The Cauchy Problem and Multi-peakons for the mCH-Novikov-CH Equation with Quadratic and Cubic Nonlinearities

Abstract

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.

Appendices

Appendix A. Basics Properties of the Littlewood-Paley Theory

Let $B(x_{0},r)$ be the open ball centered at $x_{0}$ with radius r,  ${\mathcal {C}}\equiv \{\xi \in {\mathbb {R}}^{d} | 4/3\le |\xi |\le 8/3\}$, and $\mathcal {{\tilde{C}}}\equiv B(0,2/3)+{\mathcal {C}}.$ Then there are two radial functions $\chi \in {\mathcal {D}}(B(0,4/3))$ and $\varphi \in {\mathcal {D}}({\mathcal {C}})$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \chi (\xi )+\sum _{q \ge 0} \varphi (2^{-q} \xi )=1,\quad 1/3 \le \chi ^2(\xi )+\sum _{q \ge 0} \varphi ^2(2^{-q} \xi ) \le 1 \quad (\forall \xi \in {\mathbb {R}}^{d}),\\ |q-q^{\prime }| \ge 2 \Rightarrow {\text {Supp}} \varphi (2^{-q} \cdot ) \cap {\text {Supp}} \varphi (2^{-q^{\prime }} \cdot )=\varnothing , \\ q \ge 1 \Rightarrow {\text {Supp}} \chi (\cdot ) \cap {\text {Supp}} \varphi (2^{-q^{\prime }} \cdot )=\varnothing ,\quad |q-q^{\prime }| \ge 5 \Rightarrow 2^{q^{\prime }} \widetilde{{\mathcal {C}}} \cap 2^{q} {\mathcal {C}}=\varnothing . \end{array}\right. \end{aligned}$$

The dyadic operators $\Delta _{q}$ and $S_{q}$ acting on $u(t,x)\in S'({\mathbb {R}}^d)$ are defined as

$$\begin{aligned} \begin{array}{l} \Delta _{q} u=\left\{ \begin{array}{ll} 0, &{} q \le -2, \\ \chi (D) u=\int _{{\mathbb {R}}^{d}} {\tilde{h}}(y) u(x-y)dy, &{} q=-1, \\ \varphi \left( 2^{-q} D\right) u=2^{q d} \int _{{\mathbb {R}}^{d}} h\left( 2^{q} y\right) u(x-y) dy, &{} q \ge 0, \end{array}\right. \\ S_{q} u=\sum _{q^{\prime } \le q-1} \Delta _{q^{\prime }} u, \end{array} \end{aligned}$$

where $h = {\mathcal {F}}^{-1} \varphi$ and ${\tilde{h}} = {\mathcal {F}}^{-1} \chi$ with ${\mathcal {F}}^{-1}$ denoting the inverse Fourier transform.

The Besov spaces is $B_{p, r}^{s}({\mathbb {R}}^{d})=\left\{ u \in S^{\prime }\, \big |\, \Vert u\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}=\big (\sum _{j \ge -1} 2^{r j s}\Vert \Delta _{j} u\Vert _{L^{p}({\mathbb {R}}^{d})}^{r}\big )^{1/r} <\infty \right\} .$ With the above-defined Besov spaces, we next recall some of their properties.

Lemma 12.1

(Embedding property) [2, 12, 33] Suppose $1 \le p_{1} \le p_{2} \le \infty$, $1 \le r_{1} \le r_{2} \le \infty$ and s be real. Then it holds that $B_{p_{1}, r_{1}}^{s}({\mathbb {R}}^{d}) \hookrightarrow B_{p_{2}, r_{2}}^{s-d(1/p_1-1/p_2)} ({\mathbb {R}}^{d})$. If $s>d/p$ or $s=d/p,\, r=1,$ then there holds $B_{p, r}^{s}({\mathbb {R}}^{d}) \hookrightarrow L^{\infty }({\mathbb {R}}^{d})$.

Lemma 12.2

(Interpolation) [2, 12, 33] Let $s_1,\, s_2$ be real numbers with $s_{1}<s_{2}$ and $\theta \in (0,1).$ Then there exists a constant C such that

$$\begin{aligned} \Vert u\Vert _{B_{p, r}^{\theta s_{1}+(1-\theta ) s_{2}}} \le \Vert u\Vert _{B_{p, r}^{s_{1}}}^{\theta }\Vert u\Vert _{B_{p, r}^{s_{2}}}^{(1-\theta )},\quad \Vert u\Vert _{B_{p, 1}^{\theta s_{1}+(1-\theta ) s_{2}}} \le \frac{C}{s_{2}-s_{1}}\frac{1}{\theta (1-\theta )} \Vert u\Vert _{B_{p, \infty }^{s_{1}}}^{\theta } \Vert u\Vert _{B_{p, \infty }^{s_{2}}}^{(1-\theta )}, \end{aligned}$$

where $(p, r) \in [1, \infty ]^{2}$.

