On the generalized nonlinear Camassa-Holm equation

TL;DR

This paper investigates a generalized nonlinear Camassa-Holm equation with variable coefficients, demonstrating control of dispersive terms through weighted energy and establishing solution existence and uniqueness via iterative methods.

Contribution

It introduces a framework for controlling dispersive effects in a generalized equation with variable coefficients and proves well-posedness using Picard iteration.

Findings

Control of higher order dispersive terms using weighted energy functions

Existence and uniqueness of solutions established

Applicable to equations with time- and space-dependent coefficients

Abstract

In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method.

Equations

$$\left\{\begin{array}[]{l}(1-m\partial_{x}^{2})u_{t}+a_{1}(t,x,u)u_{x}+a_{2}(t,x,u,u_{x})u_{xx}\\ \\ \qquad+a_{3}(t,x,u)u_{xxx}+a_{4}(t,x)u_{xxxx}+a_{5}(t,x)u_{xxxxx}=f\quad\hbox{ for}\quad(t,x)\in(0,T]\times\mathbb{R}\\ \\ u_{|_{t=0}}=u^{0},\end{array}\right.$$

$$|f|_{L^{p}}=\big{(}\int_{\mathbb{R}}|f(x)|^{p}dx\big{)}^{1/p}<\infty.$$

$$(f_1,f_2)={\int }_{\mathbb{R}}{f}_1(x)f_2(x)dx.$$

$$|f{|}_{L^{\infty }}=\sup |f(x)|<\infty .$$

$$\displaystyle|u|_{H^{s+2}}\$$

$${\Lambda }^s{\Lambda }_m^{-2}={\Lambda }^{s-2}{\Lambda }_m^0={\Lambda }_m^0{\Lambda }^{s-2},$$

$$\widehat{{\Lambda }_m^0}u(\xi )=(1+{\xi }^2)(1+m{\xi }^2{)}^{-1}\hat{u}(\xi ),$$

$$\displaystyle|\Lambda^{0}_{m}|_{H^{s}\to H^{s}}\$$

$$\displaystyle|(\Lambda^{0}_{m})^{-1}|_{H^{s}\to H^{s}}\$$

$$\mathcal{L}(v,\partial )$$

$$\left\{\begin{array}[]{l}{\mathcal{L}}(v,\partial)u=f,\\ u_{|_{t=0}}=u^{0}.\end{array}\right.$$

$$E^s(u{)}^2=|w{\Lambda }^{s}u{|}_{L^2}^2,$$

$$w_1\le w(t,x)\le w_2,$$

$$\displaystyle\frac{1}{2}e^{\lambda t}\partial_{t}(e^{-\lambda t}E^{s}(u)^{2})=-\frac{\lambda}{2}E^{s}(u)^{2}-\big{(}\Lambda_{m}^{0}\Lambda^{s-2}(a_{1}u_{x}),w^{2}\Lambda^{s}u\big{)}$$

$$\displaystyle|\big{(}\Lambda^{s-2}(a_{1}u_{x}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}|$$

$$\le C(m^{-1},|a_1{|}_{C^{[s]+1}},\Vert v{\Vert }_{H^s},|w{|}_{L^{\infty }})E^s(u{)}^2.$$

$$\displaystyle|\big{(}\Lambda^{s-2}(a_{2}u_{xx}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}|$$

$$\le C(m^{-1},|a_2{|}_{C^{[s]+1}},\Vert v{\Vert }_{H^s},|w{|}_{L^{\infty }})E^s(u{)}^2.$$

$$\big{(}a_{3}\Lambda^{s}\partial_{x}u,\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}=-\frac{1}{2}\big{(}\partial_{x}(\Lambda^{0}_{m}w^{2}a_{3}),(\Lambda^{s}u)^{2}\big{)},$$

$$\displaystyle|\big{(}\Lambda^{s-2}(a_{3}u_{xxx}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}|$$

$$\big{(}a_{4}\Lambda^{s-2}\partial_{x}^{4}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}=\big{(}a_{4}\Lambda^{s-2}\partial^{2}_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}-\big{(}a_{4}\Lambda^{s}\partial^{2}_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}$$

