Response Solution to Ill-Posed Boussinesq Equation with Quasi-Periodic Forcing of Liouvillean Frequency

Abstract

In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation:

$$\begin{aligned} \begin{aligned} y_{tt}(t,x)=\mu y_{xxxx}+y_{xx}+\left( y^{3}+\varepsilon f(\omega t,x)\right) _{xx},\,\, x\in [0, \pi ],\,\, \mu >0, \end{aligned} \end{aligned}$$

subject to the hinged boundary conditions

$$\begin{aligned} \begin{aligned} y(t,0)=y(t,\pi )=y_{xx}(t,0)=y_{xx}(t,\pi )=0, \end{aligned} \end{aligned}$$

where $\omega =(1,\alpha )$ with $\alpha$ being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allow $\alpha$ to be any irrational numbers, and thus the frequency $\omega =(1,\alpha )$ is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine.

Appendix

Lemma 19

(Lemma 7.3 in Liang and You 2005). For vector fields $X_F$ and $X_G$ defined on D(r, s), if

$$\begin{aligned} \begin{aligned} \Vert X_F\Vert _{s,D(r,s)\times \mathcal {O}}\le \varepsilon ',\,\,\Vert X_G\Vert _{s,D(r,s)\times \mathcal {O}}\le \varepsilon '', \end{aligned} \end{aligned}$$

then, there exists a constant c such that

$$\begin{aligned} \begin{aligned} \Vert X_{\{F,G\}}\Vert _{\delta s,D(r-\sigma ,\delta s)\times \mathcal {O}}\le c\sigma ^{-1}\delta ^{-2}\varepsilon '\varepsilon ''. \end{aligned} \end{aligned}$$

Lemma 20

(Lemma 2.1 in Yuan 2003). Let $a>0$ and $p>\frac{d}{2}$. For $w,z,q\in \ell _{a,p},$ let $F(w,z,q)=(F_{\mathbf {n}})_{\mathbf {n} \in \mathbb {Z}^{d}}$ be a sequence where

$$\begin{aligned} \begin{aligned} F_{\mathbf {n}}=\sum _{\mathbf {i}-\mathbf {j}+\mathbf {l}=\mathbf {n}}w_{\mathbf {i}}\bar{z}_{\mathbf {j}}q_{\mathbf {l}}. \end{aligned} \end{aligned}$$

Then $\Vert F(w,z,q)\Vert _{a,p}\le c\Vert w\Vert _{a,p}\Vert z\Vert _{a,p}\Vert q\Vert _{a,p}$ for a constant c depending only on a and p.

We now present the proof for Lemma 8.

Proof

The proof of Lemma 8 is similar to Lemma 3.2 in Wang et al. (2017). However, we still present the details in order to make our paper easier to read. It suffices to consider the case that $\big |\langle k,\omega \rangle + \langle l,\Omega \rangle |\le 1$, since the result is obvious when $\big |\langle k,\omega \rangle + \langle l,\Omega \rangle |>1$.

Case 1.$\overline{Q}_{n}\le Q_{n}^{\mathbb {A}}$. In this case, we know that $\frac{\overline{Q}_{n}}{Q_{n}^{\tau }}\le \overline{Q}_{n}^{\frac{3}{\mathbb {A}}} \le Q_{n}^{3}$, so $\widetilde{K}=\left[ \overline{Q}_{n}^{\frac{3}{\mathbb {A}}}\right]$. It follows from (4.6) and $|k|<\widetilde{K}\le Q_{n}^{3}$ that

$$\begin{aligned} \begin{aligned} \big |\langle k,\omega \rangle + \langle l,\Omega \rangle | \ge \frac{\gamma }{(|k|+|l|)^{\tau }} \ge \frac{c_2(\tau )\gamma }{Q_{n}^{3\tau }}. \end{aligned} \end{aligned}$$

