Regularity Results on the Flow Maps of Periodic Dispersive Burgers Type Equations and the Gravity–Capillary Equations

Abstract

In the first part of this paper we prove that the flow associated to a dispersive Burgers equation with a non local term of the form $\left| D \right| ^{\alpha -1} \partial _x u$, $\alpha \in [1,+\infty [$ is Lipschitz from bounded sets of $H^s_0({\mathbb {T}};{\mathbb {R}})$ to $C^0([0,T],H^{s-(2-\alpha )^+}_0({\mathbb {T}};{\mathbb {R}}))$ for $T>0$ and $s>\big \lceil \frac{\alpha }{\alpha -1}\big \rceil -\frac{1}{2}$, where $H^s_0$ are the Sobolev spaces of functions with 0 mean value, proving that the result obtained in Said (A geometric proof of the quasi-linearity of the water-waves system and the incompressible Euler equations) is optimal on the torus. The proof relies on a paradifferential generalization of a complex Cole–Hopf gauge transformation introduced by Tao (J Hyperbol Differ Equ 1:27–49, 2004) for the Benjamin–Ono equation. For this we prove a generalization of the Baker–Campbell–Hausdorff formula for flows of hyperbolic paradifferential equations and prove the stability of the class of paradifferential operators modulo more regular remainders, under conjugation by such flows. For this we prove a new characterization of paradifferential operators in the spirit of Beals (Duke Math J 44:45–57, 1977). In the second part of this paper we use a paradifferential version of the previous method to prove that a re-normalization of the flow of the one dimensional periodic gravity–capillary equation is Lipschitz from bounded sets of $H^s$ to $C^0([0,T],H^{s-\frac{1}{2}})$ for $T>0$ and $s>3+\frac{1}{2}$. This proves that the result obtained in Said (A geometric proof of the quasi-linearity of the water-waves system and the incompressible Euler equations) is optimal for the water waves system.

Appendix A. Paradifferential Calculus

In this paragraph we review classic notations and results about paradifferential and pseudodifferential calculus that we need in this paper. We follow the presentations in [14, 15, 28, 42] which give an accessible and complete presentation.

Notation A.1

In the following presentation we will use the usual definitions and standard notations for regular functions $C^k$, $C^k_b$ for bounded ones and $C^k_0$ for those with compact support, the distribution space ${\mathscr {D}}'$, ${\mathscr {E}}'$ for those distribution with compact support, ${\mathscr {D}}'^k$,${\mathscr {E}}'^k$ for distributions of order k, $L^p$ Lebesgue spaces, $H^s$ and $W^{p,q}$Sobolev spaces and finally ${\mathscr {S}}$ for the Schwartz class and it’s dual ${\mathscr {S}}'$. All of those spaces are equipped with their standard topologies. We also use the Landau notation $O_{\left\| \ \right\| }(X)$.

For the definition of the periodic symbol classes we will need the following definitions and notations.

Remark A.1

For clarity in this section and the appendix we again present the symbolic calculus on $\mathbb {R}$. All the results stated here extend tautologically to the case of $\mathbb {T}$ by applying the rules of Remark 4.1.

1.1 A.1. Littlewood–Paley Theory

Definition A.1

(Littlewood–Paley decomposition) Pick $P_0\in C^\infty _0({\mathbb {R}}^d)$ so that:

$$\begin{aligned} P_0(\xi )=1 \quad \text { for } \left| \xi \right| <1 \text { and } P_0(\xi )=0 \text { for } \left| \xi \right| >2. \end{aligned}$$

We define a dyadic decomposition of unity by:

$$\begin{aligned} \text {for } k \ge 1, \ P_{\le k}(\xi )=P_0(2^{-k}\xi ), \ P_k(\xi )=P_{\le k}(\xi )-P_{\le k-1}(\xi ). \end{aligned}$$

