Review of some modified generalized Korteweg - de Vries - Kuramoto-Sivashinsky
equations (mgKdV-KS)  part I

Marie-Thér$\grave{e}$se Aimar(1),(2)

(1) CNRS, Centrale Marseille, I2M UMR 7373, Faculté des Sciences

Site St Charles Aix Marseille Université,

3 place Victor Hugo, 13331 Marseille cedex 3, France

(2) Le Prador,129, rue du commandant Rolland, 13008 Marseille, France.
E.mail<EMAIL_ADDRESS>

Abdelkader Intissar(1)
(1) Le Prador, 129, rue du commandant Rolland, 13008 Marseille-France.

E.mail<EMAIL_ADDRESS>

Abstract

This paper reviews the results of existence and uniqueness of the solutions of
these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky
equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and
the non homogeneous boundary value problem for KdV-KS equation in quarter
plane.

Mathematics Subject Classification. 35G25, 35K55.
Keywords. Existence, Uniqueness, Stability, The modified generalized Korteweg-
de Vries-Kuramto-Sivashinsky equation, non linear initial value problems,
Cauchy problem.

Content

$\S$ 0  Introduction ……………………………………………………………………………… 2
$\S$ 1  Elementary properties of Sobolev’s spaces …………………………………… 5
$\S$ 2  Some technicalities inequalities ………………………………………………….. 13
$\S$ 3  Bonna-Smith results on Korteweg-de Vries (KdV) equation (1975) ….. 20
$\S$ 4  Aimar’s results on Kuramoto-Sivashinsky (K.S) equation (1982) ………. 26
$\S$ 5  Presentation of KdV-KS equation (1996) …………………………………….. 32
$\S$ 6 On travelling wave solution of the Kuramoto-Sivashinsky equation …. 34
$\S$ 7  Existence and uniqueness of the solution of KdV- KS equation on domain
$[0,T]\times[0,2\pi],T>0$ ………………………………………………………………………… 35
$\S$ 8  A study of the non homogeneous boundary value problem for KdV-KS
equation in quarter plane  (2016) ………………………………………………………… 47

0  Introduction

In this paper , we review the results of the well-posedness of the classical
solutions for the initial-boundary value problem for modified generalized
Korteweg - de Vries-Kuramoto Shivashinsky equations (mgKdV-KS), under
appropriate boundary conditions respectively:.We begin this work by reviewing
:

$\rhd$ In first section, we present some properties of Sobolev spaces which
are important there because they are the natural spaces to study various
partial differential equations [[35]], [[29]] . in particular to study
modified generalized Korteweg - de Vries-Kuramoto Shivashinsky equations
(mgKdV-KS).

$\rhd$ In  second section, we present some technicalities inequalities which
play a fundamental role for existence and uniqueness of the solutions of these
equations [[9]].

$\rhd$ In Third section, we consider the waves on the surface of an inviscid
fluid in a flat channel. When one is interested in the propagation of one-
directional irrotational small amplitude long waves, it is classical to model
the waves by the well-known Korteweg - de Vries (KdV) equation.

$\bullet_{1}$: The expression of the Korteweg de Vries equation (KdV) equation
[[38]],  [[5]],  [[12]],  [[13]],  [[19]] is given by:

$\displaystyle{u_{t}+u_{x}+\epsilon uu_{x}+\mu u_{xxx}=0}$ (0.1)

or

$\displaystyle{u_{t}+uu_{x}+u_{xxx}=f(x,t)}$ (0.2)

where
$\displaystyle{u:\mathbb{R}^{2}\longrightarrow\mathbb{R},(t,x)\longrightarrow
u(t,x)}$ is a real function and $u_{t}$ is its derivative on $t$, $u_{x}$ is
its derivative on $x$ and $u_{xxx}$ is its theird derivative on $x$.
$\epsilon$ and $\mu$ are the real parameters.

$\rhd$ In  forth section, we consider the Kuramoto-Sivashinsky, which arises
in the description of stability of flame fronts, reaction diffusion systems
and many other physical settings [[24]], [[33]], [[21]] [[[2]] , [[3]]],
[[27]] is one of the simplest nonlinear PDEs that exhibit spatiotemporally
chaotic behavior. In the formulation adopted here, the time evolution of the
flame front velocity $u=u(t,x)$ on a periodic domain
$\displaystyle{u(t,x)=u(t,x+L),L>0}$.

$\bullet_{2}$: The expression of the Kuramoto-Sivashinsky (KS) equation is
given by:

$\displaystyle{u_{t}+\frac{1}{2}(u_{x})^{2}+u_{xx}+u_{xxxx}=0}$ (0.3)

or

$\displaystyle{u_{t}+uu_{x}+u_{xx}+u_{xxxx}=0}$ (0.4)

Here $t\geq 0$ is the time, and $x$ is the spatial coordinate. The subscripts
$x$ and $t$ denote partial derivatives with respect to $x$ and $t$.

$\rhd$ In fifth section, we consider the travelling wave solutions of the
Kuramoto-Sivashinsky equation of the form $\tilde{u}=u(x-\omega t)$ and the
steady-state solutions. Substituting $\tilde{u}=u(\xi)$ where $\xi=x-\omega
t$. Then the KS equation is transformed to

$\displaystyle{-\omega u_{\xi}+u_{\xi\xi\xi\xi}+u_{\xi\xi}+uu_{\xi}=0}$ (0.5)

This can be rewritten as a one-dimensional system by the change of variables
$\displaystyle{x_{1}=\tilde{u},x_{2}=\tilde{u}_{\xi},x_{3}=\tilde{u}_{\xi\xi}}$
and $\displaystyle{x_{4}=\tilde{u}_{\xi\xi\xi}}$

Then we deduce the Michelson system [[27]]

$\left\\{\begin{array}[]{c}\displaystyle{\frac{dx_{1}}{d\xi}=x_{2}}\\\ \\\
\displaystyle{\frac{dx_{2}}{d\xi}=x_{3}}\\\ \\\
\displaystyle{\frac{dx_{3}}{d\xi}=x_{4}}\\\ \\\
\displaystyle{\frac{dx_{4}}{d\xi}=\omega x_{2}-x_{1}x_{2}-x_{3}}\\\
\end{array}\right.\\\ $ (0.6)

$\rhd$ In sixth section, we consider the combined Korteweg-deVries equation
with Kuramoto-Sivashinsky equation called generalized Korteweg-deVries-
Kuramoto-sivashinsky equation (gKdV-KS) which arises in some interesting
physical situations.

$\bullet_{3}$: The expression of the generalized Kuramoto-Sivashinsky (gKdV-
KS) equation is given by:

$\displaystyle{u_{t}+\frac{1}{2}(u_{x})^{2}+u_{xxx}+Ru_{xx}+u_{xxxx}=0}$ (0.7)

$R>0$ is Reynold number

Or the “derivative” generalized Korteweg-deVries-Kuramoto-sivashinsky equation

$\displaystyle{u_{t}+uu_{x}+\delta u_{xxx}+\beta u_{xx}+\gamma u_{xxxx}=0}$
(0.8)

$\delta,\beta,\gamma$ are constants.

We recall that the term $u_{xx}$ correspond to energy input at large scales
(where the real positive number $R$ is called the “anti-diffusion” parameter),
the term $u_{xxxx}$ to dissipation at small scales and the term $uu_{x}$ to
nonlinear advection.

$\rhd$ In seventh section, we consider the local well-posedness and the global
well-posedness of the non homogeneous initial boundary value problem of KdV-KS
equation in quarter plane styded by Jing Li, Bing-Yu Zhang and Zhixiong Zhang
in  [[25]] .

