On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System

Abstract

We establish the local well-posedness for the viscous two-component Camassa-Holm system. Moreover, applying the energy identity, we obtain a global existence result for the system with (𝑢₀,𝜂₀)∈𝐻¹(ℝ)×𝐿²(ℝ).

1. Introduction

We are interested in the global well-pose dness of the initial value problesm associated to the viscous version of the two-component Camassa-Holm shallow water system [1–3], namely,𝑚_𝑡+𝑢𝑚_𝑥+2𝑢_𝑥𝑚−𝐴𝑢_𝑥+𝜌𝜌_𝑥=0,𝑡>0,𝑥∈ℝ,𝑚=𝑢−𝑢_𝑥𝑥,𝜌,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝜌_𝑥=0,𝑡>0,𝑥∈ℝ, (1.1) where the variable 𝑢(𝑡,𝑥) represents the horizontal velocity of the fluid or the radial stretch related to a prestressed state, and 𝜌(𝑡,𝑥) is related to the free surface elevation from equilibrium or scalar density with the boundary assumptions, 𝑢→0 and 𝜌→1 as |𝑥|→∞. The parameter 𝐴>0 characterizes a linear underlying shear flow, so that (1.1) models wave-current interactions [4–6]. All of those are measured in dimensionless units.

Set 𝑝(𝑥)∶=(1/2)𝑒^−|𝑥|, 𝑥∈ℝ. Then (1−𝜕²_𝑥)⁻¹𝑓=𝑝∗𝑓 for all 𝑓∈𝐿²(ℝ), where ∗ denotes the spatial convolution. Let 𝜂=𝜌−1, (1.1) can be rewritten as a quasilinear nonlocal evolution system of the type𝑢_𝑡+𝑢𝑢_𝑥=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝑢²_𝑥1−𝐴𝑢+2𝜂²𝜂+𝜂,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝜂_𝑥+𝜂𝑢_𝑥+𝑢_𝑥=0,𝑡>0,𝑥∈ℝ. (1.2)

The system (1.1) without vorticity, that is, 𝐴=0, was also rigorously justified by Constantin and Ivanov [1] to approximate the governing equations for shallow water waves. The multipeakon solutions of the same system have been constructed by Popivanov and Slavova [7], and the corresponding integral surface is partially ruled. Chen et al. [8] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of AKNS hierarchy. More recently, Holm et al. [9] proposed a modified two-component Camassa-Holm system which possesses singular solutions in component 𝜌. Mathematical properties of (1.1) with 𝐴=0 have been also studied further in many works. For example, Escher et al. [10] investigated local well-posedness for the two-component Camassa-Holm system with initial data (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠≥2 and derived some precise blow-up scenarios for strong solutions to the system. Constantin and Ivanov [1] provided some conditions of wave breaking and small global solutions. Gui and Liu [11] recently obtained results of local well-posedness in the Besov spaces and wave breaking for certain initial profiles. More recently, Gui and Liu [12] studied global existence and wave-breaking criteria for the system (1.2) with initial data (𝑢₀,𝜌₀−1)∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠>(3/2).

In this paper, we consider the global well-posedness of the viscous two-component Camassa-Holm system𝑢_𝑡+𝑢𝑢_𝑥−𝑢_𝑥𝑥=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝑢²_𝑥1−𝐴𝑢+2𝜂²𝜂+𝜂,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝜂_𝑥+𝜂𝑢_𝑥+𝑢_𝑥𝑢()=0,𝑡>0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥)𝜂(),𝑥∈ℝ,0,𝑥=𝜂₀(𝑥),𝑥∈ℝ. (1.3)

The goal of the present paper is to study global existence of solutions for (1.3) to better understand the properties of the two-component Camassa-Holm system (1.2). We state the main result as follows.