Thus,

$$\begin{aligned} P_{\le k}(\xi )=\sum _{0\le j \le k}P_j(\xi ) \quad \text { and }\quad 1=\sum _{j=0}^\infty P_j(\xi ). \end{aligned}$$

Introduce the operator acting on ${\mathscr {S}} '({\mathbb {R}}^d)$:

$$\begin{aligned} P_{\le k}u={\mathscr {F}}^{-1}(P_{\le k}(\xi )u) \quad \text { and }\quad u_k={\mathscr {F}}^{-1}(P_k(\xi )u). \end{aligned}$$

Thus,

$$\begin{aligned} u=\sum _k u_k. \end{aligned}$$

Finally put for $k\ge 1, C_k={{\,\textrm{supp}\,}}\ P_k$ the set of rings associated to this decomposition.

An interesting property of the Littlewood–Paley decomposition is that even if the decomposed function is merely a distribution the terms of the decomposition are regular, indeed they all have compact spectrum and thus are entire functions. On classical functions spaces this regularization effect can be "measured" by the following inequalities due to Bernstein.

Proposition A.1

(Bernstein’s inequalities) Suppose that $a\in L^p({\mathbb {R}}^d)$ has its spectrum contained in the ball $\left\{ \left| \xi \right| \le \lambda \right\}$.

Then $a\in C^\infty$ and for all $\alpha \in {\mathbb {N}}^d$ and $1\le p \le q \le +\infty$, there is $C_{\alpha ,p,q}$ (independent of $\lambda$) such that

$$\begin{aligned} \left\| \partial ^{\alpha }_x a \right\| _{L^q} \le C_{\alpha ,p,q} \lambda ^{\left| \alpha \right| +\frac{d}{p}-\frac{d}{q}}\left\| a \right\| _{L^p}. \end{aligned}$$

In particular,

$$\begin{aligned}{} & {} \left\| \partial ^{\alpha }_x a \right\| _{L^q} \le C_{\alpha } \lambda ^{\left| \alpha \right| }\left\| a \right\| _{L^p}, \text { and for } p=2, p=\infty \\{} & {} \left\| a \right\| _{L^\infty }\le C \lambda ^{\frac{d}{2}} \left\| a \right\| _{L^2}. \end{aligned}$$

If moreover a has its spectrum is included in $\left\{ 0<\mu \le \left| \xi \right| \le \lambda \right\}$ then:

$$\begin{aligned} C_{\alpha ,q}^{-1} \mu ^{\left| \alpha \right| }\left\| a \right\| _{L^q}\le \left\| \partial ^{\alpha }_x a \right\| _{L^q} \le C_{\alpha ,q} \lambda ^{\left| \alpha \right| }\left\| a \right\| _{L^q}. \end{aligned}$$

Proposition A.2

For all $\mu >0$, there is a constant C such that for all $\lambda >0$ and for all $\alpha \in W^{\mu ,\infty }$ with spectrum contained in $\left\{ \left| \xi \right| \ge \lambda \right\}$. one has the following estimate:

$$\begin{aligned} \left\| a \right\| _{L^\infty }\le C \lambda ^{-\mu } \left\| a \right\| _{W^{\mu ,\infty }}. \end{aligned}$$

Definition A.2

(Zygmund spaces on ${\mathbb {R}}^d$) For $r\in {\mathbb {R}}$ we define the space:

$$\begin{aligned} C^r_*({\mathbb {R}}^d) \subset {\mathscr {S}}'({\mathbb {R}}^d), \ C^r_*({\mathbb {R}}^d)=\left\{ u\in {\mathscr {S}}'({\mathbb {R}}^d),\left\| u \right\| _{C_*^r}=\sup _q 2^{qr}\left\| u_q \right\| _{L^\infty }<\infty \right\} \end{aligned}$$

equipped with its canonical topology giving it a Banach space structure.

It’s a classical result that for $r\notin {\mathbb {N}}$, $C^r_*({\mathbb {R}}^d)=W^{r,\infty }({\mathbb {R}}^d)$ the classic Hölder spaces.