$$|\big{(}a_{4}\Lambda^{s-2}\partial^{2}_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}|\leq\frac{1}{m}|a_{4}\Lambda^{s-2}\partial^{2}_{x}u|_{L^{2}}E^{s}(u)\leq C(m^{-1},|a_{4}|_{L^{\infty}})E^{s}(u)^{2}.$$

$$([{\Lambda }^{s-2},a_4]{\partial }_x^{4}u,{\Lambda }_m^0{w}^2{\Lambda }^{s}u)=(s-2)({\partial }_x(a_4){\Lambda }^s{\partial }_{x}u,{\Lambda }_m^0{w}^2{\Lambda }^{s}u)+Q_2,$$

$$(s-2)({\partial }_x(a_4){\Lambda }^s{\partial }_{x}u,{\Lambda }_m^0{w}^2{\Lambda }^{s}u)=-\frac{(s-2)}{2}({\partial }_x({\partial }_x(a_4){\Lambda }_m^0{w}^2){\Lambda }^{s}u,{\Lambda }^{s}u)$$

$$|([{\Lambda }^{s-2},a_4]{\partial }_x^{4}u,{\Lambda }_m^0{w}^2{\Lambda }^{s}u)|\le C(m,s,|w{|}_{W^{2,\infty }},|a_4{|}_{H^{s+1}})E^s(u{)}^2.$$

$$a_5{\Lambda }^{s-2}{\partial }_x^{5}u=a_5{\Lambda }^{s-2}(1-{\Lambda }^2){\partial }_x^{3}u=a_5{\Lambda }^{s-2}{\partial }_x^{3}u-a_5{\Lambda }^s{\partial }_x^{3}u=a_5{\Lambda }^{s-2}{\partial }_{x}u-a_5{\Lambda }^s{\partial }_{x}u-a_5{\Lambda }^s{\partial }_x^{3}u$$

$$\displaystyle\big{(}a_{5}\partial_{x}^{3}\Lambda^{s}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}$$

$$\displaystyle-\frac{3}{2}\big{(}\partial_{x}(\Lambda_{m}^{0}w^{2}a_{5})\Lambda^{s}u,\Lambda^{s}\partial_{x}^{2}u\big{)}=\frac{3}{2}\big{(}\partial_{x}^{2}(\Lambda_{m}^{0}w^{2}a_{5})\Lambda^{s}u,\Lambda^{s}\partial_{x}u\big{)}+\frac{3}{2}\big{(}\partial_{x}(a_{5}\Lambda_{m}^{0}w^{2}),(\Lambda^{s}\partial_{x}u)^{2}\big{)}.$$

$$[{\Lambda }^{s-2},a_5]{\partial }_x^{5}u=\{{\Lambda }^{s-2},a_5{\}}_2{\partial }_x^{5}u+Q_3{\partial }_x^{5}u,$$

$$\{{\Lambda }^{s-2},a_5{\}}_2=-(s-2){\partial }_x(a_5){\Lambda }^{s-4}{\partial }_x+\frac{1}{2}[(s-2){\partial }_x^2(a_5){\Lambda }^{s-4}-(s-4)(s-2){\partial }_x^2(a_5){\Lambda }^{s-6}{\partial }_x^2]$$

$$|\big{(}Q_{3}\partial_{x}^{5}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}|\leq C(m,|\partial_{x}a_{5}|_{H^{s+1}})E^{s}(u)^{2}.$$

Full text

1. Introduction

1.1. Presentation of the problem

In this paper, we study the Cauchy problem for the general nonlinear higher order Camassa-Holm-type equation:

[TABLE]

where $u=u(t,x)$, from $[0,T]\times\mathbb{R}$ into $\mathbb{R}$, is the unknown function of the problem, $m>0$ and $a_{i}$, $1\leq i\leq 5$, are real-valued smooth given functions where their exact regularities will be precised later. This equation covers several important unidirectional models for the water waves problems at different regimes which take into account the variations of the bottom. We have in view in particular the example of the Camassa-Holm equation (see[5]), which is more nonlinear then the KdV equation (see for instance [6],[3], [4], [11], [10]). However, the most prominent example that we have in mind is the Kawahara-type approximation ( see [1]), in which case the coefficient $a_{5}$ does not vanish. The presence of the fifth order derivative term is very important, so that the equation describes both nonlinear and dispersive effects as does the Camassa-Holm equation in the case of special tension surface values (see [8]).

