A Rigidity Property for the Novikov Equation and the Asymptotic Stability of Peakons

Abstract

We consider weak solutions of the Novikov equation that lie in the energy space $H^1$ with non-negative momentum densities. We prove that a special family of such weak solutions, namely the peakons, is $H^1$-asymptotically stable. Such a result is based on a rigidity property of the Novikov solutions which are $H^1$-localized and the corresponding momentum densities are localized to the right, which extends the earlier work of Molinet (Arch Ration Mech Anal 230:185–230, 2018; Nonlinear Anal Real World Appl 50:675–705, 2019) for the Camassa–Holm and Degasperis–Procesi peakons. The main new ingredients in our proof consist of exploring the uniform in time exponential decay property of the solutions from the localization of the $H^1$ energy and redesigning the localization of the total mass from the finite speed of propagation property of the momentum densities.

Appendix A. Proofs of Lemma 3.1 and Lemma 4.1

1.1 Proof of Lemma 3.1

Proof

For a fixed $t_0\in {\mathbb {R}}$, we approximate $u(t_{0})$ by $u_{0, n}=\zeta _{n}*u(t_{0})$ which converges to $u(t_{0})$ in Y. Thanks to the continuity result in Proposition 2.3, the solution sequence $\{u_{n}\}$ induced by $\{u_{0, n}\}$ to (2.4) lies in $C\left( {\mathbb {R}} ; H^{\infty }({\mathbb {R}})\right)$. Hence for an arbitrary positive T,

$$\begin{aligned} u_{n} \rightarrow u \text { in } C\left( \left[ t_{0}-T, t_{0}+T\right] ; H^{1}({\mathbb {R}}) \right) . \end{aligned}$$

(A.1)

For this T, there exists an $n_0\geqq 0$ such that, for all $n\geqq n_0$,

$$\begin{aligned} \Vert u^2_n-u^2\Vert _{L^\infty ((t_0-T,t_0+T)\times {\mathbb {R}})}\leqq \frac{\alpha c_0}{2L}, \end{aligned}$$

which combines with (3.1) implying

$$\begin{aligned} \sup _{[t_0-T,t_0+T]}\Vert u^2_n\Vert _{L^\infty (|x-\rho (t)|>R_0)}\leqq \frac{\alpha c_0}{L}. \end{aligned}$$

(A.2)

For the sake of convenience, in the following arguments, we leave out the subscript n of $u_n$. For $t\in [t_0-T,t_0]$ and $R>R_0$, differentiating $I_{t_{0}}^{+R}(t)$, we have

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d} t} I_{t_{0}}^{+R}(t)=-\alpha \rho _t(t) \int _{{\mathbb {R}}} \Psi ^{\prime }_{K}\left( u^{2}+u_{x}^{2}\right) + \int _{{\mathbb {R}}}\Psi _{K} \frac{\mathrm {d}}{\mathrm {d} t} \left( u^{2}+u_{x}^{2}\right) . \end{aligned}$$

(A.3)

Next we estimate the second term on the right-hand side. A direct computation yields that

$$\begin{aligned} \int _{{\mathbb {R}}} \frac{\mathrm {d}}{\mathrm {d} t} \left( u^{2}+u_{x}^{2}\right) \Psi _{K}=2 \int _{{\mathbb {R}}} u u_{t} \Psi _{K}+2 \int _{{\mathbb {R}}} u_{x} u_{x t} \Psi _{K}. \end{aligned}$$

Using (2.4), we have

$$\begin{aligned} 2 \int _{{\mathbb {R}}} u u_{t} \Psi _{K}&=-2 \int _{{\mathbb {R}}} u^{3} u_{x} \Psi _{K}+2 \int _{{\mathbb {R}}} u \Psi _{K}^{\prime }h_{1}+2 \int _{{\mathbb {R}}} u_{x} \Psi _{K} h_{1}- \int _{{\mathbb {R}}} u \Psi _{K} h_{2}, \end{aligned}$$