Proposition A.3

Let $\hbox {B}$ be a ball with center 0. There exists a constant C such that for all $r>0$ and for all $(u_q)_{q\in {\mathbb {N}}}$ in ${\mathscr {S}}'({\mathbb {R}}^d)$ verifying:

$$\begin{aligned}{} & {} \forall q,{{\,\textrm{supp}\,}}\hat{u}_q \subset 2^q \hbox {B}\quad \text { and }\quad (2^{qr}\left\| u_q \right\| _\infty )_{q\in {\mathbb {N}}} \text { is bounded,} \\{} & {} \quad \text {then}, u=\sum _q u_q \in C^r_*({\mathbb {R}}^d) \text { and } \left\| u \right\| _{C_*^r} \le \frac{C}{1-2^{-r}} \displaystyle {\sup _{q \in {\mathbb {N}}}}\ 2^{qr}\left\| u_q \right\| _{L^\infty }. \end{aligned}$$

Definition A.3

(Sobolev spaces on ${\mathbb {R}}^d$) It is also a classical result that for $s\in {\mathbb {R}}$:

$$\begin{aligned} H^s({\mathbb {R}}^d)=\left\{ u\in {\mathscr {S}}'({\mathbb {R}}^d),\left\| u \right\| _s= \bigg (\sum _q 2^{2qs} {\left\| u_q \right\| _{L^2}}^2 \bigg )^{\frac{1}{2}}<\infty \right\} \end{aligned}$$

with the right hand side equipped with its canonical topology giving it a Hilbert space structure and $\left\| \ \right\| _s$ is equivalent to the usual norm on $H^s$.

Proposition A.4

Let $\hbox {B}$ be a ball with center 0. There exists a constant C such that for all $s>0$ and for all $(u_q)_{\in {\mathbb {N}}}$ in ${\mathscr {S}}'({\mathbb {R}}^d)$ verifying:

$$\begin{aligned}{} & {} \forall q, \ {{\,\textrm{supp}\,}}\hat{u}_q \subset 2^q \hbox {B}\quad \text { and }\quad (2^{qs}\left\| u_q \right\| _{L^2})_{q\in {\mathbb {N}}} \text { is in } L^2({\mathbb {N}}), \\{} & {} \quad \text {then}, \ u=\sum _q u_q \in H^s({\mathbb {R}}^d) \text { and } \left\| u \right\| _s \le \frac{C}{1-2^{-s}} \bigg (\sum _q 2^{2qs} {\left\| u_q \right\| _{L^2}}^2 \bigg )^{\frac{1}{2}}. \end{aligned}$$

We recall the usual nonlinear estimates in Sobolev spaces:

• If $u_j\in H^{s_j}({\mathbb {R}}^d), j=1,2$, and $s_1+s_2>0$ then $u_1u_2 \in H^{s_0}({\mathbb {R}}^d)$ and if

  $$\begin{aligned}{} & {} s_0\le s_j, j=1,2 \quad \text { and }\quad s_0\le s_1+s_2-\frac{d}{2}, \ \ \ \\{} & {} \quad \text {then }\exists K\in {\mathbb {R}}, \left\| u_1u_2 \right\| _{H^{s_0}}\le K \left\| u_1 \right\| _{H^{s_1}}\left\| u_2 \right\| _{H^{s_2}}, \end{aligned}$$

  where the last inequality is strict if $s_1$ or $s_2$ or $-s_0$ is equal to $\frac{d}{2}$.

• For all $C^\infty$ function F vanishing at the origin, if $u \in H^s({\mathbb {R}}^d)$ with $s>\frac{d}{2}$, then

  $$\begin{aligned} \left\| F(u) \right\| _{H^s} \le C(\left\| u \right\| _{H^s}),\end{aligned}$$

  for some non decreasing function C depending only on F.

1.2 A.2. Paradifferential Operators

We start by the definition of symbols with limited spatial regularity. Let ${\mathscr {W}}\subset {\mathscr {S}}'$ be a Banach space.

