Abstract
In this paper, a wave-breaking criterion of strong solutions is acquired in the Soblev space \(H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})\) with \(s>\frac{3}{2}\) by employing the localization analysis in the transport equation theory, which is different from that of the two-component Camassa-Holm system.
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1 Introduction
The classical two-component Camassa-Holm system takes the form
where the variable \(u(t,x)\) represents the horizontal velocity of the fluid, and \(\rho(t,x)\) is related to the free surface elevation from equilibrium with the boundary assumptions \(u\rightarrow0\) and \(\rho\rightarrow1\) as \(| x|\rightarrow\infty\). System (1) was found originally in [1], but it was firstly derived rigorously by Constantin and Ivanov [2]. The system has bi-Hamiltonian structure and is completely integrable. Since the birth of the system, a large number of literature was devoted to the study of the two-component Camassa-Holm system. Some mathematical and physical properties of the system have been obtained. Chen et al. [3] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of the AKNS hierarchy. Escher et al. [4] used Kato’s theory to establish local well-posedness for the two-component system and presented some precise blow-up scenarios for strong solutions of the system. In [2], Constantin and Ivanov described sufficient conditions for wave-breaking and global solution to the system. Dynamics in the periodic case for system (1) were considered in [5]. It is worth mentioning that the wave-breaking criteria of strong solutions is determined in the lowest Soblev space \(H^{s}\) with \(s>\frac{3}{2}\) by applying the localization analysis in the transport equation theory [6]. The other results related to the system can be found in [7–15].
Inspired by the works mentioned, in this article, we consider a modified periodic two-component Camassa-Holm system on the circle \(\mathbb{S}\) with \(\mathbb{S=R/Z}\) (the circle of unit lengh):
where \(m=(1-\partial^{2}_{x})^{2}u\), and \(\mathbb{R}\) is the real number set. In fact, system (2) is a two-component generalization of the equation (if \(\rho=0\) in system (2))
Equation (3) was first derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to the \(H^{2}\) metric [16], and it is known as a modified Camassa-Holm equation and also viewed as a geodesic equation on some diffeomorphism group [16]. It is shown in [16] that the well-posedness and dynamics of Eq. (3) on the unit circle \(\mathbb{S}\) are significantly different from that of the Camassa-Holm equation. For example, Eq. (3) does not conform with blow-up solution in finite time.
As we know, differently from the Camassa-Holm equation, Eq. (3) has not blow-up solution. The motivation of the present paper is to find out whether or not system (2) has some similar dynamics as the classical two-component Camassa-Holm equation and Eq. (3) mathematically, for example, wave-breaking and global solution. One of the difficulties is the acquisition of the a priori estimates of \(\| u_{xx}\|_{L^{\infty}}\) and \(\| u_{xxx}\|_{L^{\infty}}\). This difficulty has been overcome by Lemmas 3.4 and 3.5. We mainly use the ideas of [6] to derive a wave-breaking criterion (see Theorem 1) of strong solutions for system (2) in the low Sobolev spaces \(H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})\) with \(s>\frac{3}{2}\), where a new conservation law is necessary. We need to point out that in the Sobolev spaces \(H^{s}(\mathbb{R})\times H^{s-1}(\mathbb{R})\) with \(s>\frac{3}{2}\), the wave-breaking of the solution for system (1) only depends on the slope of the component u of the solution [6]. However, since the slope of the component u of the solution is bounded by the Sobolev imbedding theorem \(H^{1}\hookrightarrow L^{\infty}\), the wave-breaking of the solution for system (2) is determined only by the slope of the component ρ of solution definitely in the low Sobolev spaces \(H^{s}(\mathbb {S})\times H^{s-1}(\mathbb{S})\) with \(s>\frac{3}{2}\) (see Theorem 1). This implies that there exists some difference between system (2) and the two-component Camassa-Holm equation. Moreover, this is quite different from Eq. (3) because Eq. (3) does not admit a blow-up solution in infinite time.
2 The main results
We denote by ∗ the convolution. Note that if \(g(x):=1+2\sum^{\infty}_{n=1}\frac{1}{1+2n^{2}+n^{4}}\cos(nx)\), then \((1-\partial^{2}_{x})^{-2}f=g\ast f\) for all \(f\in L^{2}(\mathbb{R})\), and \(g\ast m=u\). We let C denote all of different positive constants that depend on initial data. To investigate dynamics of system (2), we can rewrite system (2) in the form
The main result of the present paper is as follows.
Theorem 1
Let \(z_{0}=(u_{0},\rho_{0})\in H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})\), \(s>\frac{3}{2}\), and T be the maximal existence time of the solution \(z=(u,\rho)\) to system (4). Assume that \(m_{0}\in L^{2}(\mathbb{S})\) and \(T<\infty\). Then
3 Preliminaries
In order to prove Theorem 1, we first give some lemmas.