(A.4)

where $h_1 :=p*\left( \frac{3}{2}uu_{x}^{2}+u^{3}\right)$ and $h_2 :=p*u^{3}_{x}$. Furthermore,

$$\begin{aligned} 2 \int _{{\mathbb {R}}} u_{x} u_{x t} \Psi _{K} =&\ 2 \int _{{\mathbb {R}}} \partial _{x}(u_{x} \Psi _{K})u^{2}u_{x} - 2 \int _{{\mathbb {R}}} u_{x} \Psi _{K} h_{1} + 2\int _{{\mathbb {R}}} u_{x} \Psi _{K} \left( \frac{3}{2}uu^{2}_{x}+u^{3} \right) \nonumber \\&+ \int _{{\mathbb {R}}} u\left( \Psi _{K}\partial ^{2}_{x}h_{2}+\Psi _{K}^{\prime }\partial _{x}h_{2} \right) \nonumber \\ =&\int _{{\mathbb {R}}} u^{2}u_{x}^{2}\Psi _{K}^{\prime }+2 \int _{{\mathbb {R}}} u_{x} \Psi _{K} h_{1} + 2\int _{{\mathbb {R}}} u^3 u_{x} \Psi _{K} \nonumber \\&+ \int _{{\mathbb {R}}} u\left( h_{2}\Psi _{K} + \Psi _{K}^{\prime }\partial _{x}h_{2} \right) . \end{aligned}$$

(A.5)

Now combining (A.4) and (A.5), we arrive at

$$\begin{aligned} \int _{{\mathbb {R}}} \frac{\mathrm {d}}{\mathrm {d} t} \left( u^{2}+u_{x}^{2}\right) \Psi _{K}=\int _{{\mathbb {R}}} u^{2} u_{x}^{2} \Psi _{K}^{\prime }+ \int _{{\mathbb {R}}} u\Psi _{K}^{\prime } \partial _{x} h_{2}+2 \int _{{\mathbb {R}}} uh_{1}\Psi _{K}^{\prime }, \end{aligned}$$

(A.6)

which, together with (A.3) and (A.6), implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d} t} I_{t_{0}}^{+R}(t)=&-\alpha \rho _t(t) \int _{{\mathbb {R}}} \Psi _{K}^{\prime }\left( u^{2}+u_{x}^{2}\right) +\int _{{\mathbb {R}}}u^{2} u_{x}^{2}\Psi _{K}^{\prime }+ \int _{{\mathbb {R}}} \left( u \partial _{x} h_{2}+2uh_{1}\right) \Psi _{K}^{\prime } \nonumber \\ =:&-\alpha \rho _t(t) \int _{{\mathbb {R}}} \Psi _{K}^{\prime }\left( u^{2}+u_{x}^{2}\right) +J_{1}+J_{2}. \end{aligned}$$

Next we estimate $J_1$ and $J_2$. We write $J_1$ as

$$\begin{aligned} J_{1}&=\int _{|x-\rho (t)|<R_{0}}u^{2} u_{x}^{2}\Psi _{K}^{\prime }+\int _{|x-\rho (t)|>R_{0}}u^{2} u_{x}^{2}\Psi _{K}^{\prime } =: J_{11}+J_{12}. \end{aligned}$$

From $R\geqq R_0$, for $|x-\rho (t)|<R_{0}$ and $t\in [t_0-T,t_0]$, we have

$$\begin{aligned}&x-\rho (t_{0})- R-\alpha (\rho (t)-\rho (t_{0})) = x-\rho (t) \\&\quad -R+(\rho (t)-\alpha \rho (t))-\left( \rho (t_{0})-\alpha \rho (t_{0})\right) \\&\quad \leqq R_{0}-R-(1-\alpha ) c_0\left( t_{0}-t\right) , \end{aligned}$$

which, together with (3.5) and the non-negativity of $\Psi _{K}$, shows that

$$\begin{aligned} J_{11}(t)&\leqq 2 R_0 C_0 \Vert u(t)\Vert ^{2}_{L^{\infty }}\left\| u_{x}(t)\right\| _{L^{2}}^{2}e^{R_{0} / K} e^{-R / K} e^{-\frac{(1-\alpha )}{K} c_0\left( t_{0}-t\right) } \nonumber \\&\lesssim \Vert u(t)\Vert ^4_{H^{1}} e^{R_{0} / K} e^{-R / K} e^{-\frac{(1-\alpha )}{K} c_0\left( t_{0}-t\right) }. \end{aligned}$$