Theorem 1.1. For (𝑢₀,𝜂₀)∈𝐻¹(ℝ)×𝐿²(ℝ), there exists a unique global solution (𝑢,𝜂) of (1.3) such that 𝑢(𝑡,𝑥)∈𝐶[0,∞);𝐿²_𝑥(ℝ)∩𝐶(0,∞);𝐻¹_𝑥(ℝ,𝜂()[))𝑡,𝑥∈𝐶0,∞;𝐿²_𝑥(ℝ). (1.4)

To proof Theorem 1.1, we will first establish global well-posedness of the following regularized two-component system with 𝜀>0 given:𝑢_𝑡+𝑢𝑢_𝑥−𝑢_𝑥𝑥=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝑢²_𝑥1−𝐴𝑢+2𝜂²𝜂+𝜂,𝑡>0,𝑥∈ℝ,_𝑡−𝜀𝜂_𝑥𝑥+𝑢𝜂_𝑥+𝜂𝑢_𝑥+𝑢_𝑥𝑢()=0,𝑡>0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥)𝜂(),𝑥∈ℝ,0,𝑥=𝜂₀(𝑥),𝑥∈ℝ, (1.5)

that is,𝑚_𝑡−𝑚_𝑥𝑥+𝑢𝑚_𝑥+2𝑢_𝑥𝑚−𝐴𝑢_𝑥+𝜂𝜂_𝑥+𝜂_𝑥𝜂=0,𝑡>0,𝑥∈ℝ,_𝑡−𝜀𝜂_𝑥𝑥+𝑢𝜂_𝑥+𝜂𝑢_𝑥+𝑢_𝑥=0,𝑡>0,𝑥∈ℝ,𝑚=𝑢−𝑢_𝑥𝑥𝑢(),𝑡≥0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥),𝜂(0,𝑥)=𝜂₀(𝑥),𝑥∈ℝ. (1.6) Due to the Duhamel’s principle, we can also rewrite (1.6) as an integral equation𝑢(𝑡,𝑥)=𝑒^{𝑡𝜕2𝑥}𝑢₀+^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝜂()𝑢,𝜂𝑑𝜏,𝑡,𝑥=𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀+^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂𝑑𝜏, (1.7) where𝑔(𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂)=−𝑢𝜕_𝑥𝜂−𝜂𝜕_𝑥𝑢−𝜕_𝑥𝑢, 𝑓𝑢,𝜕_𝑥𝑢,𝜂=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝜕_𝑥𝑢²1−𝐴𝑢+2𝜂²−1+𝜂2𝜕_𝑥𝑢², (1.8)𝑒^{𝑡𝜕2𝑥}𝑢₀=(𝑒^{−4𝜋2𝑡𝜉}𝑢₀(𝜉))^∨, 𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀=(𝑒^{−4𝜋2𝜀𝑡𝜉}𝜂₀(𝜉))^∨, here and in what follows, we denote the Fourier (or inverse Fourier) transform of a function 𝑓 by 𝑓 (or 𝑓^∨).

The remainder of the paper is organized as follows. In Section 2, we will set up and introduce some estimates for the nonlinear part of (1.5). In Section 3, we will get the local well-posedness of (1.3) by constructing the global well-posedness of (1.5) using the contraction argument and energy identity. The last section is devoted to the proof of Theorem 1.1.

2. Preliminaries

We will list some lemmas needed in Section 3. First, we state the following lemma which consists of the crucial inequality involving the operator 𝜕_𝑥(1−𝜕²_𝑥)⁻¹.

Lemma 2.1 (see [13]). For 𝑔,ℎ∈𝐿²(ℝ), 𝜕_𝑥1−𝜕²_𝑥⁻¹𝑔ℎ_𝐿2𝑥≤𝑐𝑔_𝐿2𝑥‖ℎ‖_𝐿2𝑥, (2.1) or more generally 𝜕_𝑥^𝑠1−𝜕²_𝑥⁻¹𝑔ℎ_𝐿2𝑥≤𝑐𝑔_𝐿2𝑥‖ℎ‖_𝐿2𝑥, (2.2) for all 𝑠<3/2.

The next two lemmas are regarding the nonlinear part of (1.5).