Definition A.4

Given $m \in {\mathbb {R}}$, $\Gamma ^m_{\mathscr {W}}({\mathbb {R}})$ denotes the space of locally bounded functions $a(x,\xi )$ on ${\mathbb {R}}\times ({\mathbb {R}}{\setminus } 0)$, which are $C^\infty$ with respect to $\xi$ for $\xi \ne 0$ and such that, for all $\alpha \in {\mathbb {N}}$ and for all $\xi \ne 0$, the function $x \mapsto \partial ^\alpha _\xi a(x,\xi )$ belongs to ${\mathscr {W}}$ and there exists a constant $C_\alpha$ such that, for all $\epsilon >0$:

$$\begin{aligned} \forall \left| \xi \right| >\epsilon , \left\| \partial ^\alpha _\xi a(.,\xi ) \right\| _{{\mathscr {W}}}\le C_{\alpha ,\epsilon } (1+\left| \xi \right| )^{m-\left| \alpha \right| }. \end{aligned}$$

(A.1)

The spaces $\Gamma ^m_{\mathscr {W}}({\mathbb {R}})$ are equipped with their natural Fréchet topology induced by the semi-norms defined by the best constants in (A.1).

For quantitative estimates we introduce as in [28]:

Definition A.5

For $m\in {\mathbb {R}}$ and $a \in \Gamma ^m_{\mathscr {W}}({\mathbb {R}})$, we set

$$\begin{aligned}M^m_{\mathscr {W}}(a;n)=\sup _{\left| \alpha \right| \le n} \ \sup _{\left| \xi \right| \ge \frac{1}{2}}\left\| (1+\left| \xi \right| )^{m-\left| \alpha \right| }\partial ^\alpha _\xi a(.,\xi ) \right\| _{{\mathscr {W}}}, \text { for } n\in {\mathbb {N}}.\end{aligned}$$

For ${\mathscr {W}}=W^{\rho ,\infty },\rho \ge 0$, we write:

$$\begin{aligned} \Gamma ^m_{W^{\rho ,\infty }}({\mathbb {R}})=\Gamma ^m_\rho ({\mathbb {R}}) \quad \text { and }\quad M^m_\rho (a)=M^m_{W^{\rho ,\infty }}(a;1). \end{aligned}$$

Moreover we introduce the following spaces equipped with their natural Fréchet space structure:

$$\begin{aligned}{} & {} C^{\infty }_b({\mathbb {R}})=\cap _{\rho \ge 0}W^{\rho ,\infty }, \ \Gamma ^m_\infty ({\mathbb {R}})=\cap _{\rho \ge 0}\Gamma ^m_\rho ({\mathbb {R}}), \ \Gamma ^{-\infty }_\rho ({\mathbb {R}})=\cap _{m\in {\mathbb {R}}}\Gamma ^m_\rho ({\mathbb {R}}) \text { and,} \\{} & {} \Gamma ^{-\infty }_\infty ({\mathbb {R}})=\cap _{\rho \ge 0}\cap _{m\in {\mathbb {R}}}\Gamma ^m_\rho ({\mathbb {R}}). \end{aligned}$$

In higher dimensions the 1 in the definition of $M^m_\rho$ should be replaced by $1+\lfloor \frac{d}{2}\rfloor$.

Definition A.6

Define an admissible cutoff function as a function $\psi ^{B,b}\in C^\infty (\hat{D}^2)$, $B>1,b>0$ that verifies:

• (1)
  $$\begin{aligned} \psi ^{B,b}(\xi ,\eta )=0 \quad \text { when } \left| \xi \right| < B\left| \eta \right| +b, \text { and } \psi ^{B,b}(\xi ,\eta )=1 \text { when } \left| \xi \right| >B\left| \eta \right| +b+1. \end{aligned}$$