Lemma 12.3

(Product law) [2, 12, 33] Let $(p, r)\in [1, \infty ]^{2}$ and s be real. Then $\Vert u v\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})} \le C(\Vert u\Vert _{L^{\infty }({\mathbb {R}}^{d})}\Vert v\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})} +\Vert u\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})} \Vert v\Vert _{L^{\infty }({\mathbb {R}}^{d})}),$ namely, the space $L^{\infty }({\mathbb {R}}^{d}) \cap B_{p, r}^{s}({\mathbb {R}}^{d})$ is an algebra. Moreover, if $s>d/p$ or $s=d/p,\, r=1,$ then there holds $\Vert u v\Vert _{B_{p, r}^{s}\left( {\mathbb {R}}^{d}\right) } \le C\Vert u\Vert _{B_{p, r}^{s}\left( {\mathbb {R}}^{d}\right) }\Vert v\Vert _{B_{p, r}^{s}\left( {\mathbb {R}}^{d}\right) }.$

Lemma 12.4

(Moser-type estimates) [2, 32] Let $s>\max \{d/p,\, d/2\}$ and $(p, r)\in [1, \infty ]^{2}$. Then, for any $a \in B_{p, r}^{s-1}({\mathbb {R}}^{d})$ and $b \in B_{p, r}^{s}({\mathbb {R}}^{d}),$ there holds $\Vert a b\Vert _{B_{p, r}^{s-1}({\mathbb {R}}^{d})} \le C\Vert a\Vert _{B_{p, r}^{s-1}({\mathbb {R}}^{d})}\Vert b\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}.$

The following Lemma is useful for proving the blow-up criterion.

Lemma 12.5

(Moser-type estimates) [46, 47] Let $s \ge 0.$ Then one has

$$\begin{aligned} \begin{aligned} \Vert f g\Vert _{H^{s}({\mathbb {R}})}&\le C(\Vert f\Vert _{H^{s}({\mathbb {R}})}\Vert g\Vert _{L^{\infty }({\mathbb {R}})} +\Vert f\Vert _{L^{\infty }({\mathbb {R}})}\Vert g\Vert _{H^{s}({\mathbb {R}})}), \\ \Vert f \partial _{x} g\Vert _{H^{s}({\mathbb {R}})}&\le C(\Vert f\Vert _{H^{s+1}({\mathbb {R}})}\Vert g\Vert _{L^{\infty }({\mathbb {R}})} +\Vert f\Vert _{L^{\infty }({\mathbb {R}})}\Vert \partial _{x}g\Vert _{H^{s}({\mathbb {R}})}), \end{aligned} \end{aligned}$$

where C’s are constants independent of f and g.

Lemma 12.6

[2] Let $s \in {\mathbb {R}}$ and $1 \le p, r \le \infty$. Then the Besov spaces have the following properties:

• $B_{p, r}^{s}({\mathbb {R}}^{d})$ is a Banach space and continuously embedding into ${\mathcal {S}}^{\prime }({\mathbb {R}}^{d}),$ where ${\mathcal {S}}^{\prime }({\mathbb {R}}^{d})$ is the dual space of the Schwartz space ${\mathcal {S}}({\mathbb {R}}^{d})$;

• If $p, r<\infty ,$ then ${\mathcal {S}}({\mathbb {R}}^{d})$ is dense in $B_{p, r}^{s}({\mathbb {R}}^{d})$;

• If $u_{n}$ is a bounded sequence of $B_{p, r}^{s}({\mathbb {R}}^{d}),$ then an element $u \in B_{p, r}^{s}({\mathbb {R}}^{d})$ and a subsequence $u_{n_{k}}$ exist such that $\lim _{k \rightarrow \infty } u_{n_{k}}=u \text{ in } {\mathcal {S}}^{\prime }({\mathbb {R}}^{d}) \text{ and } \Vert u\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})} \le C \liminf _{k \rightarrow \infty }\Vert u_{n_{k}}\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}.$

Appendix B. Some Lemmas in the Theory of the Transport Equation

We recall some a priori estimates [2, 32] for the following transport equation

$$\begin{aligned} \phi _{t}+v\cdot \nabla \phi =\omega , \quad \left. \phi \right| _{t=0}=\phi _{0}. \end{aligned}$$

(B.1)