Lemma 2.2. Consider the following: ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥},_𝑡0𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤𝐶𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}. (2.3)(2.4)

Proof. Let us prove (2.3) firstly. Thanks to Lemma 2.1, the Sobolev embedding theorem 𝐻¹_𝑥(ℝ)↪𝐿^∞_𝑥(ℝ), and the Hölder’s inequality, we have ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤sup_{[]𝑡∈0,𝑇}^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)_𝐿2𝑥≤𝑑𝜏_𝑇0‖𝑢‖²_𝐿2𝑥1𝑑𝑡+2^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+𝐴^𝑇₀‖𝑢‖_𝐿2𝑥1𝑑𝑡+2^𝑇₀𝜂²_𝐿2𝑥+𝑑𝑡_𝑇0𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝑢𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡, (2.5) which yields that ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤^𝑇₀‖𝑢‖²_𝐿2𝑥1𝑑𝑡+2^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+𝐴𝑇^1/2𝑇₀‖𝑢‖²_𝐿2𝑥𝑑𝑡^1/2+12^𝑇₀𝜂²_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜂²_𝐿2𝑥𝑑𝑡^1/2+^𝑇₀‖𝑢‖_𝐻1𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑢‖𝑑𝑡≤𝐶‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥}. (2.6) Similarly, we can get ^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤sup_{[]𝑡∈0,𝑇}^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)_𝐿2𝑥≤𝑑𝜏_𝑇0𝑢𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥≤𝑑𝑡_𝑇0‖𝑢‖_𝐿∞𝑥𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂_𝐿∞𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡^1/2, (2.7) and then ^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤^𝑇₀‖𝑢‖_𝐻1𝑥𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂_𝐻1𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡^1/2≤^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝜂_𝐿2𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡^1/2≤2^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+2^𝑇₀𝜕_𝑥𝜂²_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡^1/2, (2.8) which implies (2.4).

Lemma 2.3. Consider the following: ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤𝐶𝑇^1/2‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥},_𝑡0𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤𝐶𝑇^1/2𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}. (2.9)(2.10)

Proof. We mainly prove (2.9). For this, we have that ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤^𝑇_0ℝ^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓(𝜏,𝑥)𝑑𝜏²𝑑𝑥_𝐿∞𝑇𝑑𝑡^1/2≤𝑇^1/2𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}. (2.11) Therefore, applying Lemma 2.2, we can easily obtain (2.9).
Similarly, we can also obtain (2.10).

Let us state the following lemma, which was obtained in [13] (up to a slight modification).

Lemma 2.4. For any 𝑢₀∈𝐻¹(ℝ) and 𝛿>0, there exists 𝑇₁=𝑇₁(𝑢₀)>0 such that 𝜕_𝑥𝑒^{𝑡𝜕2𝑥}𝑢_{0𝐿2𝑇𝐿2𝑥}=^𝑇1_0ℝ𝜕_𝑥𝑒^{𝑡𝜕2𝑥}𝑢₀²𝑑𝑥𝑑𝑡^1/2≤𝛿. (2.12) For any 𝜂₀∈𝐿²(ℝ), 𝜀>0, and 𝛿>0, there exists 𝑇₂=𝑇₂(𝜂₀,𝜀)>0 such that 𝜕_𝑥𝑒^{𝜀𝑡𝜕2𝑥}𝜂_{0𝐿2𝑇𝐿2𝑥}=^𝑇2_0ℝ𝜕_𝑥𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀²𝑑𝑥𝑑𝑡^1/2≤𝛿. (2.13)

Next, we consider the nonlinear part of (1.7). When written as 𝑣=𝑢−𝑒^{𝑡𝜕2𝑥}𝑢₀, 𝜇=𝜂−𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀, then we have𝜕_𝑡𝑣=𝜕²_𝑥𝑣+𝑓𝑢,𝜕_𝑥,𝜕𝑢,𝜂_𝑡𝜇=𝜀𝜕²_𝑥𝜇+𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂,𝑣()𝜇()𝑥,0=0,𝑥,0=0, (2.14) that is,𝑣(𝑥,𝑡)=^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥()𝜇()=𝑢,𝜂𝜏,𝑥𝑑𝜏,𝑥,𝑡^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏. (2.15)

First, we have the following basic estimates.