Lemma 3.1
([6, 17]) (1-D Moser-type estimates)
The following estimates hold:
-
(i)
For \(s\geq0\),
$$\begin{aligned} \| fg\|_{H^{s}}\leq C\bigl(\| f\| _{H^{s}}\| g\|_{L^{\infty}}+ \| f \|_{L^{\infty}}\| g\|_{H^{s}}\bigr). \end{aligned}$$(5) -
(ii)
For \(s>0\),
$$\begin{aligned} \| f\partial_{x}g\|_{H^{s}}\leq C\bigl(\| f\| _{H^{s+1}}\| g \|_{L^{\infty}}+ \| f\|_{L^{\infty}}\| \partial_{x}g \|_{H^{s}}\bigr). \end{aligned}$$(6) -
(iii)
For \(s_{1}\leq\frac{1}{2}\), \(s_{2}>\frac{1}{2}\), and \(s_{1}+s_{2}>0\),
$$\begin{aligned} \| f\partial_{x}g\|_{H^{s_{1}}}\leq C\| f\| _{H^{s_{1}}}\| g \|_{H^{s_{2}}}, \end{aligned}$$(7)
where C is a constant independent of f and g.
Lemma 3.2
Suppose that \(s>-\frac{d}{2}\). Let v be a vector field such that ∇v belongs to \(L^{1}([0,T]; H^{s-1})\) if \(s>1+\frac{d}{2}\) or to \(L^{1}([0,T]; H^{\frac{d}{2}}\cap L^{\infty})\) otherwise. Suppose also that \(f_{0}\in H^{s}\), \(F\in L^{1}([0,T]; H^{s})\), and that \(f\in L^{\infty}([0,T];H^{s})\cap C([0,T]; S')\) solves the d-dimensional linear transport equation
Then \(f\in C([0,T]; H^{s})\). More precisely, there exists a constant C depending only s, p, and d, and such that the following statements hold:
(1) If \(s\neq1+\frac{d}{2}\), then
or
with \(V(t)=\int_{0}^{t}\|\nabla v(\tau)\|_{H^{\frac{d}{2}}\cap L^{\infty}}\,d\tau\) if \(s<1+\frac{d}{2}\) and \(V(t)=\int_{0}^{t}\|\nabla v(\tau)\|_{H^{s-1}}\,d\tau\) else.
(2) If \(f=v\), then for all \(s>0\), estimates (9) and (10) hold with \(V(t)=\int_{0}^{t}\|\partial_{x}u (\tau)\|_{L^{\infty}}\,d\tau\).
Lemma 3.3
([6])
Let \(0<\sigma<1\). Suppose that \(f_{0}\in H^{\sigma}\), \(g\in L^{1}([0,T]; H^{\sigma})\), \(\nu, \partial_{x}\nu\in L^{1}([0,T]; L^{\infty})\), and \(f\in L^{\infty}([0,T]; H^{\sigma})\cap C([0,T];S')\) solves the 1-dimensional linear transport equation
Then \(f\in C([0,T]; H^{\sigma})\). More precisely, there exists a constant C depending only σ and such that the following statement holds:
or
with \(V(t)=\int_{0}^{t}(\| \nu(\tau)\|_{L^{\infty}}+\| \partial_{x}\nu(\tau)\|_{L^{\infty}})\,d\tau\).
Lemma 3.4
For all \(x\in\mathbb{R}\), the following statements hold:
and
Proof
Let \(g(x)\) be the Green function for the operator \((1-\partial^{2}_{x})^{2}\). Then from
we get
Hence,
which results in
From Cauchy integral test we have
It follows that
Now, we prove (ii). From the Fourier series we have
from which we get
On the other hand,
Hence, we have
□
Lemma 3.5
Let \(z_{0}=(u_{0},\rho_{0})\in H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})\) with \(s>\frac{3}{2}\). Suppose that T is the maximal existence time of solution \(z=(u,\rho)\) of system (4) with the initial data \(z_{0}\). Then, for all \(t\in[0, T)\), the following conservation law holds:
Moreover, assume that \(m_{0}\in L^{2}\). Then
and
Proof
Multiplying the first equation of system (2) by u and integrating by parts, we get
Multiplying the second equation of system (2) by ρ and integrating by parts, we get
which, together with (19), yields
which implies (16).
Next, we prove (17). Multiplying the first equation of system (2) by m and integrating by parts, we have
which results in
By the Hölder inequality we get from (23) that
Applying Gronwall’s inequality, we obtain
which, together with (16), yields
On the other hand, from Lemma 3.4 we deduce
It follows from (24) that
Similarly, we can obtain (18).
This completes the proof of Lemma 3.5. □
4 Proof of main theorem
Proof of Theorem 1
Using the maximal principle to the transport equation about ρ,
we have
Applying Gronwall’s inequality yields
Using the Sobolev embedding theorem \(H^{s}\hookrightarrow L^{\infty}\) (\(s>\frac{1}{2}\)), we get from Lemma 3.5 that
Therefore, we have
Next, we split the remaining proof of Theorem 1 into five steps.