(A.7)

As for $J_{12}$, it holds that

$$\begin{aligned} J_{12}&\leqq \Vert u(t)\Vert ^{2}_{L^{\infty }\left( |x-\rho (t)|>R_{0}\right) } \int _{|x-\rho (t)|>R_{0}}u_{x}^{2}\Psi _{K}^{\prime }. \end{aligned}$$

(A.8)

From $|\Psi _{K}^{\prime \prime }|<\frac{1}{K}\Psi _{K}^{\prime }$ and (3.5), for $J_2$ we have

$$\begin{aligned} J_2=&\int _{{\mathbb {R}}} \left( u \partial _{x} h_{2}+2uh_{1}\right) \Psi _{K}^{\prime } =- \int _{{\mathbb {R}}} u_x h_{2}\Psi _{K}^{\prime }- \int _{{\mathbb {R}}}u h_{2}\Psi _{K}^{\prime \prime } +\int _{{\mathbb {R}}} 2uh_{1}\Psi _{K}^{\prime } \\ \leqq&\int _{{\mathbb {R}}} u (p*u^3)\Psi _{K}^{\prime }+\int _{{\mathbb {R}}}\frac{u}{K}(p*u^3)\Psi _{K}^{\prime }+\int _{{\mathbb {R}}} 2u \left( p*\left( \frac{3}{2}u^{3}+u^{3}\right) \right) \Psi _{K}^{\prime } \\ \leqq&\left( 6+\frac{1}{K}\right) \Vert u(t)\Vert _{L^{\infty }} \int _{{\mathbb {R}}}(p*u^3)\Psi _{K}^{\prime } \leqq \ L\Vert u(t)\Vert ^{2}_{L^{\infty }} \int _{{\mathbb {R}}}u^2 \Psi _{K}^{\prime }, \end{aligned}$$

where $L :=\left( 6+\frac{1}{K}\right) \left( \frac{K^2}{K^{2}-1}\right)$. Therefore applying similar treatment to $J_2$, we also have

$$\begin{aligned} J_2\leqq&L\Vert u(t)\Vert ^{2}_{L^{\infty }\left( |x-\rho (t)|<R_{0}\right) } \int _{|x-\rho (t)|<R_{0}}u^2 \Psi _{K}^{\prime }+L\Vert u(t)\Vert ^{2}_{L^{\infty }\left( |x-\rho (t)|>R_{0}\right) } \\&\quad \int _{|x-\rho (t)|>R_{0}}u^2 \Psi _{K}^{\prime } \\ =:&\ J_{21}+J_{22}. \end{aligned}$$

Similarly, we have

$$\begin{aligned}&J_{21}\lesssim \Vert u(t)\Vert ^{4}_{H^{1}} e^{R_{0} / K} e^{-R / K} e^{-\frac{(1-\alpha )}{K} c_0\left( t_{0}-t\right) }, \end{aligned}$$

(A.9)

$$\begin{aligned}&J_{22} \leqq L \Vert u(t)\Vert ^{2}_{L^{\infty \left( |x-\rho (t)|>R_{0}\right) }} \int _{|x-\rho (t)|>R_{0}}u^{2}\Psi _{K}^{\prime }. \end{aligned}$$

(A.10)