Lemma 2.5. Consider the following: 𝑓_{𝐿1𝑇𝐿2𝑥}≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥},𝑔_{𝐿1𝑇𝐿2𝑥}≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}. (2.16)(2.17)

Proof. First, let us prove (2.16), 𝑓_{𝐿1𝑇𝐿2𝑥}=12𝜕_𝑥𝑢²+𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝜕_𝑥𝑢²1−𝐴𝑢+2𝜂²+𝜂_{𝐿1𝑇𝐿2𝑥}≤^𝑇₀12𝜕_𝑥(𝑢²)+𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝜕_𝑥𝑢²1−𝐴𝑢+2𝜂²+𝜂_𝐿2𝑥≤1𝑑𝑡2^𝑇₀𝜕_𝑥1−𝜕²_𝑥⁻¹𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥(1−𝜕²_𝑥)⁻¹(𝑢²)_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥1−𝜕²_𝑥⁻¹(𝐴𝑢)_𝐿2𝑥1𝑑𝑡+2^𝑇₀𝜕_𝑥1−𝜕²_𝑥⁻¹𝜂²_𝐿2𝑥+𝑑𝑡_𝑇0𝑢𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥1−𝜕²_𝑥⁻¹𝜂_𝐿2𝑥𝑑𝑡, (2.18) which implies 𝑓_{𝐿1𝑇𝐿2𝑥}≤12^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+^𝑇₀‖𝑢‖²_𝐿2𝑥𝑑𝑡+^𝑇₀‖𝑢‖_𝐿∞𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+𝐴^𝑇₀‖𝑢‖_𝐿2𝑥+1𝑑𝑡2^𝑇₀𝜂²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂_𝐿2𝑥𝑑𝑡, (2.19) then we can get that 𝑓_{𝐿1𝑇𝐿2𝑥}≤12^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+𝐶^𝑇₀‖𝑢‖²_𝐿2𝑥𝑑𝑡+^𝑇₀‖𝑢‖_𝐻1𝑥𝜕_𝑥𝑢_𝐿2𝑥+1𝑑𝑡2^𝑇₀𝜂²_𝐿2𝑥𝑑𝑡+𝑇^1/2𝑇₀𝜂²_𝐿2𝑥𝑑𝑡^1/2+𝐴𝑇^1/2𝑇₀‖𝑢‖²_𝐿2𝑥𝑑𝑡^1/2≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥}, (2.20) where we applied Lemma 2.1, Sobolev embedding theorem 𝐻¹_𝑥(ℝ)↪𝐿^∞_𝑥(ℝ), and Hölder’s inequality. This proves (2.16).
Next, we prove (2.17), 𝑔_{𝐿1𝑇𝐿2𝑥}=−𝑢𝜕_𝑥𝜂−𝜂𝜕_𝑥𝑢−𝜕_𝑥𝑢_{𝐿1𝑇𝐿2𝑥}≤^𝑇₀𝑢𝜕_𝑥𝜂+𝜂𝜕_𝑥𝑢+𝜕_𝑥𝑢_𝐿2𝑥≤𝑑𝑡_𝑇0𝑢𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥≤𝑑𝑡_𝑇0‖𝑢‖_𝐿∞𝑥𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂_𝐿∞𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡, (2.21) which yields that 𝑔_{𝐿1𝑇𝐿2𝑥}≤^𝑇₀‖𝑢‖_𝐻1𝑥𝜕_𝑥𝜂_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂_𝐻1𝑥𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥𝑑𝑡≤𝐶^𝑇₀‖𝑢‖²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜂²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝜂²_𝐿2𝑥𝑑𝑡+^𝑇₀𝜕_𝑥𝑢_𝐿2𝑥‖𝑢‖𝑑𝑡≤𝐶²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}, (2.22) and this ends the proof of (2.17).

Then we have some estimates for 𝜕_𝑥𝑣 and 𝜕_𝑥𝜇.