In 2016, they was considered the following problem:

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+u_{xxxx}+\delta
u_{xxx}+u_{xx}+uu_{x}=0,\quad(t,x)\in(0,T)\times\mathbb{R}^{+}}\\\ \\\
u(0,x)=\phi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
x\in\mathbb{R}^{+}\\\ \\\ u(t,0)=h_{1}(t)\quad
u_{x}(t,0)=h_{2}(t)\quad\quad\quad\quad\quad t\in(0,T),\\\ \end{array}\right.$
(0.9)

where $T>0$, $\displaystyle{\delta\in\mathbb{R}}$ and $\displaystyle{\phi\in
H^{s}(\mathbb{R}^{+})}$.

They was conjectured that the wellposedness will fail when $s<-2.$.

Some their main results was based on $s$-compatibility conditions.

$\star$ In another paper, we review some results concerned with the local
well-posedness of the initial-value problems (IVP) for the Kawahara equation

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+\alpha u_{xxx}+\beta
u_{xxxxx}+uu_{x}=0\quad x,t\in\mathbb{R}}\\\ \\\ u(0,x)=u_{0}(x)\\\
\end{array}\right.$ (0.10)

and for the modified Kawahara equation

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+\alpha u_{xxx}+\beta
u_{xxxxx}+u^{2}u_{x}=0\quad x,t\in\mathbb{R}}\\\ \\\ u(0,x)=u_{0}(x)\\\
\end{array}\right.$ (0.11)

where $\alpha$ and $\beta$ are real constants and $\beta\neq 0$. These fifth-
order KdV type equations arise in modeling gravity-capillary waves on a
shallow layer and magneto-sound propagation in plasmas (see e.g. [[18]],
[[20]]).

The wellposedness issue on these fifth-order KdV type equations has previously
been studied by several authors. In [[30]], Ponce considered a general fifth-
order KdV equation

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+c_{1}uu_{x}+c_{2}u_{xxx}+c_{3}u_{x}u_{xx}+c_{4}uu_{xxx}+c_{5}u_{xxxxx}=0\quad
x,t\in\mathbb{R}}\\\ \\\ u(0,x)=u_{0}(x)\\\ \end{array}\right.$ (0.12)

and established the global well-posedness of the corresponding IVP for any
initial data in $H^{4}(\mathbb{R})$.

##  1  Elementary properties of Sobolev’s spaces

######  Definition 1.1.

(Classical Lebesgue and Sobolev spaces)

Let $I$ be an interval of $\mathbb{R}$ and $p\in\mathbb{R}$, $1\leq
p\leq\infty$ we define the Sobolecv’s space as following :

$\displaystyle{W^{1,p}(I)=\\{u\in L^{p}(I),\exists\,\,g\in
L^{p}(I);\int_{I}u(s)\varphi^{{}^{\prime}}(s)ds=-\int_{I}g(s)\varphi(s)ds\quad\forall\,\,\varphi\in
C_{c}^{1}(I)\\}}$

where

$\displaystyle{L^{p}(I),1\leq p<\infty}$ : Space of the measurable functions
$u$ on $I$ such that $\displaystyle{\int_{I}|u(s)|^{p}ds<\infty}$

$\displaystyle{||u||_{L^{p}(I)},1\leq p<\infty}$: norm of $u$ in $L^{p}(I)$
given by
$\displaystyle{||u||_{L^{p}(I)}=(\int_{I}|u(s)|^{p}ds)^{\frac{1}{p}}}$

$C_{c}^{1}(I)$ is space of differentiable functions such that their derivative
is continuous with compact support.

We denote:

$\displaystyle{H^{1}(I)=W^{1,{\color[rgb]{1,0,0}2}}(I)}$.

$\displaystyle{C^{n}(I)}$ Space of $n$-times differentiable functions defined
on $I$ such that $k$-th derivative is continuous for $k=0,.....,n$

$\displaystyle{L^{\infty}(I)}$: Space of measurable functions $u$ on $I$ such
that are essentially bounded.

$\displaystyle{||u||_{L^{\infty}(I)}}$: Norm of $u$ in the
space$L^{\infty}(I)$ i.e,
$\displaystyle{||u||_{L^{\infty}(I)}=sup\\{|u(s)|;s\in I\\}}$.

######  Remark 1.2.

$\bullet_{1}$ $\displaystyle{W^{1,p}(I)}$ can be defined by :

$\displaystyle{W^{1,p}(I)=\\{u\in L^{p}(I),u^{\prime}\in L^{p}(I)\text{ (in
weakly sense that correspond to}\,\,g\text{ of above definition})\\}}$

$\bullet_{2}$ $\displaystyle{W^{1,p}(I)}$ is equipped by norm:

$\displaystyle{||u||_{W^{1,p}(I)}=||u||_{L^{p}(I)}+||u^{{}^{\prime}}||_{L^{p}(I)}}$
(1.1)

$\bullet_{3}$ $\displaystyle{H^{1}(I)}$ is equipped by scalar product :

$\displaystyle{<u,v>_{H^{1}(I)}=<u,v>_{L^{2}(I)}+<u^{{}^{\prime}},v^{{}^{\prime}}>_{L^{2}(I)}}$
(1.2)

i.e

$\displaystyle{H^{1}(I)=\\{u\in L^{2}(I);u^{\prime}\in
L^{2}(I)\\};||u||_{H^{1}(I)}=(||u||_{L^{2}}^{2}+||u^{\prime}||_{L^{2}(I)}^{2})^{\frac{1}{2}}}$
(1.3)

$\bullet_{4}$ Let $k\in\mathbb{N}$, the Sobolev’s space $H^{k}(I)$ is defined
by:

$\displaystyle{H^{k}(I)=\\{u\in L^{2}(I);u^{\alpha}\in
L^{2}(I),\forall\,\,\alpha\leq
k\\};\,\,||u||_{H^{k}(I)}=(\sum_{\alpha=0}^{k}||\partial^{\alpha}u||_{L^{2}})^{\frac{1}{2}}}$
(1.4)

$\bullet_{5}$ $\displaystyle{W^{1,p}(I)}$ is Banach space for $1\leq
p\leq\infty$ and reflexive for $1<p<\infty$

$\bullet_{6}$ $\displaystyle{H^{1}(I)}$ is Hilbert space with respect above
scalar product.

$\bullet_{7}$ $\displaystyle{H_{0}^{1}(I)}$ is the closure of $C_{c}^{1}(I)$
in $H^{1}(I)$.

$\bullet_{8}$ $\displaystyle{u\in L_{loc}^{1}(I)}$ $\iff$ $u\in L^{1}(K)$ for
all compact $K\subset I$.

######  Theorem 1.3.

(continuous representation)

(i) Let $u\in L_{loc}^{1}(I)$, if
$\displaystyle{\int_{I}u\varphi=0\quad\forall\varphi\quad\in C_{c}(I)}$ then
$\displaystyle{u=0\,\,\text{ almost everywhere on}\,\,I}$

(ii) Let $u\in L_{loc}^{1}(I)$ which satisfies
$\displaystyle{\int_{I}u\varphi^{{}^{\prime}}=0\quad\forall\,\,\varphi\in
C_{c}^{1}(I)}$ then

$\displaystyle{\,\,\exists\,\,C\text{(a constant)}\,\,\text{such that
}\,\,u=C\,\,\text{ almost everywhere}}$

(iii) Let $u\in L_{loc}^{1}(I)$, $x_{0}\in I$ and
$\displaystyle{v(x)=\int_{x_{0}}^{x}g(s)ds\quad\forall x\in I}$ then

$v\in C(I)$ and
$\displaystyle{\int_{I}v\varphi^{{}^{\prime}}=-\int_{I}g\varphi}$

where $C(I)$ be the space of continuous functions on $I$.

