KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.

We warmly thank W. Craig for many discussions about the reduction approach of the linearized operators and the reversible structure, and P. Bolle for deep observations about the Hamiltonian case. We also thank T. Kappeler, M. Procesi for many useful comments.

Authors and Affiliations

1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 , Naples, Italy

   Pietro Baldi & Massimiliano Berti

2. SISSA, Via Bonomea 265, 34136 , Trieste, Italy

   Massimiliano Berti & Riccardo Montalto

Corresponding author

Correspondence to Massimiliano Berti.

This research was supported by the European Research Council under FP7 and partially by the PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.

Appendix: Tame and Lipschitz estimates

In this Appendix we present standard tame and Lipschitz estimates for composition of functions and changes of variables which are used in the paper.

Let \( H^s := H^{s}({\mathbb {T}}^d,{\mathbb {C}}) \) (with norm \( \Vert \ \Vert _s \)) and \( W^{s, \infty } := W^{s, \infty }({\mathbb {T}}^d,{\mathbb {C}}) \), \( d \ge 1\).

Lemma 6.1

Let \( s_0 > d/2\). Then

1. (i)

   Embedding. \(\Vert u \Vert _{L^\infty } \le C(s_0) \Vert u \Vert _{s_0}\) for all \(u \in H^{s_0} \).

2. (ii)

   Algebra. \(\Vert uv \Vert _{s_0} \le C(s_0) \Vert u \Vert _{s_0} \Vert v \Vert _{s_0}\) for all \(u, v \in H^{s_0}\).

3. (iii)

   Interpolation. For \(0 \le s_1 \le s \le s_2\), \(s = \lambda s_1 + (1-\lambda ) s_2\),

   $$\begin{aligned} \Vert u \Vert _{s} \le \Vert u \Vert _{s_1}^\lambda \Vert u \Vert _{s_2}^{1-\lambda } , \quad \forall u \in H^{s_2}. \end{aligned}$$

   (6.1)

   Let \( a_0, b_0 \ge 0\) and \( p,q > 0 \). For all \( u \in H^{a_0 + p + q} \), \( v \in H^{b_0 + p + q} \),

   $$\begin{aligned} \Vert u \Vert _{a_0 + p} \Vert v \Vert _{b_0 + q} \le \Vert u \Vert _{a_0 + p + q} \Vert v \Vert _{b_0} + \Vert u \Vert _{a_0} \Vert v \Vert _{b_0 + p + q}. \end{aligned}$$

   (6.2)

   Similarly, for the \(|u|_{s, \infty } := \sum _{|\beta | \le s} | D^\beta u |_{L^\infty } \) norm,

   $$\begin{aligned} | u |_{s, \infty } \le C(s_1, s_2) | u |_{s_1, \infty }^\lambda | u |_{s_2, \infty }^{1-\lambda } , \quad \forall u \in W^{s_2, \infty } , \end{aligned}$$

   (6.3)

   and \( \forall u \in W^{a_0 + p + q, \infty } \), \( v \in W^{b_0 + p + q, \infty } \),

   $$\begin{aligned}&| u |_{a_0 + p, \infty } | v |_{b_0 + q, \infty }\nonumber \\&\quad \le C(a_0, b_0, p,q)\left( | u |_{a_0 + p + q, \infty } | v |_{b_0, \infty } + | u |_{a_0, \infty } | v |_{b_0 + p + q, \infty }\right) . \end{aligned}$$

   (6.4)
4. (iv)

   Asymmetric tame product. For \(s \ge s_0\),

   $$\begin{aligned} \Vert uv \Vert _s \le C(s_0) \Vert u\Vert _s \Vert v\Vert _{s_0} + C(s) \Vert u\Vert _{s_0} \Vert v \Vert _s , \quad \forall u,v \in H^s. \end{aligned}$$

   (6.5)
5. (v)

   Asymmetric tame product in \(W^{s,\infty }\). For \(s \ge 0\), \(s \in {\mathbb {N}}\),

   $$\begin{aligned} | uv |_ {s, \infty } \le \tfrac{3}{2} \, | u |_ {L^\infty } |v|_ {s, \infty } + C(s) |u|_ {s, \infty } |v|_ {L^\infty } , \quad \forall u,v \in W^{s,\infty }. \end{aligned}$$

   (6.6)
6. (vi)