Lemma 2.6. Consider the following: 𝜕_𝑥𝑣_{𝐿2𝑇𝐿2𝑥}≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥}𝜕_𝑥,𝜇_{𝐿2𝑇𝐿2𝑥}≤𝐶𝜀^−1/2‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}. (2.23)(2.24)

Proof. We mainly prove (2.24). We have that 𝜇_{𝐿∞𝑇𝐿2𝑥}=^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔(𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂)(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤^𝑇₀𝑔_𝐿2𝑥𝑑𝑡=𝑔_{𝐿1𝑇𝐿2𝑥}. (2.25)
Multiply 𝜇 to the second equation of (2.14) and integrate with respect to 𝑥 over ℝ. After integration by parts, we have 𝑑𝑑𝑡_ℝ𝜇²2𝑑𝑥+𝜀_ℝ𝜕_𝑥𝜇²𝑑𝑥=_ℝ1𝜇𝑔𝑑𝑥,2_ℝ𝜇²(𝑇)1𝑑𝑥−2_ℝ𝜇²(0)𝑑𝑥+𝜀^𝑇_0ℝ𝜕_𝑥𝜇²𝑑𝑥𝑑𝑡=^𝑇_0ℝ𝜇𝑔𝑑𝑥𝑑𝑡, (2.26) for any 𝜀>0, which implies 𝜀^𝑇_0ℝ𝜕_𝑥𝜇²𝑑𝑥𝑑𝑡≤^𝑇_0ℝ𝜕𝜇𝑔𝑑𝑥𝑑𝑡,_𝑥𝜇²_{𝐿2𝑇𝐿2𝑥}≤1𝜀^𝑇₀𝜇_𝐿2𝑥𝑔_𝐿2𝑥1𝑑𝑡≤𝜀𝜇_{𝐿∞𝑇𝐿2𝑥}𝑔_{𝐿1𝑇𝐿2𝑥}. (2.27) By (2.25), together with (2.27), we have that 𝜕_𝑥𝜇_{𝐿2𝑇𝐿2𝑥}≤1𝜀^1/2𝑔_{𝐿1𝑇𝐿2𝑥}. (2.28) By Lemma 2.5, (2.24) follows.
Similarly, we can also obtain (2.23).

3. Local Well-Posedness

Let 𝑧∶=^𝑢_𝜂, 𝐴(𝑧)∶=^{𝑒2𝑥𝑡𝜕𝑢0}_{𝑒2𝑥𝜀𝑡𝜕𝜂0},𝐵(𝑧)∶=^𝑡0𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_𝑡0𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(),𝐷𝜏,𝑥𝑑𝜏₁∶=𝐶[0,𝑇);𝐿²_𝑥(ℝ)∩𝐶(0,𝑇);𝐻¹_𝑥(ℝ),𝐷₂∶=𝐶[0,𝑇);𝐿²_𝑥(ℝ),𝑋^𝑇_𝑎∶=𝑧∈𝐷₁×𝐷₂∶⟦𝑧⟧=‖𝑧−𝐴(𝑧)‖_𝐿^∞_{𝑇𝐿2𝑥}+𝜕_𝑥𝑧_{𝐿2𝑇𝐿2𝑥}+‖𝑧‖_𝐿²_{𝑇𝐿2𝑥}≤𝑎, (3.1) and define the mapping Φ∶𝑋^𝑇_𝑎→𝑋^𝑇_𝑎 byΦ(𝑧)=𝐴(𝑧)+𝐵(𝑧). (3.2)

Theorem 3.1. For any 𝜀>0, there exist 𝑇=𝑇_𝜀>0 and 𝑎>0 such that Φ(𝑋^𝑇_𝑎)⊆𝑋^𝑇_𝑎. In addition, Φ∶𝑋^𝑇_𝑎→𝑋^𝑇_𝑎 is a contraction mapping.