(iv) Let $u\in W^{1,p}(I)$ then

$\displaystyle{\exists\,\,\tilde{u}\in C(I);u=\tilde{u}\,\,\text{almost
everywhere on}\,I}$ and
$\displaystyle{\tilde{u}(x)-\tilde{u}(y)=\int_{y}^{x}u^{\prime}(s)ds\,\,\forall\,\,x,y\in
I}$

Proof

(i) Assume, at contrary, we suppose there exists a set $A\subset I$ such that
$\mu(A)\neq 0$ and $u>0$ on $I$ (where $\mu$ is measure of Lebesgue.

$\rhd_{1}$ If $I$ is bounded we put $a=infI$ and $b=supI$ and we consider the
sequence $\displaystyle{([a+\frac{1}{n},b-\frac{1}{n}])_{n}}$.
$\displaystyle{\mu(A\cap[a+\frac{1}{n},b-\frac{1}{n}])\longrightarrow\mu(A)\neq
0}$ as $n\longrightarrow+\infty$, then there exists $n_{0}$ such that
$\displaystyle{\mu(A\cap[a+\frac{1}{n_{0}},b-\frac{1}{n_{0}}])\neq 0}$.

$\rhd_{2}$ If $I$ is not semi-bounded then we consider the sequence
$\displaystyle{([-n,n])_{n}}$ and it follows there exists $n_{0}$ such that
$\displaystyle{\mu(A\cap[-n_{0},n_{0}])\neq 0}$.

$\rhd_{3}$ If $I$ has another form we can consider the following sequence
$\displaystyle{([a-\frac{1}{n},n])_{n}}$. So we can always find an interval
$[c,d]\subset I$ such that $\displaystyle{\mu(A\cap([c,d])\neq 0}$.

We can take the following function $v$ such that

$\left\\{\begin{array}[]{c}v=1\,\,\text{ on}\,\,[c,d]\\\ \\\ v=0\,\,\text{
on}\,\,^{C}[c-\frac{1}{n},d+\frac{1}{n}]\\\ \end{array}\right.$

From $v\in C_{c}(I)$ we deduce that $\displaystyle{\int_{I}uv=0}$ and as $u>0$
on $A$ it follows that
$\displaystyle{\int_{I}uv\geq\int_{A\cap[c,d]}uv=\int_{A\cap[c,d]}u>0}$ which
is not possible . Then $u=0$ almost everywhere on $I$.

(ii) Let $\psi\in L_{loc}^{1}(I);\int_{I}\psi=1$.

$\displaystyle{\forall\,\,w\in
C_{c}^{1}(I),\,\,\text{let}h=w-(\int_{I}w)\psi\in C_{c}(I)}$ then
$\exists\,\,\varphi\in C_{c}(I)$;
$\displaystyle{\varphi^{{}^{\prime}}=w-(\int_{I}w)\psi}$ it follows that

$\displaystyle{\int_{I}u\varphi^{{}^{\prime}}=0}$

$\displaystyle{\int_{I}u(x)(w(x)-(\int_{I}w(t)dt)\psi(x))dx=0}$

$\displaystyle{\int_{I}u(t)w(t)dt-\int_{I}\int_{I}w(t)\psi(x)u(x)dtdx=0}$

$\displaystyle{\int_{I}w(t)(u(t)-\int_{I}\psi(x)u(x)dx)dt=0}$

As this last equality holds for all $w\in C_{c}(I)$ then by the property (i)
we have $\displaystyle{u(t)-\int_{I}\psi(x)u(x)dx=0}$ almost everywhere on $I$

It follows that $u=C$ almost everywhere on $I$ where
$\displaystyle{C=\int_{I}\psi(x)u(x)dx}$.

(iii) Let $a=infI$ and $b=supI$ then we have

$\displaystyle{\int_{I}v\varphi^{{}^{\prime}}=\int_{a}^{b}(\int_{x_{0}}^{x}g(t)dt)\varphi^{{}^{\prime}}(x)dx=\int_{a}^{x_{0}}\int_{x}^{x_{0}}g(t)\varphi^{{}^{\prime}}(x)dtdx+\int_{x_{0}}^{b}\int_{x_{0}}^{x}g(t)\varphi^{{}^{\prime}}(x)dtdx}$

By Fubini’s theorem and as $\varphi(a)=\varphi(b)=0$ then we obtain

$\displaystyle{\int_{I}v\varphi^{{}^{\prime}}=\int_{a}^{x_{0}}g(t)\int_{a}^{t}\varphi^{{}^{\prime}}(x)dxdt+\int_{x_{0}}^{b}g(t)\int_{t}^{b}\varphi^{{}^{\prime}}(x)dxdt=-\int_{I}g\varphi}$

(iv) Let a fixed $x_{0}\in I$ and
$\displaystyle{\tilde{u}(x)=\int_{x_{0}}^{x}u^{\prime}(t)dt}$ then by (iii) we
deduce that

$\displaystyle{\int_{I}\tilde{u}\varphi^{{}^{\prime}}=-\int_{I}u^{{}^{\prime}}\varphi\quad\forall\,\,\varphi\in
C_{c}^{1}(I)}$

Using an integration by parts, we deduce that

$\displaystyle{\int_{I}(u-\tilde{u})\varphi^{{}^{\prime}}=0\quad\forall\,\,\varphi\in
C_{c}^{1}(I)}$

So by (iii) we deduce that $\displaystyle{u-\tilde{u}=C}$ almost everywhere on
$I$ and the function $\displaystyle{\tilde{u}(x)=u(x)+C}$ satisfies the
request properties.

######  Definition 1.4.

Let $\Omega$ an open domain of $\mathbb{R}^{n}$ and let $p\in\mathbb{R}$,
$1\leq p\leq\infty$. The Sobolev space $\displaystyle{W^{1,p}(\Omega)}$ can be
defined by :

$\bullet_{1}$ $\displaystyle{W^{1,p}(\Omega)=\\{u\in
L^{p}(\Omega),\exists\,\,g_{1},g_{2},.....,g_{n}\in
L^{p}(\Omega);\int_{\Omega}u\frac{\partial\varphi}{\partial
x_{i}}=-\int_{\Omega}g_{i}\varphi\,\,\forall\,\,\varphi\in
C_{c}^{1}(\Omega)\,\,\forall\,\,i\in[1,n]\\}}$

or

$\displaystyle{W^{1,p}(\Omega)=\\{u\in
L^{p}(\Omega),\,\,\forall\,\,i\in[1,n]\,\,\frac{\partial u}{\partial x_{i}}\in
L^{p}(\Omega)\text{ (in weakly sense that correspond to}\,\,g_{i})\\}}$

We denote $\displaystyle{\nabla u=}$ $\left(\begin{array}[]{c}\frac{\partial
u}{\partial x_{1}}\\\ \frac{\partial u}{\partial x_{2}}\\\ .\\\ .\\\ .\\\ \\\
\frac{\partial u}{\partial x_{n}}\\\ \end{array}\right)$ =
$\left(\begin{array}[]{c}g_{1}\\\ g_{2}\\\ .\\\ .\\\ .\\\ \\\ g_{n}\\\
\end{array}\right)$

$\bullet_{2}$ $\displaystyle{W^{1,p}(\Omega)}$ is equipped by norm:

$\displaystyle{||u||_{W^{1,p}(\Omega)}=||u||_{L^{p}(\Omega)}+||\nabla
u||_{L^{p}(\Omega)}}$ (1.5)

$\bullet_{3}$ We denote $\displaystyle{H^{1}(\Omega)=W^{1,2}(\Omega)}$ which
is equipped by scalar product :

$\displaystyle{<u,v>_{H^{1}(\Omega)}=<u,v>_{L^{2}(\Omega)}+\sum_{i=1}^{n}<\frac{\partial
u}{\partial x_{i}},\frac{\partial v}{\partial x_{i}}>_{L^{2}(\Omega)}}$ (1.6)

associated to the following norm:

$\displaystyle{||u||_{H^{1}(\Omega)}=(||u||_{L^{2}(\Omega)}^{2}+||\nabla
u||_{L^{2}(\Omega)}^{2})^{\frac{1}{2}}}$ (1.7)

$\bullet_{4}$ $\displaystyle{H_{0}^{1}(\Omega)}$ is defined as the closure of
$C_{c}^{1}(\Omega)$ in $H^{1}(\Omega)$.

