, it has global strong solutions modeling permanent waves [20, 24–27] but also blow-up solutions modeling wave breaking [20–28]. On the other hand, it has global weak solutions with initial data u₀ ∈ H¹ [29–35]. Moreover, the initial-boundary value problem for the Camassa-Holm equation on the half-line and on a finite interval was discussed in [36, 37]. It is observed that if u is the solution of the Camassa-Holm equation with the initial data u₀ in H¹(ℝ), we have for all t > 0,

(1.2)

It is worth pointing out that the advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models wave breaking [2, 20, 21].

In general, it is difficult to avoid energy dissipation mechanisms in a real world. Ott and Sudan [38] investigated how the KdV equation was modified by the presence of dissipation and the effect of such dissipation on the solitary solution of the KdV equation. Ghidaglia [39] investigated the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system.

The Camassa-Holm equation with dissipative term is

(1.3)

where f(t, x) is the forcing term, L(u) is a dissipative term, L can be a differential operator or a quasi-differential operator according to different physical situations.

With f = 0 and

, (1.3) becomes weakly dissipative Camassa-Holm equation

(1.4)

where γ > 0 is a constant.

The local well-posedness, global existence, and blow-up phenomena of the Cauchy problem of (1.4) on the line [40] and on the circle [41] were studied. A new global existence result and a new blow-up result for strong solutions to this equation with certain profiles are presented recently [42]. We found that the behaviors of (1.4) are similar to the Camassa-Holm equation in a finite interval of time, such as the local well-posedness and the blow-up phenomena, and that there are considerable differences between (1.4) and the Camassa-Holm equation in their long-time behaviors. The global solutions of (1.4) decay to zero as time goes to infinity provided the potential is of one sign (see [40, 41]). This long-time behavior is an important feature that the Camassa-Holm equation does not possess. It is well known that the Camassa-Holm equation has peaked traveling wave solutions. But the fact that any global solution of (1.4) decays to zero means that there are no traveling wave solutions of (1.4).

Another difference between (1.4) and the Camassa-Holm equation is that (1.4) does not have the following conservation laws

(1.5)

which play an important role in the study of the Camassa-Holm equation.

Equation (1.4) has the same blow-up rate as the Camassa-Holm equation does when the blow-up occurs [41]. This fact shows that the blow-up rate of the Camassa-Holm equation is not affected by the weakly dissipative term, but the occurrence of blow-up of (1.4) is affected by the dissipative parameter [40, 41].

In the paper, we would like to consider the following weakly dissipative Camassa-Holm equation

(1.6)

(1.7)

(1.8)

where

is the weakly dissipative term, γ > 0 is a constant, and the forcing term f is ω-periodic in time t and L-periodic in spatial x. Without loss of generality, we assume further ∫_Ωf(t, x)dx = 0, where Ω = [0, L]. When system is periodically dependent on time t, we want to know whether there exists time-periodic solution with the same period for the system. In many nonlinear evolution equations, the study of time-periodic solution has attracted considerable interest (e.g., [43–45]). In this paper, we will prove that (1.6)–(1.8) have a solution by using the Galerkin method [46], and Leray-Schauder fixed point theorem [44].

Our paper is organized as follows. In Section 2, we give some notations and definition of some space used in this paper. In Section 3, we prove the existence of the approximate solution and give uniform a priori estimates needed where we prove the convergence of a sequence of the approximate solution. Section 4 is devoted to the study of the existence and uniqueness of time-periodic solution for (1.6)–(1.8).

2. Preliminaries

Before starting our work, it is appropriate to introduce some notations and inequalities that will be used in the paper.

Let X be a Banach space, we denote by C^k(ω; X) the set of ω-periodic X-valued measurable functions on ℝ¹ with continuous derivatives up to order k. The norm in the space C^k(ω; X) is .

For 1 ≤ p ≤ ∞, the space L^p(ω; X) is the set of ω-periodic X-valued measurable functions on ℝ such that

(2.1)

The space W^k,p(ω; X) denote the set of functions which belong to L^p(ω; X) together with their derivatives up to order k, and we write W^k,2(ω; X) = H^k(ω; X) in particular when X is a Hilbert space.

L^p(Ω) and H^m(Ω) are classical Sobolev spaces. For simplicity, we write by ∥·∥_p as p ≠ 2 and by ∥·∥.

The following inequalities (see [47]) will be used in the proofs later

(2.2)

(2.3)

where D^ju = (∂^ju)/(∂x^j), 1/p = j + θ(1/2 − m) + (1 − θ)(1/2) as 0 ≤ j < m, j/m ≤ θ ≤ 1.