Case 2.$\overline{Q}_{n}> Q_{n}^{\mathbb {A}}\ge 9Q_{n}^{\tau +1}$. In this case, we have $\frac{\overline{Q}_{n}}{Q_{n}^{\tau }}>\overline{Q}_{n}^{\frac{3}{\mathbb {A}}}$, thus $\widetilde{K}=\left[ \frac{\overline{Q}_{n}}{Q_{n}^{\tau }}\right]$. For any $|k|<\widetilde{K}$, we decompose it as

$$\begin{aligned} \begin{aligned} k=(k_{1},k_{2})=(\tilde{k}_{1},\tilde{k}_{2})+m(P_{n},-Q_{n}), \end{aligned} \end{aligned}$$

where $|\tilde{k}_{2}|<Q_{n}.$ Since

$$\begin{aligned} \begin{aligned} |mQ_{n}|-|\tilde{k}_{2}|\le |\tilde{k}_{2}-mQ_{n}|=|k_{2}|\le |k|< \widetilde{K}=\left[ \frac{\overline{Q}_{n}}{Q_{n}^{\tau }}\right] , \end{aligned} \end{aligned}$$

we get

$$\begin{aligned} \begin{aligned} |mQ_{n}|<\frac{\overline{Q}_{n}}{Q_{n}^{\tau }}+Q_{n}, \end{aligned} \end{aligned}$$

which means

$$\begin{aligned} \begin{aligned} |m|<\frac{\overline{Q}_{n}}{Q_{n}^{\tau +1}}+1. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} 1\ge \big |\langle k,\omega \rangle + \langle l,\Omega \rangle |&=\big |(\tilde{k}_{1}+\tilde{k}_{2}\alpha ) +m(P_{n}-Q_{n}\alpha )\pm \langle l,\Omega \rangle |\\&\ge |\tilde{k}_{1}|-\alpha |\tilde{k}_{2}| -|m||P_{n}-Q_{n}\alpha |- |l||\Omega |. \end{aligned} \end{aligned}$$

This implies that

$$\begin{aligned} \begin{aligned} |\tilde{k}_{1}|&\le 1+|\tilde{k}_{2}|+|m||(P_{n}-Q_{n}\alpha )|+2|\Omega |\\&\le 1+Q_{n}+(\frac{\overline{Q}_{n}}{Q_{n}^{\tau +1}}+1)\frac{1}{\overline{Q}_{n}}+4\\&\le 1+Q_{n}+\frac{1}{Q_{n}^{\tau +1}}+\frac{1}{\overline{Q}_{n}}+4\\&\le 7+Q_{n}. \end{aligned}. \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned} |\tilde{k}|=|\tilde{k}_{1}|+|\tilde{k}_{2}| \le 7+2Q_{n}\le 9Q_{n}\le \left[ \frac{\overline{Q}_{n}}{Q_{n}^{\tau }}\right] =\widetilde{K}. \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} \big |\langle \tilde{k},\omega \rangle +\langle l,\Omega \rangle |\ge \frac{\gamma }{(|\tilde{k}|+|l|)^{\tau }}\ge \frac{\gamma }{(9Q_{n}+2)^{\tau }} \ge \frac{\gamma }{11^{\tau }Q_{n}^{\tau }}. \end{aligned} \end{aligned}$$

In conclusion, we get

$$\begin{aligned} \begin{aligned} \big |\langle k,\omega \rangle + \langle l,\Omega \rangle |&=\big |\langle \tilde{k},\omega \rangle +\langle l,\Omega \rangle +m(P_{n}-Q_{n}\alpha )|\\&\ge \big |\langle \tilde{k},\omega \rangle + \langle l,\Omega \rangle |-|m(P_{n}-Q_{n}\alpha )|\\&\ge \frac{\gamma }{11^{\tau }Q_{n}^{\tau }}-(\frac{\overline{Q}_{n}}{Q_{n}^{\tau +1}}+1)\frac{1}{\overline{Q}_{n}}\\&\ge \frac{\gamma }{11^{\tau }Q_{n}^{\tau }}-\frac{2}{Q_{n}^{\tau +1}}\\&\ge c_2(\tau )\gamma Q_{n}^{-\tau }. \end{aligned} \end{aligned}$$

$\square$