Proof. We first need to show that the map is well defined for some appropriate 𝑎 and 𝑇. Let 𝑧∈𝑋^𝑇_𝑎, then we have ⟦Φ𝑧⟧=‖Φ𝑧−𝐴(𝑧)‖_𝐿^∞_{𝑇𝐿2𝑥}+‖Φ𝑧‖_𝐿²_{𝑇𝐿2𝑥}+𝜕_𝑥(Φ𝑧)_{𝐿2𝑇𝐿2𝑥}. (3.3)
Considering the terms in (3.3) one by one, from Lemma 2.2, the first term in (3.3) can be estimated as follows: ‖Φ𝑧−𝐴(𝑧)‖_𝐿^∞_{𝑇𝐿2𝑥}=^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}+^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿∞𝑇𝐿2𝑥}≤𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝐶𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥}≤𝐶⟦𝑧⟧²+𝐶𝑇^1/2⟦𝑧⟧. (3.4) From Lemma 2.3, the second term in (3.3) can be estimated as follows: ‖Φ𝑧‖_𝐿²_{𝑇𝐿2𝑥}=𝑒^{𝑡𝜕2𝑥}𝑢₀+^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}+𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀+^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤𝑒^{𝑡𝜕2𝑥}𝑢_{0𝐿2𝑇𝐿2𝑥}+^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}+𝑒^{𝜀𝑡𝜕2𝑥}𝜂_{0𝐿2𝑇𝐿2𝑥}+^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}, (3.5) which implies ‖Φ𝑧‖_𝐿²_{𝑇𝐿2𝑥}≤𝑇^1/2𝑢_0𝐿2𝑥+^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜂_0𝐿2𝑥+^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤𝑇^1/2𝑧_0𝐿2𝑥+𝐶𝑇^1/2⟦𝑧⟧²+𝐶𝑇⟦𝑧⟧. (3.6) From Lemma 2.6, the third term in (3.3) can be estimated as follows: 𝜕_𝑥(Φ𝑧)_{𝐿2𝑇𝐿2𝑥}=𝜕_𝑥𝑒^{𝑡𝜕2𝑥}𝑢₀+𝜕_𝑥^𝑡₀𝑒^{(𝑡−𝜏)𝜕2𝑥}𝑓𝑢,𝜕_𝑥𝑢,𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑒^{𝜀𝑡𝜕2𝑥}𝜂₀+𝜕_𝑥^𝑡₀𝑒^{𝜀(𝑡−𝜏)𝜕2𝑥}𝑔𝑢,𝜕_𝑥𝑢,𝜂,𝜕_𝑥𝜂(𝜏,𝑥)𝑑𝜏_{𝐿2𝑇𝐿2𝑥}≤𝜕_𝑥𝑒^{𝑡𝜕2𝑥}𝑢_{0𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑣_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑒^{𝜀𝑡𝜕2𝑥}𝜂_{0𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜇_{𝐿2𝑇𝐿2𝑥}≤2𝛿+𝐶‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2‖𝑢‖_𝐿²_{𝑇𝐿2𝑥}+𝜂_{𝐿2𝑇𝐿2𝑥}+𝐶𝜀^1/2‖𝑢‖²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}+𝜂²_{𝐿2𝑇𝐿2𝑥}+𝜕_𝑥𝜂²_{𝐿2𝑇𝐿2𝑥}+𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇𝐿2𝑥}1≤2𝛿+𝐶1+𝜀^1/2⟦𝑧⟧²+𝐶𝑇^1/211+𝜀^1/2⟦𝑧⟧. (3.7) Combining (3.4)–(3.7), we have that ⟦Φ𝑧⟧≤2𝛿+𝑇^1/2𝑧_0𝐿2𝑥+𝐶⟦𝑧⟧²+𝐶𝑇^1/2⟦𝑧⟧²1+𝐶1+𝜀^1/2⟦𝑧⟧²+𝐶𝑇⟦𝑧⟧+𝐶𝑇^1/212+𝜀^1/2⟦𝑧⟧≤2𝛿+𝑇^1/2𝑧_0𝐿2𝑥+𝐶2+𝑇^1/211+𝜀^1/2𝑎²+𝐶𝑇𝑎+𝐶𝑇^1/212+𝜀^1/2𝑎. (3.8)
With appropriate values of 𝛿, 𝑎, and 𝑇, we are able to have that ⟦Φ𝑧⟧≤𝑎, that is, Φ∶𝑋^𝑇_𝑎→𝑋^𝑇_𝑎 is well defined.
Similar to the above argument, we can show that Φ∶𝑋^𝑇_𝑎→𝑋^𝑇_𝑎 is a contraction mapping, Φ𝑧₁−Φ𝑧₂≤𝐶′𝑧₁−𝑧₂, (3.9) where 𝐶′=𝐶′(𝑇,𝑎,𝜀,‖𝑧₁‖_{𝐿2𝑇𝐿2𝑥},‖𝑧₂‖_{𝐿2𝑇𝐿2𝑥},‖𝜕_𝑥𝑧₁‖_{𝐿2𝑇𝐿2𝑥},‖𝜕_𝑥𝑧₂‖_{𝐿2𝑇𝐿2𝑥}) can be chosen as 0<𝐶′<1 with appropriate values of 𝑇 and 𝑎.