######  Remark 1.5.

There is no continuous representation like that one dimension of above theorem

######  Definition 1.6.

Let $m\geq 2$ and $1\leq p\leq\infty$, by recurrence we define:

$\displaystyle{W^{m,p}(\Omega)=\\{u\in W^{m-1,p}(\Omega),\frac{\partial
u}{\partial x_{i}}\in W^{m-1,p}(\Omega)\,\,\forall\,\,i\in[1,n]\\}}$ (1.8)

and

$\displaystyle{H^{m}(\Omega)=W^{m,2}(\Omega)}$ (1.9)

######  Remark 1.7.

$\rhd_{1}$ The Sobolev spaces [[35]] were introduced mostly for the use of the
theory of partial differential equations. Differential operators are often
closable in such spaces. Of course, Sobolev spaces being examples of Banach
or, sometimes, Hilbert spaces are interesting object for themselves. But their
importance is connected with the fact that the theory of partial differential
equations can be, and even most easily, developed just in such a spaces. The
reason is because partial differential operators are very well situated in
Sobolev spaces.. The spaces of continuous (or of class $C^{k}$) functions is
not very suitable for the studies of partial differential equations.

Why spaces of (only) continuous functions are not very suitable? The answer is
connected with the following observation [[29]]

Namely, for every $k\in\mathbb{N}$ the Laplace operator

$\displaystyle{\Delta;C^{k+2}(\Omega)\longrightarrow
C^{k}(\Omega);u=0\,\,\text{on}\,\,\partial\Omega}$ (1.10)

is continuous, but its image is not closed in $C^{k}(\Omega)$. In particular,
for continuous right hand side $f\in C^{k}(\Omega)$ the solution of the
equation:

$\displaystyle{\Delta u=f,\quad u=0\,\,\text{on}\,\,\partial\Omega}$ (1.11)

in general must not be a $C^{2}(\Omega)$ function. Similar situation we face
for other elliptic operators.

$\rhd_{2}$ Consider now a partial differential operator with constant
coefficients $a_{\alpha}$:

$\displaystyle{P(D)=\sum_{|\alpha|\leq m}a_{\alpha}D^{\alpha}}$ (1.12)

It can be considered as an operator from $C^{\infty}(\Omega)$ into itself, or
as an operator from $C^{k}(\Omega)$ into $C^{k-m}(\Omega)$, for any $m\leq k$.
But we prefer to define it as:

$\displaystyle{P(Du)(x)=\sum_{|\alpha|\leq m}a_{\alpha}D^{\alpha}u(x)}$ (1.13)

on the linear subspace $\mathfrak{C}^{m}(\Omega)$ consisting of all the
functions in $C^{m}(\Omega)$ that have a finite norm:

$\displaystyle{||u||_{W^{m,p}(\Omega)}=(\sum_{|\alpha|\leq
m}\int_{\Omega}|D^{\alpha}u|^{p})^{\frac{1}{p}}}$ (1.14)

It is often desirable to extend $P(D)$ to a closed linear operator in
$L^{p}(\Omega),(1\leq p<\infty)$. It is well known that this is possible:

######  Theorem 1.8.

The operator $P(D)$ from $L^{p}(\Omega)\,\,\,p\in[1;\infty)$ into itself with
a domain $\mathfrak{C}^{m}(\Omega)$ has a closure.

Proof

The following is a well known equivalent condition for an operator $T$ to have
a closure:

$\rhd$ Let $T$ be a linear operator from a linear subspace $D(T)$ of a Banach
space $\mathbb{X}$ into a Banach space $\mathbb{Y}$ . $T$ has a closure
$\tilde{T}$ if and only if the following condition is satisfied:

$\displaystyle{u_{n}\in D(T),u_{n}\longrightarrow 0,Tu_{n}\longrightarrow
v\,\,\text{ imply}\,\,v=0}$ (1.15)

$\rhd$ Now, in view of the necessary condition above it suffices to show that
if $\displaystyle{u_{n}\in C^{m}(\Omega);||u_{n}||_{W^{0,p}}\longrightarrow
0,||P(D)u_{n}-v||_{W^{0,p}}\longrightarrow 0}$ then $v=0$.

Let $\varphi\in C_{0}^{\infty}(\Omega)$. Integration by parts gives:

$\displaystyle{\int_{\Omega}P(D)u_{n}\varphi
dx={\int_{\Omega}u_{n}P(-D)}\varphi dx}$ (1.16)

As $n\longrightarrow\infty$, the integrals on the right converge to zero
whereas the integrals on the left converge to
$\displaystyle{\int_{\Omega}v\varphi dx}$. Therefore,

$\displaystyle{\int_{\Omega}v\varphi dx=0\,\,\text{for all}\,\,\varphi\in
C_{0}^{\infty}(\Omega)}$ (1.17)

Thus, by density, we conclude that $v=0$ and this result show that the Sobolev
spaces are natural for the studies of differential operators.

For more informations concerning dense subsets of Sobolev spaces and the
classes of domains, described in terms of the “smoothness” of their boundary
$\partial\Omega$

considered in this Sobolev theory consult ([[1]], Chapter III).
In particular the three classes of domains which are most often considered:

$\bullet$ Domains $\Omega\subset\mathbb{R}^{n}$ having the cone property.

$\bullet$ Domains having the local Lipschitz property.

$\bullet$ Domains having the $C^{m}$-regularity property.

The fractional Sobolev space $\displaystyle{W^{s,p}}$: Let $\Omega$ be a
general open set in $\mathbb{R}^{n}$. For any real $s>0$ and for any
$p\in[1,+\infty[$, we want to define the fractional Sobolev spaces
$\displaystyle{W^{s,p}(\Omega)}$. In the literature, fractional Sobolev-type
spaces are also called Aronszajn, Gagliardo or Slobodeckij spaces, by the name
of the ones who introduced them, almost simultaneously (see  [[4]], [[11]] ,
[[34]]).

We start by fixing the fractional exponent $s$ in $]0,1[$. For any
$p\in[1,+\infty[$, we define $\displaystyle{W^{s,p}(\Omega)}$ as follows

$\displaystyle{W^{s,p}(\Omega)=\\{u\in
L^{p}(\Omega);\frac{|u(x)-u(y)|^{p}}{|x-y|^{\frac{n}{p}+s}}dxdy\in
L^{p}(\Omega\times\Omega)}$ (1.18)

i.e., an intermediary Banach space between $L^{p}(\Omega)$ and
$W^{1,p}(\Omega)$ endowed with the natural norm :

$\displaystyle{||u||_{W^{s,p}(\Omega)}:=(\int_{\Omega}|u|^{p}+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+ps}}dxdy)^{\frac{1}{p}}}$
(1.19)

where the term

$\displaystyle{[u]_{W^{s,p}(\Omega)}:=(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+ps}}dxdy)^{\frac{1}{p}}}$

is the so-called Gagliardo semi-norm of $u$.