Looking for solutions of (1.1) plays an important and significant role in the study of unidirectional limits for water wave problems with variable depth and topographies. To our knowledge the problem (1.1) has not been analyzed previously. In the present paper, we prove the local well-posedness of the initial value problem (1.1) by a standard Picard iterative scheme and the use of adequate energy estimates under a condition of nondegeneracy of the higher dispersive coefficient $a_{5}$.

1.2. Notations and Main result

In the following, $C_{0}$ denotes any nonnegative constant whose exact expression is of no importance. The notation $a\lesssim b$ means that $a\leq C_{0}\ b$.

We denote by $C(\lambda_{1},\lambda_{2},\dots)$ a nonnegative constant depending on the parameters $\lambda_{1}$, $\lambda_{2}$,…and whose dependence on the $\lambda_{j}$ is always assumed to be nondecreasing.

For any $s\in\mathbb{R}$, we denote $[s]$ the integer part of $s$.

Let $p$ be any constant with $1\leq p<\infty$ and denote $L^{p}=L^{p}(\mathbb{R})$ the space of all Lebesgue-measurable functions $f$ with the standard norm

[TABLE]

The real inner product of any two functions $f_{1}$ and $f_{2}$ in the Hilbert space $L^{2}(\mathbb{R})$ is denoted by

[TABLE]

The space $L^{\infty}=L^{\infty}(\mathbb{R})$ consists of all essentially bounded and Lebesgue-measurable functions $f$ with the norm

[TABLE]

We denote by $W^{1,\infty}(\mathbb{R})=\{f,\mbox{ s.t. }f,\partial_{x}f\in L^{\infty}(\mathbb{R})\}$ endowed with its canonical norm.

For any real constant $s\geq 0$, $H^{s}=H^{s}(\mathbb{R})$ denotes the Sobolev space of all tempered distributions $f$ with the norm $|f|_{H^{s}}=|\Lambda^{s}f|_{L^{2}}<\infty$, where $\Lambda$ is the pseudo-differential operator $\Lambda=(1-\partial_{x}^{2})^{1/2}$.

For any two functions $u=u(t,x)$ and $v(t,x)$ defined on $[0,T)\times\mathbb{R}$ with $T>0$, we denote the inner product, the $L^{p}$-norm and especially the $L^{2}$-norm, as well as the Sobolev norm, with respect to the spatial variable $x$, by $(u,v)=(u(t,\cdot),v(t,\cdot))$, $|u|_{L^{p}}=|u(t,\cdot)|_{L^{p}}$, $|u|_{L^{2}}=|u(t,\cdot)|_{L^{2}}$ , and $|u|_{H^{s}}=|u(t,\cdot)|_{H^{s}}$, respectively.

We denote $L^{\infty}([0,T);H^{s}(\mathbb{R}))$ the space of functions such that $u(t,\cdot)$ is controlled in $H^{s}$, uniformly for $t\in[0,T)$: \big{\|}u\big{\|}_{L^{\infty}([0,T);H^{s}(\mathbb{R}))}\ =\ \sup_{t\in[0,T)}|u(t,\cdot)|_{H^{s}}\ <\ \infty.

Finally, $C^{k}(\mathbb{R}^{i})$, $i\geq 1$ denote the space of $k$-times continuously differentiable functions.

For any closed operator $T$ defined on a Banach space $X$ of functions, the commutator $[T,f]$ is defined by $[T,f]g=T(fg)-fT(g)$ with $f$, $g$ and $fg$ belonging to the domain of $T$. The same notation is used for $f$ as an operator mapping the domain of $T$ into itself.

Moreover, we define the following operators: $\displaystyle{\Lambda_{m}^{s}=(1-m\partial_{x}^{2})^{\frac{s}{2}}}$ and its inverse $\displaystyle{\Lambda_{m}^{-s}}$ such that $\displaystyle{\displaystyle{\hat{\Lambda}_{m}^{-s}(u)}=(1+m\xi^{2})^{-\frac{s}{2}}\hat{u}}.$

Finally, we will study the local well-posedness of the initial value problem (1.1) in $H^{s}(\mathbb{R})$ endowed with canonical norm.