Theorem 3.2. For any 𝜀>0 and (𝑢₀,𝜂₀)∈𝐻¹(ℝ)×𝐿²(ℝ), there exist a 𝑇=𝑇(𝑢₀,𝜂₀,𝜀)>0 and a unique solution (𝑢_𝜀,𝜂_𝜀) of (1.5) such that 𝑢_𝜀(𝑥,𝑡)∈𝐶[0,𝑇);𝐿²_𝑥(ℝ)∩𝐶(0,𝑇);𝐻¹_𝑥(ℝ,𝜂_𝜀)(𝑥,𝑡)∈𝐶[0,𝑇);𝐿²_𝑥(ℝ). (3.10)

Proof. Theorem 3.2 is merely Theorem 3.1 with a standard uniqueness argument.

Theorem 3.3. For any 𝜀>0 and (𝑢₀,𝜂₀)∈𝐻¹(ℝ)×𝐿²(ℝ), there exists a unique global solution (𝑢_𝜀,𝜂_𝜀) of (1.5) such that 𝑢_𝜀(𝑥,𝑡)∈𝐶[0,∞);𝐿²_𝑥(ℝ)∩𝐶(0,∞);𝐻¹_𝑥(ℝ,𝜂_𝜀)(𝑥,𝑡)∈𝐶[0,∞);𝐿²_𝑥(ℝ). (3.11)

Proof. To prove Theorem 3.3, we need only to establish the a priori energy identity. Multiplying the first equation in (1.6) by 𝑢 and integrating by parts (with respect to 𝑥 over ℝ), we have that 12𝑑𝑑𝑡_ℝ𝑢²+𝑢²_𝑥𝑑𝑥+_ℝ𝑢²_𝑥+𝑢²_𝑥𝑥1𝑑𝑥−2_ℝ𝜂²𝑢_𝑥𝑑𝑥+_ℝ𝑢𝜂_𝑥𝑑𝑥=0, (3.12) where we used the relation 𝑚=𝑢−𝑢_𝑥𝑥 and ∫_ℝ𝑢²𝑢_𝑥𝑑𝑥=0. Multiplying the first equation in (1.6) by 𝜂 and integrating by parts (with respect to 𝑥 over ℝ), we have that 12𝑑𝑑𝑡_ℝ𝜂²𝑑𝑥+𝜀_ℝ𝜂²_𝑥1𝑑𝑥+2_ℝ𝜂²𝑢_𝑥𝑑𝑥−_ℝ𝑢𝜂_𝑥𝑑𝑥=0. (3.13) From (3.12) and (3.13), we obtain the energy identity 12𝑑𝑑𝑡_ℝ𝑢²+𝑢²_𝑥+𝜂²𝑑𝑥+_ℝ𝑢²_𝑥+𝑢²_𝑥𝑥𝑑𝑥+𝜀_ℝ𝜂²_𝑥𝑑𝑥=0, (3.14) which gives rise to the following inequality independent of 𝜀 and 𝑇: sup_{[)𝑡∈0,𝑇}‖𝑢(𝑡,⋅)‖²_𝐻1+𝜂(𝑡,⋅)²_𝐿2+2𝑢_𝑥²_{𝐿2𝑇𝐻1}≤𝑢₀²_𝐻1+𝜂₀²_𝐿2. (3.15) According to Theorem 3.2 and the energy inequality (3.15), one can extend the local solution to the global one by a standard contradiction argument, which completes the proof of Theorem 3.3.