• (2) for all $(\alpha ,\beta )\in {\mathbb {N}}^d \times {\mathbb {N}}^d,$ there is $C_{\alpha _\beta }$, with $C_{0,0}\le 1$, such that:

  $$\begin{aligned} \forall (\xi ,\eta ): \left| \partial _\xi ^\alpha \partial _\eta ^\beta \psi ^{B,b}(\xi ,\eta ) \right| \le C_{\alpha ,\beta } (1+\left| \xi \right| )^{-\left| \alpha \right| -\left| \beta \right| }. \end{aligned}$$

  (A.2)

Definition A.7

Consider a real numbers $m\in {\mathbb {R}}$, a symbol $a\in \Gamma ^m_{\mathscr {W}}$ and an admissible cutoff function $\psi ^{B,b}$ define the paradifferential operator $T_a$ by:

$$\begin{aligned} \widehat{T_a u}(\xi )=(2\pi )^{-1}\int \limits _{{\mathbb {R}}}\psi ^{B,b}(\xi -\eta ,\eta )\hat{a}(\xi -\eta ,\eta )\hat{u}(\eta )\textrm{d}\eta , \end{aligned}$$

where $\hat{a}(\eta ,\xi )=\int e^{-ix\cdot \eta }a(x,\xi )\textrm{d}x$ is the Fourier transform of a with respect to the first variable. In the language of pseudodifferential operators:

$$\begin{aligned} T_a u={{\,\textrm{Op}\,}}(\sigma _a)u, \quad \text { where } {\mathscr {F}}_x\sigma _a(\xi ,\eta )=\psi ^{B,b}(\xi ,\eta ) {\mathscr {F}}_x a(\xi ,\eta ). \end{aligned}$$

Let $G_{\psi ^{B,b}}(x,\eta )={\mathscr {F}}^{-1}_x\psi ^{B,b}(\cdot ,\eta )$ then $\sigma _a(\cdot ,\eta )=G_{\psi ^{B,b}}(\cdot ,\eta )*a(\cdot ,\eta )$,in particular for a Fourier multiplier m(D), $T_{m(D)}\ne 0$.

An important property of paradifferential operators is their action on functions with localized spectrum.

Lemma A.1

Consider two real numbers $m\in {\mathbb {R}}$, $\rho \ge 0$, a symbol $a\in \Gamma ^m_0({\mathbb {R}})$, an admissible cutoff function $\psi ^{B,b}$ and $u \in {\mathscr {S}}({\mathbb {R}}^d)$.

• For $R>>b$, if ${{\,\textrm{supp}\,}}{\mathscr {F}}u \subset \left\{ \left| \xi \right| \le R\right\} ,$ then:

  $$\begin{aligned} {{\,\textrm{supp}\,}}{\mathscr {F}}T_a u \subset \left\{ \left| \xi \right| \le (1+\frac{1}{B})R-\frac{b}{B}\right\} , \end{aligned}$$

  (A.3)

• For $R>>b$, if ${{\,\textrm{supp}\,}}{\mathscr {F}}u \subset \left\{ \left| \xi \right| \ge R\right\} ,$ then:

  $$\begin{aligned} {{\,\textrm{supp}\,}}{\mathscr {F}}T_a u \subset \left\{ \left| \xi \right| \ge \left( 1-\frac{1}{B}\right) R+\frac{b}{B}\right\} , \end{aligned}$$

  (A.4)

The main features of symbolic calculus for paradifferential operators are given by the following theorems taken from [28] and [34].

Theorem A.1

Let $m \in {\mathbb {R}}$. if $a\in \Gamma ^m_0({\mathbb {R}})$, then $T_a$ is of order m. Moreover, for all $\mu \in {\mathbb {R}}$ there exists a constant K such that:

$$\begin{aligned}{} & {} \left\| T_a \right\| _{H^\mu \rightarrow H^{\mu -m}}\le K M^m_0(a),\text { and,}\\{} & {} \left\| T_a \right\| _{W^{\mu ,\infty } \rightarrow W^{\mu -m,\infty }}\le K M^m_0(a), \mu \notin {\mathbb {N}}. \end{aligned}$$

Theorem A.2

Let $m,m' \in {\mathbb {R}}$, and $\rho >0$, $a \in \Gamma ^m_\rho ({\mathbb {R}})$and $b \in \Gamma ^{m'}_\rho ({\mathbb {R}})$.