######  Proposition 1.9.

.Let $p\in[1,+\infty[$ and $0<s\leq{\color[rgb]{1,0,0}s^{{}^{\prime}}}<1$. Let
$\Omega$ be an open set in $\mathbb{R}^{n}$ and
$u:\Omega\longrightarrow\mathbb{R}$ be a measurable function. Then

$\displaystyle{||u||_{W^{s,p}(\Omega)}\leq||u||_{W^{{\color[rgb]{1,0,0}s^{{}^{\prime}}},p}(\Omega)}}$
(1.20)

for some suitable positive constant $C=C(n,p,s)\geq 1$. In particular

$\displaystyle{W^{{\color[rgb]{1,0,0}s^{{}^{\prime}}},p}(\Omega)\subset
W^{s,p}(\Omega)}$

Proof

Since $n+sp>n$ then the kernel $\displaystyle{\frac{1}{|z|^{n+ps}}}$ is
integrable. It follows that

$\displaystyle{\int_{\Omega}\,\int_{\Omega\cap\\{|x-y|\geq
1\\}}\frac{|u(x)|^{p}}{|x-y|^{n+sp}}dxdy\leq\int_{\Omega}(\int_{|z|\geq
1}\frac{1}{|z|^{n+ps}}dz)|u(x)|^{p}dx}$

$\displaystyle{\leq C(n,p,s)||u||_{L^{p}(\Omega)}^{p}}$.

Taking into account the above estimate, it follows

$\displaystyle{\int_{\Omega}\,\int_{\Omega\cap\\{|x-y|\geq
1\\}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq
2^{p-1}\int_{\Omega}\,\int_{\Omega\cap\\{|x-y|\geq
1\\}}\frac{|u(x)|^{p}+|u(y)|^{p}}{|x-y|^{n+sp}}dxdy}$

$\displaystyle{\leq 2^{p}C(n,p,s)||u||_{L^{p}(\Omega)}^{p}}$ (1.21)

On the other hand,

$\displaystyle{\int_{\Omega}\,\int_{\Omega\cap\\{|x-y|<1\\}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq\int_{\Omega}\,\int_{\Omega\cap\\{|x-y|<1\\}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+{\color[rgb]{1,0,0}s^{{}^{\prime}}}p}}dxdy}$
(1.22)

Thus, combining the two last inequalities, we get
$\displaystyle{\int_{\Omega}\,\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq
2^{p}C(n,p,s)||u||_{L^{p}(\Omega)}^{p}+\int_{\Omega}\,\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+{\color[rgb]{1,0,0}s^{{}^{\prime}}}p}}dxdy}$

and so

$\displaystyle{||u||_{W^{s,p}(\Omega)}\leq(C(n,p,s)+1)||u||_{L^{p}(\Omega)}^{p}+\int_{\Omega}\,\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+{\color[rgb]{1,0,0}s^{{}^{\prime}}}p}}dxdy}$

i.e.

$\displaystyle{||u||_{W^{s,p}(\Omega)}\leq\tilde{C}(n,p,s)||u||_{W^{{\color[rgb]{1,0,0}s^{{}^{\prime}}},p}(\Omega)}}$

which gives the desired estimate.

######  Remark 1.10.

It is also possible to define spaces $W^{s,p}(\Omega)$ for
$s\in\mathbb{R}-\mathbb{N},s>1$. This requires appearing the derivatives of
integer orders less than s in the Gagliardo semi-norm, as presented in [[7,
chapter 4.6]] or [[8]].

In the following sections, we frequently use following inequalities.

##  2  Some technicalities inequalities

######  Lemma 2.1.

$\displaystyle{\forall x,y\geq 0,\forall\,\,0\leq\alpha\leq 1}$ then we have

$\displaystyle{x^{\alpha}y^{1-\alpha}\leq\alpha x+(1-\alpha)y}$ (2.1)

Proof

If $x=y=0$, then the inequality holds. So we assume that $(x,y)\neq(0,0)$ and
as the inequality is symmetric with respect
$\alpha\longleftrightarrow(1-\alpha)$ and $x\longleftrightarrow y$, we can
assume that $y\neq 0$.

Now since the set of couples $(x,y)\in\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$
is the same as the set of couples ($xt,y)$ with $y\neq 0$, we are reduced to
establishing:

$\displaystyle{(xy)^{\alpha}y^{1-\alpha}\leq\alpha xy+(1-\alpha)y}$ (2.2)

which, after division by y, becomes:

$\displaystyle{x^{\alpha}\leq^{{\color[rgb]{1,0,0}?}}\alpha x+(1-\alpha)}$
(2.3)

Since the inequality to be established is obvious for $\alpha=0,1$, we can
assume that $0<\alpha<1$. We are therefore to consider here the function of a
single variable $x\in\mathbb{R}_{+}$ :

$\displaystyle{f(x)=x^{\alpha}-\alpha x+\alpha-1}$ (2.4)

So its derivative is:

$\displaystyle{f^{\prime}(x)=\alpha(x^{\alpha-1}-1)}$ (2.5)

Now we have $f(0)=\alpha-1<0$, $f^{\prime}(x)\geq 0$ for $0\leq x\leq 1$ and
$f^{\prime}(x)\leq 0$ for $1\leq x\leq\infty$ which forces $f$ to reach its
maximum at the point $x=1$ then as $f(1)=0$ and therefore $f$ always takes
values $\leq 0$, which establishes the desired inequality.

From this lemma we deduce

######  Theorem 2.2.

Holder inequality)

Let $1<p<\infty$ and $q$ its conjugate exponent i.e
$\displaystyle{\frac{1}{p}+\frac{1}{q}=1}$.

Let $u\in L^{p}(\mathbb{R}^{n})$ and $v\in L^{p}(\mathbb{R}^{n})$ then we have

$\displaystyle{||uv||_{L^{1}(\mathbb{R}^{n})}\leq||u||_{L^{p}(\mathbb{R}^{n})}||v||_{L^{q}(\mathbb{R}^{n})}\,\,(uv\in
L^{1}(\mathbb{R}^{n}))}$ (2.6)

In particular

$\displaystyle{||uv||_{L^{1}(\mathbb{R}^{n})}\leq||u||_{L^{2}(\mathbb{R}^{n})}||v||_{L^{2}(\mathbb{R}^{n})}\,\,(if\,\,p=q=2)}$
(2.7)

and

$\displaystyle{||uv||_{L^{1}(\mathbb{R}^{n})}\leq||u||_{L^{\infty}(\mathbb{R}^{n})}||v||_{L^{1}(\mathbb{R}^{n})}\,\,(if\,\,p=\infty\,\,\text{therefore}\,\,q=1)}$
(2.8)

Proof

We can assume that $\displaystyle{||u||_{L^{p}(\mathbb{R}^{n})}\neq 0}$ and
$\displaystyle{||v||_{L^{q}(\mathbb{R}^{n})}\neq 0}$ and divide $u$ and $v$ by
their norms:

$\displaystyle{u\longrightarrow\frac{u}{||u||_{L^{p}(\mathbb{R}^{n})}}}\,\,\text{and}\,\,\displaystyle{v\longrightarrow\frac{v}{||v||_{L^{p}(\mathbb{R}^{n})}}}$

in order to bring us back, in the inequality to be established, to the case
where $u$ and $v$ are both of unit norm:

$\displaystyle{||u||_{L^{p}(\mathbb{R}^{n})}=1}\,\,\text{
and}\,\,\displaystyle{||v||_{L^{q}(\mathbb{R}^{n})}=1}$

For $t\in\mathbb{R}^{n}$ applying above lemmas to numbers $x=|u(t)|^{p}$ and
$y=|v(t)|^{q}$ with $\alpha=\frac{1}{p}$ and $1-\alpha=\frac{1}{q}$ to get:

$\displaystyle{|u(t)||v(t)|\leq\frac{1}{p}|u(t)|^{p}+\frac{1}{q}|v(t)|^{q}}$

Or

$\bullet_{1}$ Young’s inequality : $\displaystyle{\forall a,b\geq
0\,\,,\forall\epsilon>0,\quad 1<p<\infty,\frac{1}{p}+\frac{1}{q}=1}$

$\displaystyle{ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q}}$ (2.9)

$\bullet_{2}$ (Young’s inequality with $\epsilon$ [[9]]).