Let us now state our main result:

Theorem 1.1**.**

Let $s>\frac{5}{2}$ and $f\in C([0,T];H^{s}(\mathbb{R}))$. We suppose that:

• •

$a_{1},a_{3}$* in $C([0,T],C^{[s]+1}(\mathbb{R}^{2}))$, $a_{2}$ in $C([0,T],C^{[s]+1}(\mathbb{R}^{3}))$ and $\partial^{k}_{x}a_{j}$ are bounded with respect to $x$ for all $0\leq k\leq[s]+1$ and $1\leq j\leq 3$.*

• •

$a_{4}\in C([0,T];H^{s+1}(\mathbb{R}))$, $\partial_{t}a_{4}\in L^{\infty}(0,T,L^{\infty}(\mathbb{R}))$

• •

$a_{5}\in C([0,T];H^{s+2}(\mathbb{R}))\cap C([0,T],L^{\infty}(\mathbb{R}))$* ** with** $\partial_{t}a_{5}\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}))$,*

• •

$F(t,x):=\int_{0}^{x}\frac{a_{4}}{a_{5}}dy\in C([0,T];L^{\infty}(\mathbb{R}))~{}\text{and}~{}\partial_{t}F\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}))$,

Assume moreover that there is a positive constant $c_{1}>0$ such that $c_{1}\leq|a_{5}(t,x)|\;\forall\,(t,x)\in[0,T]\times\mathbb{R}.$ Then for all $u^{0}\in H^{s}(\mathbb{R})$, there exist a time $T^{\star}>0$ and a unique solution $u$ to (1.1) in $C([0,T^{\star}];H^{s})$.

2. Proof of the Main results

Before we start the proof, we give the following useful lemma:

Lemma 2.1**.**

*Let $m>0$, $s\in\mathbb{R}^{+}$ then the linear operator $\Lambda^{2}_{m}$: $H^{s+2}(\mathbb{R})\to H^{s}(\mathbb{R})$ is well defined, continuous, one-to-one and onto. If we suppose that $u=\Lambda^{-2}_{m}f$ for $f\in H^{s}(\mathbb{R})$ then

[TABLE]

Moreover,

[TABLE]

where $\Lambda^{0}_{m}$: $H^{s}(\mathbb{R})\to H^{s}(\mathbb{R})$ is linear continuous one-to-one and onto operator defined by

[TABLE]

with

[TABLE]

Proof.

$\|\Lambda^{-2}_{m}f\|_{H^{s+2}}=\|(1+\xi^{2})^{\frac{s}{2}+1}(1+m\xi^{2})^{-1}\hat{f}\|_{L^{2}}$.

If $m\geq 1$, then $1+m\xi^{2}\geq 1+\xi^{2}$ and $\frac{1+\xi^{2}}{1+m\xi^{2}}\leq 1$, then $\|(1+\xi^{2})^{\frac{s}{2}+1}(1+m\xi^{2})^{-1}\hat{f}\|_{L^{2}}=\|(1+\xi^{2})^{\frac{s}{2}}(1+\xi^{2})(1+m\xi^{2})^{-1}\hat{f}\|_{L^{2}}\leq\|(1+\xi^{2})^{\frac{s}{2}}\hat{f}\|_{L^{2}}$.

If $0<m<1$, we have $\frac{1+\xi^{2}}{1+m\xi^{2}}=1+(1-m)\frac{\xi^{2}}{1+m\xi^{2}}\leq 1+\frac{(1-m)}{m}=\frac{1}{m}$, then

$\|\Lambda^{-2}_{m}f\|_{H^{s+2}}\leq\frac{1}{m}\|f\|_{H^{s}}$.

Now $\|\Lambda^{0}_{m}f\|_{H^{s}}=\|\Lambda^{2}\Lambda^{-2}_{m}f\|_{H^{s}}=\|\Lambda^{-2}_{m}f\|_{H^{s+2}}\leq\max(1,\frac{1}{m})\|f\|_{H^{s}}$.

$\|(\Lambda^{0}_{m})^{-1}f\|_{H^{s}}=\|(1+m\xi^{2})(1+\xi^{2})^{-1}(1+\xi^{2})^{{\frac{s}{2}}}\hat{f}\|_{L^{2}}$.