From Theorem 3.3, one has the local well-posedness of system (1.3).

Theorem 3.4. For (𝑢₀,𝜂₀)∈𝐻¹(ℝ)×𝐿²(ℝ), there exist a 𝑇=𝑇(𝑢₀,𝜂₀)>0 and a unique local solution (𝑢,𝜂) of (1.3) such that 𝑢(𝑥,𝑡)∈𝐶[0,𝑇);𝐿²_𝑥(ℝ)∩𝐶(0,𝑇);𝐻¹_𝑥(ℝ,𝜂()[))𝑥,𝑡∈𝐶0,𝑇;𝐿²_𝑥(ℝ). (3.16)

Similar to the proof of Theorem  4.1 in [12] (up to a slight modification), we may get the following.

Theorem 3.5. Let 𝑧₀=(𝑢₀,𝜂₀)∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥1, and let 𝑧=(𝑢,𝜂) be the corresponding solution to (1.3). Assume that 𝑇^∗_𝑧0>0 is the maximal time of existence, then 𝑇^∗_𝑧0<∞⟹^𝑇∗𝑧0₀𝜕_𝑥𝑢(𝜏)_𝐿∞𝑑𝜏=∞. (3.17)

4. Proof of Theorem 1.1

Proof of Theorem 1.1. From Theorem 3.4, we have got the local solution of (1.3). On the other hand, according to the second equation of (1.3), we have 12𝑑𝜂𝑑𝑡^2𝐿2𝑥≤2𝜕_𝑥𝑢_𝐿∞𝑥𝜂²_𝐿2𝑥+𝜕_𝑥𝑢_𝐿2𝑥𝜂_𝐿2𝑥≤4𝜕_𝑥𝑢_𝐿∞𝑥𝜂²_𝐿2𝑥+𝜕_𝑥𝑢_𝐿2𝑥𝜂_𝐿2𝑥≤4𝜕_𝑥𝑢_𝐿∞𝑥𝜂²_𝐿2𝑥+𝜂²_𝐿2𝑥+14𝜕_𝑥𝑢²_𝐿2𝑥=4𝜕_𝑥𝑢_𝐿∞𝑥+1𝜂²_𝐿2𝑥+14𝜕_𝑥𝑢²_𝐿2𝑥. (4.1) An application of Gronwall’s inequality yields 𝜂²_𝐿2𝑥≤𝜂₀²_𝐿2𝑥+14𝜕_𝑥𝑢²_{𝐿2𝑇𝐿2𝑥}𝑒^{∫𝑡0(1+4‖𝜕𝑥𝑢‖𝐿∞𝑥)𝑑𝜏}. (4.2) Similar to the proof of Theorem 3.3, we may get the energy identity 12𝑑𝑑𝑡_ℝ𝑢²+𝑢²_𝑥+𝜂²𝑑𝑑𝑥+𝑑𝑡_ℝ𝑢²_𝑥+𝑢²_𝑥𝑥𝑑𝑥=0, (4.3) which implies sup_{[)𝑡∈0,𝑇}‖𝑢(𝑡,⋅)‖²_𝐻1+𝜂(𝑡,⋅)²_𝐿2+2𝑢_𝑥²_{𝐿2𝑇𝐻1}≤𝑢₀²_𝐻1+𝜂₀²_𝐿2. (4.4) Due to the Sobolev embedding theorem 𝐻¹(ℝ)↪𝐿^∞(ℝ) and (4.4), we obtain that for any 𝑇<+∞, ^𝑇₀𝜕_𝑥𝑢_𝐿∞𝑥𝑑𝑡≤^𝑇₀𝜕_𝑥𝑢_𝐻1𝑥𝑑𝑡≤𝑇^1/2𝜕_𝑥𝑢_{𝐿2𝑇(𝐻1𝑥)}<+∞. (4.5) Therefore, from Theorem 3.5, we deduce that the maximal existence time 𝑇=+∞. This proves Theorem 1.1.

Acknowledgments

The work of the author is supported in part by the NSF of China under Grant no. 11001111 and no. 11141003. The author would like to thank the referees for constructive suggestions and comments.