• Composition: Then $T_a T_b$ is a paradifferential operator with symbol:

  $$\begin{aligned}{} & {} a \otimes b\in \Gamma ^{m+m'}_\rho ({\mathbb {R}}),\text { more precisely,} \\{} & {} T^{\psi ^{B,b}}_a T^{\psi ^{B',b}}_b= T^{\psi ^{\frac{BB'}{B+B'+1},b}}_{a\otimes b}. \end{aligned}$$

  Moreover $T_a T_b- T_{a\#b}$ is of order $m+m'-\rho$ where $a \#b$ is defined by:

  $$\begin{aligned}a \#b=\sum _{\left| \alpha \right| <\rho }\frac{1}{i^{\left| \alpha \right| }\alpha !} \partial ^\alpha _\xi a \partial ^\alpha _x b, \end{aligned}$$

  and there exists $r\in \Gamma ^{m+m'-\rho }_0({\mathbb {R}})$ such that:

  $$\begin{aligned} M^{m+m'-\rho }_0(r) \le K (M^m_\rho (a) M^{m'}_0(b)+M^m_0 (a) M^{m'}_\rho (b)), \end{aligned}$$

  and we have

  $$\begin{aligned} T^{\psi ^{B,b}}_a T^{\psi ^{B',b}}_b- T^{\psi ^{\frac{BB'}{B+B'+1},b}}_{a\#b}=T^{\psi ^{\frac{BB'}{B+B'+1},b}}_r. \end{aligned}$$

• Adjoint: The adjoint operator of $T_a$, $T_a^*$ is a paradifferential operator of order m with symbol $a^*$ defined by:

  $$\begin{aligned} a^*=\sum _{\left| \alpha \right| <\rho } \frac{1}{i^{\left| \alpha \right| }\alpha !}\partial ^\alpha _\xi \partial ^\alpha _x \bar{a}. \end{aligned}$$

  Moreover, for all $\mu \in {\mathbb {R}}$ there exists a constant K such that

  $$\begin{aligned} \left\| T_a^*-T_{a^*} \right\| _{H^\mu \rightarrow H^{\mu -m+\rho }} \le K M^m_\rho (a). \end{aligned}$$

If $a=a(x)$ is a function of x only then the paradifferential operator $T_a$ is called a paraproduct. It follows from Theorem A.2 and the Sobolev embedding that:

• If $a \in H^\alpha ({\mathbb {R}})$ and $b \in H^\beta ({\mathbb {R}})$ with $\alpha ,\beta >\frac{d}{2}$, then

  $$\begin{aligned}T_aT_b-T_{ab} \text { is of order } -\bigg ( \min \left\{ \alpha ,\beta \right\} -\frac{d}{2} \bigg ).\end{aligned}$$

• If $a \in H^\alpha ({\mathbb {R}})$ with $\alpha >\frac{d}{2}$, then

  $$\begin{aligned}T_a^*-T_{a^*} \text { is of order } -\bigg (\alpha -\frac{d}{2} \bigg ).\end{aligned}$$

• If $a \in W^{r,\infty }({\mathbb {R}})$, $r\in {\mathbb {N}}$ then:

  $$\begin{aligned}\left\| au-T_au \right\| _{H^r} \le C \left\| a \right\| _{W^{r,\infty }} \left\| u \right\| _{L^2}.\end{aligned}$$

An important feature of paraproducts is that they are well defined for function $a=a(x)$ which are not $L^\infty$ but merely in some Sobolev spaces $H^r$ with $r<\frac{d}{2}$.