Lemma 12.7

[2, 32] Let $1 \le p \le p_{1} \le \infty ,$ $1 \le r \le \infty$ and $s \ge -d \min (1/p_1,\, 1-1/p)$. Let $\phi _{0} \in B_{p, r}^{s}({\mathbb {R}}^{d})$. $\omega \in L^{1}([0, T] ; B_{p, r}^{s}({\mathbb {R}}^{d}))$ and $\nabla v \in L^{1}([0, T] ; B_{p, r}^{s}({\mathbb {R}}^{d}) \cap L^{\infty }({\mathbb {R}}^{d})),$ then there exists a unique solution $\phi \in L^{\infty }([0, T] ; B_{p, r}^{s}({\mathbb {R}}^{d}))$ to Eq. (B.1) satisfying:

$$\begin{aligned} \Vert \phi \Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}\le & {} \Vert \phi _{0}\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})} +\int _{0}^{t}[\Vert \omega (t^{\prime })\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}+C U_{p_{1}}(t^{\prime })\Vert \phi (t^{\prime })\Vert _{B_{p, r}^{s} ({\mathbb {R}}^{d})}] d t^{\prime },\nonumber \\ \end{aligned}$$

(B.2)

$$\begin{aligned} \Vert \phi \Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}\le & {} \left[ \Vert \phi _{0}\Vert _{B_{p, r}^{s}({\mathbb {R}}^{d})}+\int _{0}^{t}\Vert \omega (t^{\prime })\Vert _{B_{p, r}^{s} ({\mathbb {R}}^{d})}e^{-C U_{p_{1}}(t^{\prime })} d t^{\prime }\right] e^{C U_{p_1}(t)}, \end{aligned}$$

(B.3)

where $U_{p_{1}}(t)=\int _{0}^{t}\Vert \nabla v\Vert _{B_{p_{1}, \infty }^{d/p_{1}}({\mathbb {R}}^{d}) \cap L^{\infty }({\mathbb {R}}^{d})} d t^{\prime }$ if $s<1+d/p_{1}, U_{p_{1}}(t)=\int _{0}^{t}\Vert \nabla v\Vert _{B_{p_{1}, r}^{s-1}\left( {\mathbb {R}}^{d}\right) } d t^{\prime }$ if $s>1+d/p_{1}$ or $s=1+d/p_{1},\, r=1,$ and C is a constant depending only on $s, p, p_{1}$, and r.

Lemma 12.8

[2] Let $s \ge -d \min (1/p_1, 1-1/p).$ Let $\phi _{0} \in B_{p, r}^{s}({\mathbb {R}}^{d})$, $\omega \in L^{1}([0, T] ; B_{p, r}^{s} ({\mathbb {R}}^{d}))$ and $v \in L^{\rho }([0, T]; B_{\infty , \infty }^{-M}({\mathbb {R}}^{d}))$ for some $\rho >1$ and $M>0$ be a time-dependent vector field satisfying

$$\begin{aligned} \nabla v\in \left\{ \begin{array}{ll} L^{1}([0, T] ; B_{p_{1}, \infty }^{d/p}({\mathbb {R}}^{d})), &{} \mathrm{if }\,\, s<1+d/p_{1}, \\ L^{1}\left( [0, T] ; B_{p_{1}, \infty }^{s-1}\left( {\mathbb {R}}^{d}\right) \right) , &{} \mathrm{if }\,\, s>1+d/p_1\,\, \mathrm{or } \,\, s=1+d/p_1 \,\, \mathrm{and } \,\, r=1. \end{array}\right. \end{aligned}$$

Then, Eq. (B.1) has a unique solution $\phi \in {\mathcal {C}}([0, T] ; B_{p, r}^{s}({\mathbb {R}}^{d}))$ for $r<\infty$, or $\phi \in (\bigcap _{s^{\prime }<s} {\mathcal {C}}([0, T]; B_{p, \infty }^{s^{\prime }}({\mathbb {R}}^{d}))) \cap {\mathcal {C}}_{w}([0, T] ; B_{p, \infty }^{s}({\mathbb {R}}^{d})))$ for $r=\infty$. Furthermore, the inequalities (B.2)-(B.3) hold.

Lemma 12.9

(A priori estimate in the Sobolev spaces) [2, 46] Let $0\le \sigma <1$. Let $\phi _{0} \in H^{\sigma },\, \omega \in L^{1}(0, T ; H^{\sigma })$ and $\partial _{x} v \in L^{1}(0, T ; L^{\infty }).$ Then the solution $\phi$ to Eq. (B.1) belongs to $C([0, T] ; H^{\sigma }).$ More precisely, there is a constant C depending only on $\sigma$ such that

$$\begin{aligned} \Vert \phi \Vert _{H^{\sigma }}\le \Vert \phi _{0}\Vert _{H^{\sigma }} +\int _{0}^{t}[\Vert \omega (\tau )\Vert _{H^{\sigma }}+CU^{\prime }(\tau )\Vert \phi (\tau )\Vert _{H^{\sigma }}] d\tau ,\quad U(t)=\int _{0}^{t}\Vert \partial _{x} v(\tau )\Vert _{L^{\infty }} \text{ d } \tau . \end{aligned}$$