On the singularity formation for a class of periodic higher-order Camassa–Holm equations

Abstract

The primary goal of this paper is to understand how higher-order nonlinearities affect the dispersive dynamics. As a prototype, we study a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation. We develop a delicate analysis to investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution.

Introduction

Dispersive dynamics are modeled by hyperbolic equations with regularizing higher order derivative terms corresponding to dispersion rather than dissipation. These equations describe a number of physical models including water waves, nonlinear optics, and Bose-Einstein condensates, and so on. One of the most prominent features of these systems is that the gradient catastrophe induced by progressively nonlinear steepening at the purely hyperbolic level can be resolved into an expanding, oscillatory wavetrain due to dispersive regularization, the best known example being the Korteweg-de Vries (KdV) equation [22].

Another source responsible for the regularization is the nonlocal effect, which may introduce enhanced smoothing and hence controls higher-order energy norms; see, for instance, the Benjamin-Bona-Mahoney (BBM) equation [3]. Such an effect may also de-singularize the resonance structure of the system by averaging and thus leads to global-in-time regularity. Examples include the rotating Navier-Stokes and Euler equations [1], [2], [16].

As the nonlinearity becomes more pronounced, on the other hand, the hyperbolic nature tends to dominate and triggers singularity. One of the well-known types of the singularity formation is wave breaking, as can be seen in the Whitham equation [11], [27], the Camassa–Holm (CH) equation [19], [6], [11], [13], the Degasperis-Procesi (DP) equation [15], [13], and the Novikov equation [24], [9], etc. Interestingly, higher nonlinearity can even lead to curvature blow-up, i.e. the second derivative of solution becomes unbounded in finite time while the solution and its gradient remain bounded, for example the modified Camassa–Holm (mCH) equation [18], [25], [20], [8] and the generalized modified Camassa–Holm (gmCH) equation [17], [9].

It is worth pointing out that the above mentioned equations possess a remarkable mathematical structure that can support both breaking and peaking waves – the phenomena emphasized by Whitham as “one of the most intriguing long-standing problems” [27]. Following the discoveries of these equations, in [26] the authors are motivated to seek generalizing models which admit multi-peaked solitary wave solutions and maintain a similar form expressed in terms of the momentum variable. In particular, they derive two families of equations

$$m_t+{(u^2-u_x^2)}^{p-1}{u}_{x}m+{\left(u{(u^2-u_x^2)}^{p-1}m\right)}_x=0,\text{ and }$$

$$m_t+{[{(u^2-u_x^2)}^{p}m]}_x=0,$$

where $m=u-u_{xx}$ is the momentum variable, and $p\in \mathbb{N}$. The first equation (1.1) contains the CH equation ($p=1$), and (1.2) generalizes the mCH equation ($p=1$). These two equations also share one of the Hamiltonian structures possessed by both the CH and mCH equations, as well as conservation laws for momentum and energy ($H^1$-norm). Apart from the focus on the singularity formation, the literature on the CH and mCH equations is oriented towards questions of relevance in the realm of integrable systems, geometric formulation and fluid dynamics. An exhaustive list is beyond the scope of this article, and so we only mention a few, for example [6], [17], [20], [25].

The main purpose of this paper is to extend the work of [8], [9] on quasi-linear dispersive equation with cubic nonlinearity to equations with higher order nonlinearity, and to understand the breakdown mechanism of the corresponding solutions. More specifically, we will now focus on the Cauchy problem for the periodic higher-order Camassa–Holm equations, i.e.

$$\{\begin{array}{c}{m}_t+{(u^2-u_x^2)}^{p-1}{u}_{x}m+{(u{(u^2-u_x^2)}^{p-1}m)}_x=0,t>0,x\in \mathbb{S},\hfill \\ u(0,x)=u_0(x),x\in \mathbb{S},\hfill \end{array}$$

where $\mathbb{S}=\mathbb{R}/\mathbb{Z}=[0,1)$ is the unit circle, and aim to leverage ideas from our previous work to further study the formation of singularities to (1.3). Since $p=1$ reduces to the well-known CH equation, we will consider here only the cases when $p\ge 2$. The local well-posedness of (1.3) follows from the ideas of [14] in dealing with the CH equation in the framework of nonhomogeneous Besov spaces $B_{q,r}^s$, cf. Theorem 2.1, which extends the standard Sobolev spaces $H^s=B_{2,2}^s$.