Proposition A.5

Let $m>0$. If $a\in H^{\frac{d}{2}-m}({\mathbb {R}})$ and $u \in H^\mu ({\mathbb {R}})$ then $T_au \in H^{\mu -m}({\mathbb {R}})$. Moreover,

$$\begin{aligned} \left\| T_a u \right\| _{H^{\mu -m}}\le K \left\| a \right\| _{H^{\frac{d}{2} -m}}\left\| u \right\| _{H^{\mu }} \end{aligned}$$

A main feature of paraproducts is the existence of paralinearisation theorems which allow us to replace nonlinear expressions by paradifferential expressions, at the price of error terms which are smoother than the main terms.

Theorem A.3

Let $\alpha , \beta ,\kappa \in {\mathbb {R}}$ be such that $\alpha ,\beta > \frac{d}{2}$ and $\kappa \ge 0$, then

• Bony’s Linearization Theorem: For all $C^\infty$ function F, if $a \in H^\alpha ({\mathbb {R}})$ then;

  $$\begin{aligned} F(a)- F(0)-T_{F'(a)}a \in H^{2\alpha -\frac{d}{2}} ({\mathbb {R}}). \end{aligned}$$

• If $a\in H^\alpha ({\mathbb {R}})$, $b\in H^\beta ({\mathbb {R}})$ and $c\in W^{\kappa ,\infty }$, then $R(a,b)=ab-T_ab-T_ba \in H^{\alpha + \beta -\frac{d}{2}} ({\mathbb {R}})$ and $R(a,c)=ac-T_ac-T_ca \in H^{\alpha + \kappa } ({\mathbb {R}})$. Moreover there exists a positive constant K independent of a, b and c such that:

  $$\begin{aligned} {\left\{ \begin{array}{ll} \left\| R(a,b) \right\| _{H^{\alpha + \beta -\frac{d}{2}} }=\left\| ab-T_ab-T_ba \right\| _{H^{\alpha + \beta -\frac{d}{2}} }\le K \left\| a \right\| _{H^\alpha } \left\| b \right\| _{H^\beta },\\ \left\| R(a,c) \right\| _{H^{\alpha + \kappa } }\le K \left\| a \right\| _{H^\alpha } \left\| c \right\| _{W^{\kappa ,\infty }}. \end{array}\right. } \end{aligned}$$

  (A.5)

1.3 A.3. Paracomposition

We recall the main properties of the paracomposition operator first introduced by Alinhac in [7] to treat low regularity change of variables. Here we present the results we reviewed and generalized in some cases in [34].

Theorem A.4

Let $\chi :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$ be a $C^{1+r}$ diffeomorphism with $D\chi \in W^{r,\infty }$, $r>0, r\notin {\mathbb {N}}$ and take $s \in {\mathbb {R}}$ then the following map is continuous:

$$\begin{aligned} H^s({\mathbb {R}}^d)&\rightarrow H^s({\mathbb {R}}^d)\\ u&\mapsto \chi ^* u=\sum _{k\ge 0} \sum _{\begin{array}{c} l\ge 0 \\ k-N \le l \le k+N \end{array}}P_l(D)u_k\circ \chi , \end{aligned}$$

where $N \in {\mathbb {N}}$ is such that $2^{N}>\sup _{k,{\mathbb {R}}^d} \left| \Phi _k D\chi \right| ^{-1}$ and $2^{N}>\sup _{k,{\mathbb {R}}^d} \left| \Phi _k D\chi \right|$.

Taking $\tilde{\chi }:{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$ a $C^{1+\tilde{r}}$ diffeomorphism with $D\chi \in W^{\tilde{r},\infty }$ map with $\tilde{r}>0$, then the previous operation has the natural fonctorial property:

$$\begin{aligned}{} & {} \forall u \in H^s({\mathbb {R}}^d), \chi ^* \tilde{\chi }^* u= ({\chi \circ \tilde{\chi }})^* u +Ru,\\{} & {} \quad \text {with, } R:H^{s}({\mathbb {R}}^d) \rightarrow H^{s+min(r,\tilde{r})}({\mathbb {R}}^d) \text { continous}. \end{aligned}$$

We now give the key paralinearization theorem taking into account the paracomposition operator.