Let $\displaystyle{a>0,b>0,1<p,q<\infty;\frac{1}{p}+\frac{1}{q}=1}$, Then

$\displaystyle{ab\leq\epsilon
a^{p}+C_{\epsilon}b^{q}},\quad\text{for}\quad\displaystyle{C_{\epsilon}=(\epsilon
p)^{-q/p}q^{-1}}$ (2.10)

So

$\displaystyle{\int_{\mathbb{R}^{n}}|u(t)||v(t)|dt\leq\frac{1}{p}\int_{\mathbb{R}^{n}}|u(t)|^{p}dt+\frac{1}{q}\int_{\mathbb{R}^{n}}|v(t)|^{q}dt}$

0r

$\displaystyle{\int_{\mathbb{R}^{n}}|u(t)||v(t)|dt\leq\frac{1}{p}||u||_{L^{p}(\mathbb{R}^{n})}^{p}+\frac{1}{q}||v||_{L^{q}(\mathbb{R}^{n})}^{q}=\frac{1}{p}+\frac{1}{q}=1}$

i.e

$\displaystyle{||uv||_{L^{1}(\mathbb{R}^{n})}\leq
1=1^{2}=||u||_{L^{p}(\mathbb{R}^{n})}.||v||_{L^{q}(\mathbb{R}^{n})}}$

######  Corollary 2.3.

( interpolation inequalities for $L^{p}(\Omega)$ and for $W^{m,p}(\Omega)$)

As a consequence of the Holder inequality the following inequalities are
satisfied :
$\rhd_{1}$

$\displaystyle{||u||_{L^{q}(\Omega)}\leq||u||_{L^{o}(\Omega)}^{\beta}||u||_{L^{r}(\Omega)}^{1-\beta}}$
(2.11)

valid for $\displaystyle{u\in L^{q}(\Omega)}$ with $\displaystyle{p\leq q\leq
r}$ and $\displaystyle{\frac{1}{q}=\frac{\beta}{p}+\frac{1-\beta}{r}}$

$\rhd_{2}$

$\displaystyle{||u||_{j,p}\leq
K\epsilon||u||_{m,p}+K\epsilon^{-\frac{j}{m-j}}||u||_{0,p}}$ (2.12)

for any $\displaystyle{u\in W^{m,p}(\Omega)}$ where $\Omega$ is a suitably
regular domain.

Here $\displaystyle{0\leq j\leq m-1}$ and

$\displaystyle{||u||_{j,p}=\\{\sum_{|\alpha|=j}\int_{\Omega}|D^{\alpha}u|^{p}\\}^{\frac{1}{p}}}$

where
$\displaystyle{\alpha=(\alpha_{1},\alpha_{2},....,\alpha_{n})\in\mathbb{N}^{n},|\alpha|=\alpha_{1}+\alpha_{2}+....+\alpha_{n}}$
and
$\displaystyle{D^{\alpha}u=D_{1}^{\alpha_{1}}D_{2}^{\alpha_{2}}....D_{n}^{\alpha_{n}}u}$

$\bullet$ Can relating the integral of a function to the integral of its
gradient ?

######  Lemma 2.4.

(The Dirichlet Poincaré Inequality for the case $n=1$)

If $\displaystyle{u:]-r,r[\longrightarrow\mathbb{R},r>0}$ is a $C^{1}$
function satisfying $u(-r)=u(r)=0$.then

$\displaystyle{\int_{-r}^{r}u^{2}dx\leq
4r^{2}\int_{-r}^{r}(u^{\prime})^{2}dx}$ (2.13)

Proof

By the Fundamental Theorem of Calculus we have
$\displaystyle{u(s)=\int_{-r}^{s}u^{\prime}(x)dx}$ Therefore
$\displaystyle{|u(s)|\leq\int_{-r}^{s}|u^{\prime}(x)|dx}$.

Applying the CauchySchwarz inequality to
$\displaystyle{\int_{-r}^{s}u^{\prime}(x)dx}$ to get:

$\displaystyle{\int_{-r}^{s}u^{\prime}(x)dx\leq(\int_{-r}^{s}1^{2}dx)^{1/2}(\int_{-r}^{s}(u^{\prime}(x))^{2}dx)^{1/2}}$

$\displaystyle{\leq(s+r)^{\frac{1}{2}}\int_{-r}^{s}(u^{\prime}(x))^{2}dx)^{1/2}}$

$\displaystyle{\leq(2r)^{1/2}\int_{-r}^{r}(u^{\prime}(x))^{2}dx)^{1/2}}$.

It follows that:

$\displaystyle{|u(s)|\leq(2r)^{1/2}\int_{-r}^{r}(u^{\prime}(x))^{2}dx)^{1/2}}$.

Squaring both sides gives

$\displaystyle{|u(s)|^{2}\leq 2r\int_{-r}^{r}(u^{\prime}(x))^{2}dx}$.

and finally we integrate over $[-r,r]$ to give

$\displaystyle{\int_{-r}^{r}u^{2}dx\leq
4r^{2}\int_{-r}^{r}(u^{\prime})^{2}dx}$

It is not difficult to extend these proofs to higher dimensional cubes
$B_{r}\subset\mathbb{R}^{n},r>0$.

######  Corollary 2.5.

If $\displaystyle{u:B_{r}\longrightarrow\mathbb{R}}$ is a $C^{1}$ function
with $u=0$ on $\partial B_{r}$ then there exists a constant $C(n)>0$ such that

$\displaystyle{\int_{B_{r}}u^{2}dx\leq C(n)r^{2}\int_{B_{r}}|\nabla u|^{2}dx}$
(2.14)

######  Lemma 2.6.

(The Nueman-Poincaré Inequality for the case $n=1$)

If $\displaystyle{u:]-r,r[\longrightarrow\mathbb{R},r>0}$ is a $C^{1}$
function and we define $\displaystyle{A=\frac{1}{2r}\int_{-r}^{r}udx}$ then

$\displaystyle{\int_{-r}^{r}(u-A)^{2}dx\leq
4r^{2}\int_{-r}^{r}(u^{\prime})^{2}dx}$ (2.15)

Proof

Take a differentiable function $u$ with
$\displaystyle{A=\frac{1}{2r}\int_{-r}^{r}udx}$. Note by the intermediate
value theorem that there is a point $c$ in $[-r,r]$ with $u(c)=A$. We have

$\displaystyle{u(s)=u(c)+\int_{c}^{s}u^{\prime}(x)dx}$

From this we get

$\displaystyle{|u(s)-A|\leq\int_{c}^{s}|u^{\prime}(x)|dx\leq\int_{-r}^{r}|u^{\prime}(x)|dx}$

Apply Cauchy-Schwarz again to give

$\displaystyle{|u(s)-A|\leq
2r(\int_{-r}^{r}(u^{\prime}(x))^{2}dx)^{\frac{1}{2}}}$

Squaring and integrating then gives :

$\displaystyle{\int_{-r}^{r}|u(s)-A|^{2}ds\leq
4r^{2}\int_{-r}^{r}(u^{\prime}(s))^{2}ds}$

It is not difficult to extend these proofs to higher dimensional cubes
$B_{r}\subset\mathbb{R}^{n},r>0$.