(2.4)

3. A Priori Estimates

In this section, we first prove that (1.6)–(1.8) have a sequence of approximate solutions , then give a prior, estimates about .

We denote the unbounded linear operator Au = −u_xx on X = L²∩{u∣u(x + L) = u(x), ∫_Ωu dx = 0}, then the set of all linearly independent eigenvectors

of A, that is, Aw_j = λ_jw_j, with 0 < λ₁ ≤ λ₂ ≤ ⋯≤λ_j → ∞, is an orthonormal basis of L²(Ω). For any n and a group of function

, where a_jn(t)(j = 1,2, …, n) ∈ C¹(ω; ℝ), the function

is called an approximate solution to (1.6)–(1.8) if it satisfies the equation as follows:

(3.1)

where Nu_n = −3u_nu_nx + 2u_nxu_nxx + u_nu_nxxx and H_n = span {w₁, w₂, …, w_n}. By the classical theory of ordinary differential equations, for any fixed

, the equation (u_nt − u_nxxt + γ(u_n − u_nxx), w_j) = (Nv_n + f, w_j), j = 1, …, n has a unique ω-periodic solution u_n and the mapping F : v_n → u_n is continuous and compact in C¹(ω; H_n). Hence by Leray-Schauder fixed point theorem, we want to prove the existence of an approximate solution only to show

for all possible solution of (3.1) replaced by λNu_n(0 ≤ λ ≤ 1) instead of nonlinear term Nu_n, where c is a constant which only depends on L, ε, ω, γ, and f.

Lemma 3.1. If f ∈ C¹(ω; H⁻¹(Ω)), then

(3.2)

where c₁ is a constant which only depends on L, ω, ε, γ, k₃, and f,

and d₁ = min {2γ, 2γ − ε} > 0.

Proof. Multiplying (3.1) by a_jn(t) and summing up over j from 1 to n, we obtain

(3.3)

Then, we can get

(3.4)

Notice that , .

From Young’s inequality, we have , where ε > 0 is a constant.

According to the above relations, we can derive from (3.4) that

(3.5)

where d₁ = min {2γ, 2γ − ε} > 0.

Considering the time periodicity of u_n and integrating (3.5) over [0, ω], we get

(3.6)

Hence, there exists t^* ∈ [0, ω) such that .

From (3.5), we have .

Integrating the above inequality with respect to t from t^* to t ∈ [t^*, t^* + ω], we deduce that

(3.7)

Hence, we infer

(3.8)

which concludes our proof.

From Lemma 3.1 and Leray-Schauder fixed point theorem, (3.1) has solution , which is also a sequence of approximate solutions of (1.6)–(1.8). In order to obtain the convergence of sequence , we need to give a priori estimates for the high-order derivers of .

Lemma 3.2. If f ∈ C¹(ω; H⁻¹(Ω)), then

(3.9)

where c₂ is a constant which only depends on L, ω, ε, γ, λ_n, k₁, k₂, k₃, and f,

and d₂ = min {2γ − (13/2)ελ_n, 2γ − (21/2)ε} > 0.

Proof. Multiplying (3.1) by −λ_ja_jn(t) and summing up over j from 1 to n, we have

(3.10)

The above equation yields

(3.11)

From Young’s inequality, we have

(3.12)

where ε > 0 is a constant.

From (2.2), (3.8), and Young’s inequality, we can deduce that

(3.13)

From (2.3), (3.8), Cauchy-Schwarz inequality, Young’s inequality, and Lemma 3.1, we get

(3.14)

(3.15)

Taking (3.11)–(3.15) into account, we can infer that

(3.16)

where d₂ = min {2γ − (13/2)ελ_n, 2γ − (21/2)ε} > 0.

Integrating (3.16) about t from 0 to ω and considering the time periodicity of u_n, we get

(3.17)

Hence, there exists t^* ∈ [0, ω) such that

(3.18)

From (3.16), we have

(3.19)

Then we can obtain

(3.20)

which concludes our proof.

In the following, we continue to establish a priori estimates for high-order derivers of the approximate solution by an inductive argument.

Lemma 3.3. For any k ≥ 0, if f ∈ C¹(ω; H^k−1(Ω)), then

(3.21)

where c is a constant which only depends on L, ω, ε, γ, λ_n, k, k₁, k₂, k₃, f and d₃ = {2γ − 16ελ_n, 2γ − 14ε} > 0.

Proof. By Lemma 3.2, we know the conclusion of Lemma 3.3 holds for k = 0.

Assume that for k ≤ m − 1(m ≥ 2) the conclusion of Lemma 3.3 holds, we want to prove that the same statement holds for k = m also.