A blow-up criterion can then be obtained in a rather standard way, cf. Theorem 2.2. A refined criterion can be deduced assuming additional information on the sign of the momentum density; see Theorem 2.3. The blow-up criterion found in Theorem 2.2 depends on the $L^{\infty }$-norm of the momentum m, yet the refinement (2.21) states in a more detailed way that what is truly responsible for the blow-up of solutions is ${\inf }_{x\in \mathbb{S}}{({(u^2-u_x^2)}^{p-1}u)}_x$, provided that the initial momentum density $m_0$ does not change sign. In other words, taking into consideration the conservation of the $H^1$-norm of solutions and the sign preservation of m, singularities can arise only in the form of $|m|\to \infty$, and hence the curvature blow-up: $|u_{xx}|\to \infty$. It is this blow-up phenomenon that distinguishes this family of higher-order CH equations from the classical CH equation.

In search for the blow-up data, one often makes essential use of the “global” information like the conservation of energy, preservation of (anti-)symmetry, and so on. On the other hand, it was established in [4], [5], [10], [12] that the CH-family of equation exhibits a very strong non-diffusive character that extremely “localized” information (even a pointwise information) about the data is enough to lead to finite time blow-up of solutions. This hyperbolic kind of feature seems to be slightly counter balanced by the stronger non-local effects due to higher nonlinearity of the equations, as was explored in [8], [9]. By imposing a sign condition on the initial momentum density, we were able to show that finite time singularity can be induced by mild local oscillations (measured by $u_x/u$), independent of the energy. Here in (1.3) the order of nonlinearity is higher, making the analysis much more challenging.

Our approach is to trace the dynamics of the solution and its gradient along the characteristics. Tracking dynamics along characteristics proves to be an effective technique in the study of singularity formation for hyperbolic equations, for example [21], [23], as well as the quasi-linear CH-type equations. However, adaption of this elementary method to nonlocal models with high-order nonlinearity as such considered in this paper is rather subtle. Special features of the structure of the equation have to be incorporated to yield a corresponding blow-up data.

The key ingredient in our analysis is the preservation of the sign of m along the characteristics. In particular, we will choose initial data such that $m_0$ does not change sign. This eliminates fast local oscillation of solutions, i.e.

$$u\pm u_x\ge 0.$$

The other novelty in our method is that our analysis does not rely on the dynamics of the blow-up quantity $M:={(u{(u^2-u_x^2)}^{p-1})}_x$. This is very different from our previous work on the mCH equation [8]. In fact, the situation we consider here is substantially more difficult than those in the mCH [8] and gCH equations [9]. First the nonlocal terms have complicated algebraic structure and the increasing the order of nonlinearity substantially complicates the estimates for the convolution terms. To address the issue, we find that the terms in the convolution enjoy a special structure that immediately generates an optimal lower bound; see the discussion in Section 4.1. The second difficulty is more serious. The dynamics of M along the characteristics (cf. (3.4)) involves competition between the solution u and its gradient $u_x$ in a way that seems difficult to be resolved by the suppression of fast oscillation. The complicated interaction between u and $u_x$ in the evolution equation of M prevents one from getting a Ricatti-type inequality for M. To get around this, we will focus instead on the dynamics of $u_x$, where the interaction between u and $u_x$ is much easier to analyze. With the help of the convolution estimates, we can show that $u_x$ satisfies a Riccati-type inequality along the characteristics provided that the mild local oscillation can propagate. This indicates a finite time blow-up of $u_x$. However, conservation of the $H^1$-energy and the suppression of fast local oscillation (1.4) confirm that $u_x$ stays bounded as long as the solution u exists. Thus the solution must cease to exist in finite time. Given the refined blow-up criterion (2.21), it then follows that $u_{xx}$ blows up in finite time. Our main result is the following.

Theorem 1.1

Let $2\le p\le 10$ and $u_0\in H^s(\mathbb{S})$ for $s>5/2$. Suppose that $m_0(x)=u_0(x)-u_{0,xx}(x)\ge 0$ for all $x\in \mathbb{S}$ and there is some $x_0\in \mathbb{S}$ such that

$$m_0(x_0)>0,\frac{u_{0,x}}{u_0}(x_0)<A_p:=\{\begin{array}{cc}-\frac{\sqrt{77}}{11},\hfill & p=2,\hfill \\ -\sqrt{\frac{28-\sqrt{233}}{29}},\hfill & p=3,\hfill \\ -0.5857,\hfill & p=4,\hfill \\ -0.5340,\hfill & p=5,\hfill \\ -0.4958,\hfill & p=6,\hfill \\ -0.4660,\hfill & p=7,\hfill \\ -0.4418,\hfill & p=8,\hfill \\ -0.4216,\hfill & p=9,\hfill \\ -0.4043,\hfill & p=10.\hfill \end{array}$$

Then the corresponding solution to (1.3) blows up in finite time. In particular, the blow up occurs in the form that the curvature $u_{xx}$ tends to ∞ in finite time.