Theorem A.5

Let u be a $W^{1,\infty }({\mathbb {R}}^d)$ map and $\chi :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$ be a $C^{1+r}$ diffeomorphism with $D\chi \in W^{r,\infty }$, $r>0, r\notin {\mathbb {N}}$. Then:

$$\begin{aligned} u \circ \chi (x)=\chi ^* u(x)+ T_{Du\circ \chi }\chi (x)+ R_0(x)+R_1(x)+R_2(x) \end{aligned}$$

where the paracomposition given in the previous theorem verifies the estimates:

$$\begin{aligned}{} & {} \forall s \in {\mathbb {R}}, \left\| \chi ^* u(x) \right\| _{H^s}\le C(\left\| D\chi \right\| _{L^\infty })\left\| u(x) \right\| _{H^s},\\{} & {} \quad u'\circ \chi \in \Gamma ^0_{W^{0,\infty }({\mathbb {R}}^d)}({\mathbb {R}}^d) \quad \text { for } u \text { Lipchitz,} \end{aligned}$$

and the remainders verify the estimates:

$$\begin{aligned}{} & {} \left\| R_0 \right\| _{H^{1+r +min(1+\rho ,s-\frac{d}{2})}} \le C\left\| D\chi \right\| _{C_*^r}\left\| u \right\| _{H^{1+s}} \\{} & {} \left\| R_1 \right\| _{H^{1+r+s}} \le C(\left\| D\chi \right\| _{L^\infty }) \left\| D\chi \right\| _{C_*^r}\left\| u \right\| _{H^{1+s}}. \\{} & {} \left\| R_2 \right\| _{H^{1+r+s}} \le C\left( \left\| D\chi \right\| _{L^\infty },\left\| D\chi ^{-1} \right\| _{L^\infty }\right) \left\| D\chi \right\| _{C_*^r}\left\| u \right\| _{H^{1+s}}. \end{aligned}$$

Finally the commutation between a paradifferential operator $a \in \Gamma ^m_{\beta }({\mathbb {R}}^d)$ and a paracomposition operator $\chi ^*$ is given by the following

$$\begin{aligned} \chi ^* T_a u =T_{a^*} \chi ^* u+T_{{q}^*} \chi ^* u \text { with } q \in \Gamma ^{m-\beta }_{0}({\mathbb {R}}^d), \end{aligned}$$

where $a^*$ has the local expansion:

$$\begin{aligned} a^*(x,\xi ) \sim \sum _{\begin{array}{c} \alpha \\ \left| \alpha \right| \le \lfloor min(r,\rho ) \rfloor \end{array}}\frac{1}{\alpha !}\partial ^\alpha a(\chi (x),D\chi ^{-1}(\chi (x))^\top \xi )Q_{\alpha }(\chi (x),\xi )\in \Gamma ^m_{\min (r,\beta )}({\mathbb {R}}^d),\nonumber \\ \end{aligned}$$

(A.6)

where,

$$\begin{aligned} P_{\alpha }(x',\xi )=D^\alpha _{y'}(e^{i(\chi ^{-1}(y')-\chi ^{-1}(x')-D \chi ^{-1}(x')(y'-x')).\xi })_{|y'=x'} \end{aligned}$$

and $Q_{\alpha }$ is polynomial in $\xi$ of degree $\le \frac{\left| \alpha \right| }{2}$, with $Q_{0}=1, Q_{1}=0$.

Remark A.2

The simplest example for the paracomposition operator is when $\chi (x)=Ax$ is a linear operator and in that case we see that if N is chosen sufficiently large in the definition of $\chi ^*$:

$$\begin{aligned} u(Ax) = (Ax)^*u,\text { and } T_{u'(Ax)}Ax = 0. \end{aligned}$$