Multiplying (3.1) by and summing up over j from 1 to n, we have

(3.22)

Follow the same methods discussed in Lemma 3.2, we have

(3.23)

From the conclusion of Lemma 3.3 for k ≤ m − 1, (2.2), (2.4) and Young’s inequality, we can deduce that

(3.24)

Similarly, we can also deduce that

(3.25)

From the conclusion of Lemma 3.3 for k ≤ m − 1, Young’s inequality and (2.3), we have

(3.26)

Combining (3.25) and the above inequality, we can get

(3.27)

Similarly,

(3.28)

Taking (3.22)–(3.24) and (3.27)-(3.28) into account, we can deduce that

(3.29)

From the above relation, we can infer

(3.30)

where d₃ = {2γ − 16ελ_n, 2γ − 14ε} > 0.

Integrating (3.30) about t from 0 to ω, there exists t^* ∈ [0, ω) such that

(3.31)

From (3.30), we have

(3.32)

Integrating the above inequality from t^* to t ∈ [t^*, t^* + ω] and with (3.31), we can easily obtain

(3.33)

The proof is completed.

Lemma 3.4. For any k ≥ 0, if f ∈ C¹(ω; H^k+1(Ω)), then

(3.34)

where c is a constant which only depends on L, ω, ε, γ, λ_n, k, k₁, k₂, k₃, and f.

Proof. We first prove the conclusion of Lemma 3.4 holds for k = 0. Multiplying (3.1) by and summing up over j from 1 to n, we have

(3.35)

By Lemma 3.3, if f ∈ C¹(ω; H¹(Ω)), then we have . Hence,

(3.36)

Therefore, from (3.35) and (3.36), it is easy to know that

(3.37)

Assume that the conclusion of Lemma 3.4 holds for k ≤ m(m ≥ 1), we want to prove that the conclusion of Lemma 3.4 also holds for k = m + 1.

Multiplying (3.1) by and summing up over j from 1 to n, we have

(3.38)

By Lemma 3.3, if f ∈ C¹(ω; H^m+2(Ω)), then for k ≤ m + 5. Hence,

(3.39)

Taking (3.38) and (3.39) into account, it follows

(3.40)

This completes the proof of Lemma 3.4 by an inductive argument.

4. Existence and Uniqueness of Time-Periodic Solution

We have proved that (1.6)–(1.8) have a sequence of approximate solutions . In this section, we want to prove that the sequence converges and the limit is a solution of (1.6)–(1.8).

By Lemmas 3.1–3.4 and standard compactness arguments, we conclude that there is a subsequence which we denote also by {u_n} such that for any K ≥ 0, if f ∈ C¹(ω; H^k+1(Ω)), we have

(4.1)

From the above lemmas, we know that the nonlinear terms are well defined

(4.2)

as n → ∞, uniformly in t,

(4.3)

as n → ∞, uniformly in t,

(4.4)

as n → ∞, uniformly in t.

Consequently, it follows that

(4.5)

Thanks to the estimates obtained in the previous section, we have

(4.6)

a.e. on ℝ¹ × Ω.

So we obtain that the existence of time periodic solution for (1.6)–(1.8), which is the following theorem.

Theorem 4.1. Given f ∈ C¹(ω; H^k+1(Ω)), k ≥ 0, there exists a time periodic solution u(t, x) to (1.6)–(1.8), such that u(t, x) ∈ L^∞(ω; H^k+4(Ω))∩W^1,∞(ω; H^k(Ω)).

Under the assumption of Theorem 4.1, we are unable to prove the uniqueness of the solution for (1.6)–(1.8). But if we assume that M is sufficiently small, then the result can be obtained.

Theorem 4.2. Suppose that the assumption in Theorem 4.1 holds. If M is sufficiently small, then the time periodic solution of (1.6)–(1.8) in Theorem 4.1 is unique.

Proof. Let u and be any two time periodic solutions of (1.6)–(1.8). With , we can get from (1.6) that

(4.7)

Taking the inner product of (4.7) with v, we have

(4.8)

Since,

(4.9)

(4.10)

(4.11)

Hence, if M is sufficient small such that , , then it follows from (4.8)–(4.11), we get

(4.12)

where ρ ≥ 0 is suitable constant.

Applying Gronwall’s inequality, we derive that

(4.13)

Since v is ω-periodic in t, then for any positive integer m we have

(4.14)

Then we can infer that

(4.15)

It follows from v(0) = v_x(0) = 0 that , which completes the proof of Theorem 4.2.

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Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation

Abstract

1. Introduction

2. Preliminaries

3. A Priori Estimates

4. Existence and Uniqueness of Time-Periodic Solution

References

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Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation

Abstract

1. Introduction

2. Preliminaries

3. A Priori Estimates

4. Existence and Uniqueness of Time-Periodic Solution

References

References

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