If $m\geq 1$, then $(1+m\xi^{2})(1+\xi^{2})^{-1}=1+(m-1)\frac{\xi^{2}}{1+\xi^{2}}\leq m$, then $\|(\Lambda^{0}_{m})^{-1}f\|_{H^{s}}\leq m\|f\|_{H^{s}}.$

If $0<m<1$, $(1+m\xi^{2})(1+\xi^{2})^{-1}\leq 1$, then $\|(\Lambda^{0}_{m})^{-1}f\|_{H^{s}}\leq\|f\|_{H^{s}}$.

Finally $\|(\Lambda^{0}_{m})^{-1}f\|_{H^{s}}\leq\max(1,m)\|f\|_{H^{s}}$.

We will start the proof of Theorem 1.1 by studying a linearized problem associated to (1.1):

2.1. Linear analysis:

For any smooth enough $v$, we define the “linearized” operator:

[TABLE]

and the following initial value problem:

[TABLE]

Equation (2.6) is a linear equation which can be solved by a standard method (see [9]) in any time interval in which its coefficients are defined and regular enough. We first establish some precise energy-type estimates of the solution. We define the “energy” norm,

[TABLE]

where $w$ is a weight function that will be chosen later. For the moment, we just require that there exists two positive numbers $w_{1},w_{2}$ such that for all $(t,x)$ in $(0,T]\times\mathbb{R}$,

[TABLE]

so that $E^{s}(u)$ is uniformly equivalent to the standard $H^{s}$-norm. Differentiating $\frac{1}{2}e^{-\lambda t}E^{s}(u)^{2}$ with respect to time, one gets using (2.6)

[TABLE]

We now turn to estimating the different terms of the r.h.s of the previous identity.

$\bullet$ Estimate of \big{(}\Lambda^{s-2}(a_{1}u_{x}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}. By the Cauchy-Schwarz inequality we have

[TABLE]

$\bullet$ Estimate of \big{(}\Lambda^{s-2}(a_{2}u_{xx}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}. By the Cauchy-Schwarz inequality, we have

[TABLE]

$\bullet$Estimate of \big{(}\Lambda^{s-2}(a_{3}u_{xxx}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}.

We have that: $a_{3}u_{xxx}=\partial_{x}^{2}(a_{3}\partial_{x}u)-\partial_{x}^{2}a_{3}\partial_{x}u-2a_{3}\partial_{x}^{2}u$, then $\Lambda^{s-2}(a_{3}u_{xxx})=\Lambda^{s-2}(\partial_{x}^{2}(a_{3}\partial_{x}u))-\Lambda^{s-2}(\partial_{x}^{2}a_{3}\partial_{x}u)-2\Lambda^{s-2}(a_{3}\partial_{x}^{2}u)$.

Now use the identity $\Lambda^{2}=1-\partial_{x}^{2}$ to get that \Lambda^{s-2}(\partial_{x}^{2}(a_{3}\partial_{x}u))=\Lambda^{s-2}\big{(}(1-\Lambda^{2})(a_{3}\partial_{x}u)\big{)}=\Lambda^{s-2}(a_{3}\partial_{x}u)-\Lambda^{s}(a_{3}\partial_{x}u)=\Lambda^{s-2}(a_{3}\partial_{x}u)-[\Lambda^{s},a_{3}]\partial_{x}u-a_{3}\Lambda^{s}\partial_{x}u, then we obtain:

\big{(}\Lambda^{s-2}(a_{3}u_{xxx}),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}=\big{(}\Lambda^{s-2}(a_{3}\partial_{x}u),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}-\big{(}[\Lambda^{s},a_{3}]\partial_{x}u,\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}-\big{(}a_{3}\Lambda^{s}\partial_{x}u,\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}-\big{(}\Lambda^{s-2}(\partial_{x}^{2}a_{3}\partial_{x}u),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}-2\big{(}\Lambda^{s-2}(a_{3}\partial_{x}^{2}u),\Lambda^{0}_{m}w^{2}\Lambda^{s}u\big{)}.