Combining (A.7)–(A.10) and (A.2) we see that there exists a sufficiently large $R>R_0$ such that

$$\begin{aligned}&-\alpha \rho _t(t) \int _{{\mathbb {R}}} \Psi _{K}^{\prime }\left( u^{2} +u_{x}^{2}\right) +J_{12}+J_{22} \\&\quad \leqq \left( -\alpha c_0+L \Vert u(t)\Vert ^{2}_{L^{\infty }\left( |x-\rho (t)|>R_{0}\right) } \right) \int _{{\mathbb {R}}} \Psi _{K}^{\prime }\left( u^{2} +u_{x}^{2}\right) <0, \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d} t} I_{t_{0}}^{+R}(t) \leqq {\tilde{C}} e^{-R / K} e^{-\frac{(1-\alpha )}{K} c_0\left( t_{0}-t\right) } \ \ \text {for} \ R \geqq R_{0} \ \text {and} \ t \in \left[ t_{0}-T, t_{0}\right] , \end{aligned}$$

where ${\tilde{C}}$ depends on K, $R_0$ and ${\mathcal {E}}(u)$. Since T is an arbitrary positive number, integrating from t to $t_0$ and by (A.1), we have

$$\begin{aligned} I_{t_{0}}^{+R}\left( t_{0}\right) -I_{t_{0}}^{+R}(t) \leqq C e^{-R / K}, \quad \forall \ t \leqq t_{0}, \end{aligned}$$

where C depends on K, $\alpha$, $R_0$, ${\mathcal {E}}(u)$ and $c_0$. One may follow the same steps to obtain

$$\begin{aligned} I_{t_{0}}^{-R}\left( t\right) -I_{t_{0}}^{-R}(t_{0}) \leqq C e^{-R / K}, \quad \forall \ t \geqq t_{0}, \end{aligned}$$

which completes the proof. $\quad \square$

1.2 Proof of Lemma 4.1

Proof

As in the proof of Lemma 4.1 in [23], utilizing the implicit function theorem, similarly we can derive (4.3). And as for the Novikov equation (2.4), each solution $u\in C\left( {\mathbb {R}}; H^1({\mathbb {R}})\right) \cap C^1\left( {\mathbb {R}}; L^2({\mathbb {R}})\right)$ indicates that the mapping $t \mapsto \rho (t)$ is $C^1$. Setting $R(t,x) :=\sqrt{c} \varphi (x-\rho (t))$ and $w :=u-R$, we can check that R satisfies

$$\begin{aligned}&\partial _{t} R+(\rho _t-c) \partial _{x} R+R^{2} \partial _{x} R+\left( 1-\partial _{x}^{2}\right) ^{-1} \partial _{x}\left( \frac{3}{2}RR^{2}_{x}+R^{3}\right) \\&\quad +\frac{1}{2}\left( 1-\partial _{x}^{2}\right) ^{-1}R^{3}_{x}=0, \end{aligned}$$

which gives

$$\begin{aligned} w_{t}-(\rho _t-c) \partial _{x} R&=-(w+R)^2 \partial _{x} w-\left( (w+R)^2-R^2\right) \partial _x R \\&\quad - \frac{1}{2} \left( 1-\partial _{x}^{2}\right) ^{-1} \left[ (w_x+ R_x)^3-R^3\right] \nonumber \\&\quad -\left( 1-\partial _{x}^{2}\right) ^{-1} \partial _{x}\left( \frac{3}{2} (w+R)(w_x+ R_x)^2-\frac{3}{2}RR^{2}_{x}+(w+ R)^3-R^3 \right) . \end{aligned}$$

Taking the $L^2$-scalar product of this last equality with $(\zeta _{n_0}*\varphi )^{\prime }(\cdot -\rho (t))$, similarly we have

$$\begin{aligned} \left| (\rho _t-c)\left( \int _{{\mathbb {R}}} \partial _{x} R \partial _{x}\left( \zeta _{n_{0}} * \varphi \right) (\cdot -\rho (t))+ \Vert w\Vert _{H^{1}}\right) \right| \lesssim K c \varepsilon _{0}. \end{aligned}$$

The rest of the proof is similar to [23]. $\quad \square$