######  Corollary 2.7.

If $\displaystyle{u:B_{r}\longrightarrow\mathbb{R}}$ is a $C^{1}$ function and
we define

$\displaystyle{A=\frac{1}{Vol(B_{r})}\int_{B_{r}}udx}$

then there exists a constant $C(n)>0$ such that

$\displaystyle{\int_{B_{r}}(u-A)^{2}dx\leq C(n)r^{2}\int_{B_{r}}|\nabla
u|^{2}dx}$ (2.16)

######  Remark 2.8.

(Poincaré inequality on $\mathbb{R}$)

The Poincaré’s inequality can not hold on the unbounded domain $\mathbb{R}$.
Indeed, consider the sequence of smooth functions

$\phi_{k}(x)=$ $\left\\{\begin{array}[]{c}0;\quad|x|>k+\frac{1}{10}\\\ \\\
-sign(x);\,\,|x|\in(k,k+1)\\\ \\\ 0;\quad|x|<k-\frac{1}{10}\\\
\end{array}\right.$

where the function is smooth and monotone. Then for all $1\leq p<\infty$ we
have $||\phi_{k}||_{L^{p}(\mathbb{R})}\equiv 2$ for all $k$ while the smooth
functions $\displaystyle{\psi(x):=\int_{-\infty}^{x}\phi_{k}(s)ds}$ satisfy
$||\psi_{k}||_{L^{p}(\mathbb{R})}\longrightarrow\infty$ as
$k\longrightarrow+\infty$. Thus, it is not possible to find a constant $C>0$
such that $||\psi_{k}||_{L^{p}(\mathbb{R})}\leq
C||\psi_{k}^{{}^{\prime}}||_{L^{p}(\mathbb{R})}$ for all $k$, and hence the
Poincaré inequality must fail in $\mathbb{R}$.

######  Lemma 2.9.

(Poincaré inequality for the case $n\geq 2$)

Let $\Omega\subset\mathbb{R}^{n}$ be a domain with continuous boundary. Let
$1\leq p<\infty$. Then for any $u\in W^{1,p}(\Omega)$ :

$\displaystyle{\int_{\Omega}|u-\frac{1}{|\Omega|}\int_{\Omega}u(y)dy|^{p}dx\leq
c\int_{\Omega}\sum_{i=1}^{n}|\frac{\partial u}{\partial x_{i}}|^{p}dx}$ (2.17)

Proof

The proof is equivalent with showing that:

$\displaystyle{\int_{\Omega}|u|^{p}dx\leq
c\int_{\Omega}\sum_{i=1}^{n}|\frac{\partial u}{\partial x_{i}}|^{p}dx}$

for the $W^{1,p}(\Omega)$ functions having zero main value.

We will used the following result:

######  Theorem 2.10.

(consult  [[1]])

The embedding $\displaystyle{W^{1,p}(\Omega)\subset L^{p}(\Omega)}$ is
compact.

Now, assume, at contrary, that there exists a sequence $u_{k}\in
W^{1,p}(\Omega),k\in\mathbb{N}$ such that
$\displaystyle{||u_{k}||_{W^{1,p}(\Omega)}=1}$ and

$\displaystyle{1=||u_{k}||_{W^{1,p}(\Omega)}\geq\int_{\Omega}|u_{k}|^{p}dx>k\int_{\Omega}\sum_{i=1}^{n}|\frac{\partial
u_{k}}{\partial x_{i}}|^{p}dx}$ (2.18)

Equivalently, this can be written as:

$\displaystyle{\int_{\Omega}\sum_{i=1}^{n}|\frac{\partial u_{k}}{\partial
x_{i}}|^{p}dx\leq\frac{1}{k}}$ (2.19)

Using the result of compactness of the embedding
$\displaystyle{W^{1,p}(\Omega)\subset L^{p}(\Omega)}$, we are thus able to
find a subsequence $\\{u_{k_{l}}\\}$ convergent in $L^{p}(\Omega)$ to $u$.
Thanks to above equation we have also, $u_{k_{l}}\longrightarrow u$ in
$W^{1,p}(\Omega)$. But this same estimate shows also, that
$\displaystyle{\frac{\partial u}{\partial
x_{i}}=0\,\,a.e.\,\text{in}\,\,\Omega}$. It can be shown, that $u=const$: in
$\Omega$. Because of the zero mean (the property preserved from the sequence)
$u$ must be equal zero (a.e. in $\Omega$). This contradicts the
property$\displaystyle{||u_{k}||_{W^{1,p}(\Omega)}=1}$.

$\bullet$ We finish this section by given some fundamental properties of
Sobolev spaces and we send to [[1]]  for their proofs.

######  Proposition 2.11.

( Compact embeddings of Sobolev spaces)

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ having the cone property.

Then the following embeddings are compact:

(i) $\displaystyle{W^{j+m,p}(\Omega)\subset W^{j,q}(\Omega)}$ if $0<n-mp$ and
$\displaystyle{j+m-\frac{n}{p}\geq j-\frac{n}{q}}$,

(ii) $\displaystyle{W^{j+m,p}(\Omega)\subset C^{j}(\overline{\Omega})}$
$mp>n$.

######  Lemma 2.12.

(Gagliardo-Nirenberg inequalities )

$\displaystyle{||u||_{L_{\infty}}\leq||u||^{\frac{1}{2}}||Du||^{\frac{1}{2}}\,\,\text{for}\,\,u\in
H^{s}(\mathbb{R});s\geq 1\,\,\text{where}\,\,D=\frac{d}{dx}}$ (2.20)
$\displaystyle{||D^{j}u||\leq
M||u||^{1-\frac{1}{m}}||D^{m}u||^{\frac{1}{m}}\,\,\text{for}\,\,u\in
H^{m}(\mathbb{R});0\leq j\leq m}$ (2.21)

$\bullet$ (Gagliardo-Nirenberg inequality [[10]]):

Let $\Omega$ be a bounded domain with $\partial\Omega$ in $C^{m}$, and let $u$
be any function in $\displaystyle{W^{m,r}(\Omega)\cap L^{q}(\Omega),1\leq
q,r\leq\infty}$.
For any integer $j,0\leq j<m$, and for any number $a$ in the interval
$\displaystyle{\frac{j}{m}\leq a\leq 1}$, set

$\displaystyle{\frac{1}{p}=\frac{j}{n}+a(\frac{1}{r}-\frac{m}{n}+(1-a)\frac{1}{q}}$

(i) If $\displaystyle{m-j-\frac{n}{r}}$ is not a nonnegative integer, then

$\displaystyle{||D^{j}u||_{L^{p}(\Omega)}\leq
C||u||_{W^{m,r}(\Omega)}^{a}||u||_{L^{q}(\Omega)}^{1-a}}$ (2.22)

(ii) If $\displaystyle{m-j-\frac{n}{r}}$ is a nonnegative integer, then the
above inequality holds for $\displaystyle{a=\frac{j}{m}}$. The constant $C$
depends only on $\Omega,r,q,j$, and $a$.