By integration by parts, the third term of the last equality becomes:

[TABLE]

Now by Cauchy Shwarz we have:

[TABLE]

$\bullet$ Estimate of \big{(}[\Lambda^{s-2},a_{4}]\partial_{x}^{4}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}+\big{(}a_{4}\Lambda^{s-2}\partial_{x}^{4}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}:

$a_{4}\Lambda^{s-2}\partial_{x}^{4}u=a_{4}\Lambda^{s-2}(1-\Lambda^{2})\partial^{2}_{x}u=a_{4}(\Lambda^{s-2}-\Lambda^{s})\partial^{2}_{x}u=a_{4}\Lambda^{s-2}\partial^{2}_{x}u-a_{4}\Lambda^{s}\partial^{2}_{x}u$, then:

[TABLE]

By Cauchy Shwarz, the first term of the last equality is controlled by:

[TABLE]

\big{(}a_{4}\Lambda^{s}\partial^{2}_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}=-\big{(}a_{4}\Lambda_{m}^{0}w^{2},(\partial_{x}\Lambda^{s}u)^{2}\big{)}+Q_{1},

where $|Q_{1}|\leq C(m,s,|w|_{W^{1,\infty}},|\partial_{x}a_{4}|_{L^{\infty}})E^{s}(u)^{2}$.

Now, using the first order Poisson brackets $\{\Lambda^{s-2},a_{4}\}_{1}=-(s-2)\partial_{x}(a_{4})\Lambda^{s-2}\partial_{x},$ see [7] we get:

[TABLE]

Where $|Q_{2}|\leq C(m,s,|w|_{W^{2,\infty}},|a_{4}|_{H^{s+1}})E^{s}(u)^{2}$. Now, by integration by parts we have:

[TABLE]

then

[TABLE]

$\bullet$ Estimate of \big{(}[\Lambda^{s-2},a_{5}]\partial_{x}^{5}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}+\big{(}a_{5}\Lambda^{s-2}\partial_{x}^{5}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}:

[TABLE]

Then

\big{(}a_{5}\Lambda^{s-2}\partial_{x}^{5}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}=\big{(}a_{5}\Lambda^{s-2}\partial_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}-\big{(}a_{5}\Lambda^{s}\partial_{x}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}-\big{(}a_{5}\Lambda^{s}\partial_{x}^{3}u,\Lambda_{m}^{0}w^{2}\Lambda^{s}u\big{)}

The first two terms can be easily controlled by $E^{s}(u)^{2}$ as above. Now,

[TABLE]

By integration by parts, we obtain

[TABLE]

Now:

[TABLE]

where $\{\cdot,\cdot\}_{2}$ stands for the second order Poisson brackets,

[TABLE]

and $Q_{3}$ is an operator of order $s-5$ that can be controlled by the general commutator estimates (see [7]). We thus get

[TABLE]

We now use the fact that $H^{1}(\mathbb{R})$ is continuously embedded in $L^{\infty}(\mathbb{R})$ to get

[TABLE]

This leads to the expression

[TABLE]

where $|Q_{4}|\leq C(m,s,|w|_{W^{1,\infty}},|a_{5}|_{H^{s+1}})E^{s}(u)^{2}$. Remarking now, by integration by parts

[TABLE]

We now choose $w$ such that

[TABLE]

therefore, if we take w=(\Lambda_{m}^{0})^{-1}\Big{(}|a_{5}|^{\big{(}{\frac{2s-7}{6}\big{)}}}\displaystyle{\exp(-\frac{1}{3}\int_{0}^{x}\frac{a_{4}}{a_{5}}dy)}\Big{)} we easily obtain (2.8). Finally, one has

[TABLE]

therefore,

[TABLE]

$\bullet$ Estimate of \big{(}w_{t}\Lambda^{s-2}u,\Lambda_{m}^{0}w\Lambda^{s}u\big{)}: Using the Cauchy-Schwarz inequality we obtain

[TABLE]

Gathering the information provided by the above estimates, since one has

[TABLE]

If we assemble the previous estimates and using Gronwall’s lemma we obtain the following estimate:

[TABLE]

Taking $\lambda=\lambda_{T}$ large enough (how large depends on $\sup_{t\in[0,T]}C(E^{s}(v(t))$ for the first term of the right hand side of the above inequality to be negative for all $t\in[0,T]$, we deduce that

[TABLE]

2.2. Proof of the theorem:

Thanks to this energy estimate, we classically conclude (see e.g. [2]) the existence of a time

[TABLE]

and a unique solution $u\in C([0,T^{*}];H^{s}(\mathbb{R}))\cap C^{1}([0,T^{*}];H^{s-3}(\mathbb{R}))$ to (1.1) as the limit of the iterative scheme

[TABLE]