   Mixed norms asymmetric tame product. For \(s \ge 0\), \(s \in {\mathbb {N}}\),

   $$\begin{aligned} \Vert uv \Vert _s \le \tfrac{3}{2} \, |u|_ {L^\infty } \Vert v \Vert _s + C(s)| u |_ {s, \infty } \Vert v \Vert _0 , \quad \forall u \in W^{s,\infty } , \ v \in H^s . \end{aligned}$$

   (6.7)

   If \(u := u(\lambda )\) and \(v := v(\lambda )\) depend in a Lipschitz way on \(\lambda \in \Lambda \subset {\mathbb {R}}\), all the previous statements hold if we replace the norms \(\Vert \cdot \Vert _s\), \(\vert \cdot \vert _ {s, \infty } \) with the norms \(\Vert \cdot \Vert _s^{\mathrm{{Lip}(\gamma )}}\), \(\vert \cdot \vert _ {s, \infty }^{\mathrm{{Lip}(\gamma )}}\).

Proof

The interpolation estimate (6.1) for the Sobolev norm (1.5) follows by Hölder inequality, see also [36], page 269. Let us prove (6.2). Let \( a = a_0 \lambda + a_1 (1-\lambda ) \), \( b = b_0 (1-\lambda ) + b_1 \lambda \), \( \lambda \in [0,1] \). Then (6.1) implies

by Young inequality. Applying (6.8) with \( a = a_0 + p \), \( b = b_0 + q \), \( a_1 = a_0 + p + q \), \( b_1 = b_0 + p + q \), then \( \lambda = q \slash (p+q) \) and we get (6.2). Also the interpolation estimates (6.3) are classical (see [11]) and (6.3) implies (6.4) as above.

\((iv)\): see the Appendix of [11]. \((v)\): we write, in the standard multi-index notation,

Using \( |(D^\beta u)(D^\gamma v)|_ {L^\infty } \le |D^\beta u|_ {L^\infty } |D^\gamma v|_ {L^\infty } \le |u|_{|\beta |, \infty } |v|_{|\gamma |, \infty } \), and the interpolation inequality (6.3) for every \( \beta \ne 0\) with \(\lambda := |\beta | / |\alpha | \in (0,1]\) (where \( |\alpha | \le s \)), we get, for any \(K > 0\),

Then (6.6) follows by (6.9), (6.10) taking \( K := K(s) \) large enough. \((vi)\): same proof as \((v)\), using the elementary inequality \(\Vert (D^\beta u)(D^\gamma v) \Vert _0 \le |D^\beta u|_{L^\infty } \Vert D^\gamma v \Vert _0 \). \(\square \)

We now recall classical tame estimates for composition of functions, see [36], section 2, pages 272–275, and [40, 41]-I, Lemma 7 in the Appendix, pages 202–203.

A function \( f : {\mathbb {T}}^d \times B_1 \rightarrow {\mathbb {C}}\), where \(B_1 := \{ y \in {\mathbb {R}}^m : |y| < 1\} \), induces the composition operator

where \(D^k u(x)\) denotes the partial derivatives \(\partial _x^\alpha u(x)\) of order \(|\alpha |=k\) (the number \( m \) of \( y \)-variables depends on \(p, d\)).

Lemma 6.2

(Composition of functions) Assume \( f \in C^r ({\mathbb {T}}^d \times B_1)\). Then

1. (i)

   For all \( u \in H^{r+p} \) such that \( |u |_{p, \infty } < 1 \), the composition operator (6.11) is well defined and

   $$\begin{aligned} \Vert \tilde{f}(u) \Vert _r \le C \Vert f \Vert _{C^r} (\Vert u\Vert _{r+p} + 1) \end{aligned}$$

   where the constant \(C \) depends on \( r,d,p \). If \( f \in C^{r+2} \), then, for all \( |u|_{p, \infty } \), \( | h |_{p, \infty } < 1 / 2 \),

   $$\begin{aligned} \big \Vert \tilde{f}(u+h) - \tilde{f} (u) \big \Vert _r&\le C \Vert f \Vert _{C^{r+1}} \, ( \Vert h \Vert _{r+p} + | h |_{p,\infty } \Vert u \Vert _{r+p}) , \\ \big \Vert \tilde{f}(u+h) - \tilde{f} (u) - \tilde{f}'(u) [h] \big \Vert _r&\le C \Vert f \Vert _{C^{r+2}} \, | h |_{p,\infty } ( \Vert h \Vert _{r+p} \!+\! | h |_{p,\infty } \Vert u \Vert _{r+p}). \end{aligned}$$
2. (ii)

   The previous statement also holds replacing \(\Vert \ \Vert _r\) with the norms \(| \ |_{r, \infty } \).