Step 1. For \(s\in(\frac{3}{2},2)\), applying Lemma 3.3 to the second equation, we have
From (28) we obtain
On the other hand, using Lemma 3.2, we get from the first equation of system (4) that
From Lemma 3.4(b) of [19], we have
Hence, we get
which, together with (29), ensures that
Using Gronwall’s inequality, we have
From (27) and Lemma 3.5 we get
Therefore, if the maximal existence time \(T<\infty\) satisfies \(\int^{t}_{0}\| \rho_{x}\|_{L^{\infty}}\,d\tau<\infty\), then we get from (33) that
which completes the proof of Theorem 1 for \(s\in(\frac{3}{2},2)\).
Step 2. For \(s\in[2,\frac{5}{2})\), applying Lemma 3.2 to the second equation of system (4), we get
Using (28) results in
which, together with (30), yields
where \(\varepsilon\in(0,\frac{1}{2})\), and we used the fact that \(H^{\frac{1}{2}+\varepsilon}\hookrightarrow L^{\infty}\cap H^{\frac{1}{2}}\).
Using Gronwall’s inequality, we have
From (27) and Lemma 3.5 we get
Applying the argument as in step 1, we complete the proof of Theorem 1 for \(s\in[2,\frac{5}{2})\).
Step 3. For \(s\in(2,3)\), differentiating once the second equation of system (4) with respect to x, we have
Using Lemma 3.3, we get
where we used the estimates
and
Using (39), (30), and (29) (where \(s-1\) is replaced by \(s-2\)) yields
Applying Gronwall’s inequality, we have
From (27) and Lemma 3.5 we get
Using the argument as in step 1, we complete the proof of Theorem 1 for \(s\in (2,3)\).
Step 4. For \(s=k\in\mathbf{N}\), \(k\geq3\), differentiating \(k-2\) times the second equation of system (4) with respect to x, we obtain
Using Lemma 3.2, we get from (43) that
Since \(H^{1}\) is an algebra, we have
and
It follows that
From the Gagliardo-Nirenberg inequality we have that, for \(\sigma\in(0,1)\),
On the other hand, for \(\sigma\in(0,1)\), rewrite (29) as
which, together with (45), yields
where (46) was used.
which, together with (48), results in
Using Gronwall’s inequality, we get
If \(T<\infty\) satisfies \(\int^{T}_{0}\| \rho_{x}\|_{L^{\infty}}\,d\tau<\infty\), applying step 2 and the induction assumption, we obtain from Lemma 3.5 that \(\| u\|_{H^{s-1}}+\| \rho\|_{H^{1}} +\| u_{xx}\|_{L^{\infty}}+\| u_{xxx}\| _{L^{\infty}}\) is uniformly bounded. From (51) we get
which contradicts the assumption that \(T<\infty\) is the maximal existence time. This completes the proof of Theorem 1 for \(s=k\in N\) and \(k\geq3\).
Step 5. For \(s\in(k,k+1)\), \(k\in N\), and \(k\geq3\), differentiating \(k-1\) times the second equation of system (4) with respect to x, we obtain
Using Lemma 3.3 with \(s-k\in(0,1)\), we get from (52) that
For each \(\varepsilon\in(0,\frac{1}{2})\), using Lemma 3.1 (6) and the fact that \(H^{\frac{1}{2}+\varepsilon}\hookrightarrow L^{\infty}\), we have
and
Therefore, from (53), (54), and (55) we get
Applying Lemma 3.2 to the first equation of system (4) for \(s\in(k,k+1)\) with \(k\geq3\), we obtain
which, together with (56) and (29) (where \(s-1\) is replaced by \(s-k\)), gives
Using Gronwall’s inequality, we get
Noting that \(k-\frac{1}{2}+\varepsilon< k\), \(k-\frac{3}{2}+\varepsilon< k-1\), and \(k\geq3\) and applying step 4, we obtain that \(\| u\|_{H^{k-\frac{1}{2}+\varepsilon}}+\| \rho\|_{H^{k-\frac{3}{2}+\varepsilon}}\) is uniformly bounded. Therefore, we complete the proof of Theorem 1 for \(s\in (k,k+1)\), \(k\in N\), and \(k\geq3\).
So, the proof of Theorem 1 is completed. □
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Acknowledgements
The author thanks the referees for their valuable comments and suggestions. This work was supported by Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing [grant number 2013QZJ02], [grant number 2014QYJ03], Scientific Research Foundation of the Education Department of Sichuan province Project [grant number 16ZA0265], SUSER [grant number 2014RC03].
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Wang, Y. A wave breaking criterion for a modified periodic two-component Camassa-Holm system. J Inequal Appl 2016, 85 (2016). https://doi.org/10.1186/s13660-016-1023-2
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DOI: https://doi.org/10.1186/s13660-016-1023-2