Remark 1.1

(1) Note that the case when $p=1$ was proved in [4]. To deal with the power index $p\ge 2$, naively one would like to build up blow-up data via tracking the dynamics of the quantity M, like in [8]. Here the argument can be substantially simplified by only considering the dynamics of u and $u_x$. The trade-off is that we are less clear about the blow-up time and the blow-up place, cf. Remark 4.2 and Remark 4.3.

(2) From condition (1.5) we see that the breakdown of solutions is generated from the local mild oscillation, which is similar to the case of the mCH equation [8].

(3) Theorem 1.1 only treats the case when $p\le 10$, mainly because of the complicated algebraic structure of the equation. For instance, to explore the convolution estimates it requires a very delicate analysis of the terms in the convolution. However it seems reasonable to expect that our method can be extended to the cases for bigger values of p.

(4) The result of Theorem 1.1 indicates that the introduction of stronger nonlinearity increases the possibility that solutions may have finite time blow-up. As one can see from the condition (1.5), as p increases the requirement on $u_{0,x}/u_0$ becomes more relaxed, making it “easier” to generate finite-time singularity.

(5) Theorem 1.1 also provides some criteria for the existence of global solutions. In particular, for $p\le 10$, if u is a global solution to (1.3) which initially satisfies that $m_0>0$. Then at any time t it must hold that

$$u_x(t,x)\ge A_{p}u(t,x),\text{ for all }x\in \mathbb{S}.$$

Therefore defining $\varphi (t,x):=e^{-A_{p}x}u(t,x)$ we have

$${\varphi }_x(t,x)=e^{-A_{p}x}(u_x-A_{p}u)(t,x)\ge 0.$$

This in turn leads to a one-sided estimate of $u(t,x)$ in terms of the value of u at $x=0$:

$$u(t,x)\ge e^{A_{p}x}u(t,0)\text{for }x>0,u(t,x)\le e^{A_{p}x}u(t,0)\text{for }x<0.$$

Together with the sign condition on m we further obtain that

$$e^{A_{p}x}u(t,0)\le u(t,x)<e^{x}u(t,0)\text{for }x>0,e^{x}u(t,0)<u(t,x)\le e^{A_{p}x}u(t,0)\text{for }x<0.$$

For a general p, one may still work out a blow-up result, at the cost of including the use of the conservation of the $H^1$-energy together with the requirement that $m_0$ remains one sign. In particular, we investigate the dynamics of the quantity $\mathrm{\Gamma }:=M+{(u^2-u_x^2)}^{p-1}{u}_x$ along the characteristics, and bound most of the terms using the $H^1$-conservation and the sign preservation of m. The motivation for analyzing the quantity Γ comes from the equation (1.3) for m, which, in the characteristic coordinates, reads $m^{\prime }=m\mathrm{\Gamma }$. Combining the dynamics of Γ and m leads to a second order differential inequality for $n:=\frac{1}{m}$:

$$n^{″}-C_{2}n\le C_1.$$

Therefore by imposing appropriate initial condition so that $n(0)>0$ and $n^{\prime }(0)<0$ one may infer from the maximum principle that n is bounded by $\tilde{n}$ satisfying ${\tilde{n}}^{″}-C_2\tilde{n}=C_1$ with the same initial conditions. Explicit computation indicates that $\tilde{n}$ is convex in time and decrease to zero in finite time. Thus n goes to zero in finite time, which in turn implies that m blows up to +∞; see Theorem 4.1.

The outline of the rest of the paper is as follows. In Section 2 we discuss the conservative properties of the momentum density m in Proposition 2.1 and establish a few criteria for the blow-up of solutions in Sobolev spaces. In Section 3 we provide the computation for the dynamics of u, $u_x$, M and Γ along the characteristics. In Section 4 we derive several types of conditions on initial data which could lead to finite time curvature blow-up.