Lemma 6.3

(Lipschitz estimate on parameters) Let \(d \in {\mathbb {N}}\), \(d/2 < s_0 \le s\), \(p \ge 0\), \(\gamma >0\). Let \( F \) be a \( C^1 \)-map satisfying the tame estimates: \( \forall \Vert u \Vert _{s_0+p} \le 1 \), \( h \in H^{s+p} \),

For \(\Lambda \subset {\mathbb {R}}\), let \( u(\lambda ) \) be a Lipschitz family of functions with \( \Vert u \Vert _{s_0 +p}^{\mathrm{{Lip}(\gamma )}} \le 1 \) (see (2.2)). Then

The same statement also holds when all the norms \(\Vert \ \Vert _s \) are replaced by \(| \ |_{s, \infty } \).

Proof

By (6.12) we get \( \sup _\lambda \Vert F(u(\lambda )) \Vert _s \le C(s) ( 1 + \Vert u \Vert _{s+p}^{\mathrm{{Lip}(\gamma )}}) \). Then, denoting \( u_1 := u(\lambda _1)\) and \(h := u(\lambda _2) - u(\lambda _1)\), we have

whence

because \( \Vert u \Vert _{s_0 +p}^{\mathrm{{Lip}(\gamma )}} \le 1 \), and the lemma follows. \(\square \)

The next lemma is also classical, see for example [24], Appendix G. The present version is proved in [2], except for the part on the Lipschitz dependence on a parameter, which is proved here below.

Lemma 6.4

(Change of variable) Let \(p:{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d\) be a \(2\pi \)-periodic function in \(W^{s,\infty }\), \( s \ge 1\), with \( |p|_{1, \infty } \le 1/2 \). Let \(f(x) = x + p(x)\). Then:

1. (i)

   \(f\) is invertible, its inverse is \(f^{-1}(y) = g(y) = y + q(y)\) where \(q\) is \( 2 \pi \)-periodic, \(q \in W^{s,\infty }({\mathbb {T}}^d,{\mathbb {R}}^d)\), and \(|q|_{s, \infty } \le C |p|_{s, \infty } \). More precisely,

   $$\begin{aligned} | q |_{L^\infty } \!=\! | p |_{L^\infty }, \quad \! | Dq |_{L^\infty } \le 2 | Dp |_{L^\infty } , \quad \! | Dq |_{s-1, \infty } \le C | Dp |_{s-1, \infty }.\quad \quad \end{aligned}$$

   (6.14)

   where the constant \(C\) depends on \(d, s\). Moreover, suppose that \(p = p_\lambda \) depends in a Lipschitz way by a parameter \(\lambda \in \Lambda \subset {\mathbb {R}}\), and suppose, as above, that \(|D_x p_\lambda |_ {L^\infty } \le 1/2\) for all \(\lambda \). Then \(q = q_\lambda \) is also Lipschitz in \(\lambda \), and

   $$\begin{aligned} |q|_{s, \infty }^{\mathrm{{Lip}(\gamma )}} \!\le \! C \left( |p|_ {s, \infty } ^{\mathrm{{Lip}(\gamma )}} \!+\! \big \{ \sup _{\lambda \in \Lambda } |p_\lambda |_{s+1, \infty } \big \} \, |p|_{L^\infty }^{\mathrm{{Lip}(\gamma )}} \right) \!\le \! C |p|_{s+1, \infty }^{\mathrm{{Lip}(\gamma )}}.\quad \quad \end{aligned}$$

   (6.15)

   The constant \(C\) depends on \(d, s \) (and is independent of \(\gamma \)).

2. (ii)

   If \(u \in H^s ({\mathbb {T}}^d,{\mathbb {C}})\), then \(u\circ f(x) = u(x+p(x))\) is also in \(H^s \), and, with the same \(C\) as in \((i)\),

   $$\begin{aligned} \Vert u \circ f \Vert _s&\le C (\Vert u\Vert _s + |Dp|_{s-1, \infty } \Vert u\Vert _1), \end{aligned}$$

   (6.16)
   $$\begin{aligned} \Vert u \circ f - u \Vert _s&\le C \left( | p |_{L^\infty } \Vert u \Vert _{s + 1} + |p|_{s, \infty } \Vert u \Vert _{2} \right) , \end{aligned}$$

   (6.17)
   $$\begin{aligned} \Vert u \circ f \Vert _{s}^{\mathrm{{Lip}(\gamma )}}&\le C \, \left( \Vert u \Vert _{s+1}^{\mathrm{{Lip}(\gamma )}} + |p|_{s, \infty }^{\mathrm{{Lip}(\gamma )}} \Vert u \Vert _2^{\mathrm{{Lip}(\gamma )}} \right) . \end{aligned}$$

   (6.18)

   (6.16), (6.17) (6.18) also hold for \(u \circ g\).

3. (iii)

   Part \((ii)\) also holds with \(\Vert \cdot \Vert _k\) replaced by \(| \cdot |_{k, \infty }\), and \(\Vert \cdot \Vert _{s}^{\mathrm{{Lip}(\gamma )}}\) replaced by \(\vert \cdot \vert _{s, \infty }^{\mathrm{{Lip}(\gamma )}}\), namely

   $$\begin{aligned} | u \circ f |_{s, \infty }&\le C (|u|_{s, \infty } + |Dp|_{s-1, \infty } |u|_{1, \infty }), \end{aligned}$$

   (6.19)
   $$\begin{aligned} | u \circ f |_{s, \infty }^\mathrm{{Lip}(\gamma )}&\le C (|u|_{s+1, \infty }^\mathrm{{Lip}(\gamma )}+ |Dp|_{s-1, \infty }^\mathrm{{Lip}(\gamma )}|u|_{2, \infty }^\mathrm{{Lip}(\gamma )}). \end{aligned}$$

   (6.20)

Proof

The bounds (6.14), (6.16) and (6.19) are proved in [2], Appendix B. Let us prove (6.15). Denote \(p_\lambda (x) := p(\lambda ,x)\), and similarly for \(q_\lambda , g_\lambda , f_\lambda \). Since \(y = f_\lambda (x) = x + p_\lambda (x)\) if and only if \(x = g_\lambda (y) = y + q_\lambda (y)\), one has

Let \(\lambda _{1}, \lambda _{2} \in \Lambda \), and denote, in short, \(q_1 = q_{\lambda _1}\), \(q_2 = q_{\lambda _2}\), and so on. By (6.21),

where \( G_2 h := h \circ g_2 \), \( G_t h := h \circ \left( g_1 + (t-1) [g_2 - g_1] \right) \), \( t \in [1,2]\). By (6.22), the \( L^\infty \) norm of \((q_2 - q_1)\) satisfies

whence, using the assumption \(|D_{x} p_1|_{L^\infty } \le 1/2\), we get \( |q_2 - q_1|_{L^\infty } \le 2 |p_2 - p_1|_{L^\infty } \).

By (6.22), using (6.6), the \(W^{s,\infty }\) norm of \((q_2 - q_1)\), for \(s \ge 0\), satisfies

Since \(|G_t (D_{x} p_1)|_{L^\infty } = |D_x p_1|_{L^\infty } \le 1/2\),

Using \( |q_2 - q_1|_{L^\infty } \le 2 |p_2 - p_1|_{L^\infty } \), (6.19), (6.4) and (6.14),

and (6.15) follows. The proof of (6.17), (6.18), (6.20) may be obtained similarly. \(\square \)

Lemma 6.5

(Composition) Suppose that for all \(\Vert u \Vert _{s_{0}+ \mu _i} \le 1\) the operator \(\mathcal{Q}_i(u)\) satisfies

Let \(\tau := \mathrm{max}\{ \tau _1, \tau _2 \}\), \(\mu := \mathrm{max}\{ \mu _1, \mu _2 \}\). Then, for all

the composition operator \(\mathcal{Q} := \mathcal{Q}_{1} \circ \mathcal{Q}_2\) satisfies the tame estimate

Moreover, if \(\mathcal{Q}_1\), \(\mathcal{Q}_2\), \(u\) and \(h\) depend in a Lipschitz way on a parameter \(\lambda \), then (6.25) also holds with \(\Vert \cdot \Vert _s\) replaced by \(\Vert \cdot \Vert _{s}^{\mathrm{{Lip}(\gamma )}}\).

Proof

Apply the estimates for (6.23) to \(\mathcal{Q}_1\) first, then to \(\mathcal{Q}_2\), using condition (6.24).   \(\square \)

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