$\bullet$ In the sequel, we will use the following inequalities as the
specific cases of the Gagliardo-Nirenberg inequality:

$\displaystyle{||D^{j}u||_{L_{\infty}(\Omega)}\leq
C||u||_{H^{m}(\Omega)}^{a}||u||_{L^{2}(\Omega)}^{1-a},\quad ma=j+\frac{n}{2}}$
(2.23)

$\displaystyle{||D^{j}u||_{L^{2}(\Omega)}\leq
C||u||_{H^{m}(\Omega)}^{a}||u||_{L^{2}(\Omega)}^{1-a},\quad ma=j}$ (2.24)

$\displaystyle{||D^{j}u||_{L^{4}(\Omega)}\leq
C||u||_{H^{m}(\Omega)}^{a}||u||_{L^{2}(\Omega)}^{1-a},\quad ma=j+\frac{n}{4}}$
(2.25)

######  Lemma 2.13.

(Kato’s inequality)

$\displaystyle{\text{if}\,\,k>2\,\,\text{and}\,\,u\in H^{k}(\mathbb{R})}$

$\displaystyle{|<u,uDu>_{k}|\leq C||u||_{2}||u||_{k}^{2}}\ $ (2.26)

where $\displaystyle{<.>_{k}}$ denotes the scalar product in
$H^{k}(\mathbb{R})$.

######  Lemma 2.14.

(Friedrich’s Inequality)

For any $\epsilon>0$ there exists an integer $M>0$ and a real-valued functions
$\displaystyle{w_{1},w_{2},.....,w_{M}\,\,\text{in}\,\,L^{2}(\Omega)}$, the
domain $\Omega$ bounded, that $\displaystyle{||w_{j}||_{L^{2}(\Omega)}=1}$,
and for any real-valued function $\displaystyle{u\in H_{0}^{1}(\Omega)}$:

$\displaystyle{||u||_{L^{2}(\Omega)}^{2}\leq\epsilon||u||_{H^{1}(\Omega)}^{2}+\sum_{j=1}^{M}<u,w_{j}>}$
(2.27)

######  Theorem 2.15.

(Rellich’s Theorem)

Let $\\{u_{m}\\}$ be a sequence of functions in
$\displaystyle{H_{0}^{1}(\Omega)}$ ($\Omega$ bounded) such that
$\displaystyle{||u||_{H^{1}(\Omega)}\leq const:<\infty}$.
There exists a subsequence $\\{u_{m^{\prime}}\\}$ convergent in
$L^{2}(\Omega)$.

##  3  Bonna-Smith results on KDV equation (1975)

Considered here are waves on the surface of an inviscid fluid in a flat
channel. When one is interested in the propagation of one-directional
irrotational small amplitude long waves, it is classical to model the waves by
the well-known KdV (Korteweg-de Vries) equation (see [[38]]).

$\bullet$: The expression of the Korteweg de Vries equation (KdV) equation is
given by:

$\displaystyle{u_{t}+u_{x}+\epsilon uu_{x}+\mu u_{xxx}=0}$

where
$\displaystyle{u:\mathbb{R}^{2}\longrightarrow\mathbb{R},(t,x)\longrightarrow
u(t,x)}$ is a real function and $u_{t}$ is its derivative on $t$, $u_{x}$ is
its derivative on $x$ and $u_{xxx}$ is its theird derivative on $x$.

######  Lemma 3.1.

The expression of the KdV equation can be simplified as:

$\displaystyle{u_{t}+\epsilon uu_{x}+\mu u_{xxx}=0}$

by means of the substitution: $s=x-t$.

Proof

$\rhd_{1}$
$\displaystyle{u(t,x)=v(t,s(t,x))\Longrightarrow\frac{\partial}{\partial
t}u(t,x)=\frac{\partial}{\partial t}[v(t,s(t,x))]}$

$\displaystyle{=\frac{\partial}{\partial
t}v(t,s(t,x)}+\frac{\partial}{\partial t}s(t,x)\frac{\partial}{\partial
s(t,x)}v(t,s(t,x))$

$\displaystyle{=\frac{\partial}{\partial t}v(t,s)-\frac{\partial}{\partial
s}v(t,s)}$.

$\rhd_{2}$
$\displaystyle{u(t,x)=v(t,s(t,x))\Longrightarrow\frac{\partial}{\partial
x}u(t,x)=\frac{\partial}{\partial x}[v(t,s(t,x))]}$

$\displaystyle{=\frac{\partial t}{\partial
x}v(t,s(t,x))+\frac{\partial}{\partial x}s(t,x)\frac{\partial}{\partial
s(t,x)}v(t,s(t,x))}$

$\displaystyle{=0+\frac{\partial}{\partial s}v(t,s)}$

i.e.

$\displaystyle{\frac{\partial}{\partial t}u(t,x)=\frac{\partial}{\partial
t}v(t,s)-\frac{\partial}{\partial s}v(t,s)}$.

$\displaystyle{\frac{\partial}{\partial x}u(t,x)=\frac{\partial}{\partial
s}v(t,s)}$

Hence

$\displaystyle{\frac{\partial}{\partial t}v(t,s)-\frac{\partial}{\partial
s}v(t,s)+\frac{\partial}{\partial s}v(t,s)+\epsilon v\frac{\partial}{\partial
s}v(t,s)+\mu\frac{\partial^{3}}{\partial s^{3}}v(t,s)}$

Or

$\displaystyle{\frac{\partial}{\partial t}v(t,s)+\epsilon
v\frac{\partial}{\partial s}v(t,s)+\mu\frac{\partial^{3}}{\partial
s^{3}}v(t,s)}$

$\bullet$ Consider the initial-value problem for the forced Korteweg de Vries
equation :

$\left\\{\begin{array}[]{c}\displaystyle{\frac{\partial}{\partial
t}u(t,x)+u\frac{\partial}{\partial x}u(t,s)+\frac{\partial^{3}}{\partial
x^{3}}u(t,x)=f(t,x)}\\\ \\\ u(0,x)=u_{0}(x)\\\ \end{array}\right.$ (3.1)

. (1.1)

for $t,x\in\mathbb{R}$. Here the dependent variable $u=u(t,x)$ is a real-
valued function of the independent $t$ and $x$ that in most situations where
the equation appears as model, correspond to distance measured in the
direction of the waves propagation and elapsed time. The forcing term $f$ may
be thought of as providing a rough accounting of terms that are neglected in
arriving at the tidy KDV equation bellow :

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+u_{x}+uu_{x}+u_{xxx}=0}\\\ \\\
u(0,x)=u_{0}(x)\\\ \end{array}\right.$

The expression of this equation can be simplified by above lemma to following
expression:

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+uu_{x}+u_{xxx}=0}\\\ \\\
u(0,x)=u_{0}(x)\\\ \end{array}\right.$ (3.2)

. (1.2)

by means of the substitution: $s=x-t$.

The well-posedness of above equation in classical, $L_{2}$-base Sobolev spaces
$H^{s}(\mathbb{R})$ for $\displaystyle{s>\frac{3}{2}}$ was well established in
the mid-1970’s (see [[5]], [[14]] [[12]] and the references therein).

In the early 1980’s Kato [[13]] discovered a subtle and rather general
smoothing effect for above equation.

######  Definition 3.2.

( Sharp Kato smoothing property and Kato smoothing property)

The solutions of the Cauchy problem $u_{t}(t,x)+Au(t,x)=0$ where $A$ is an
operator linear or nonlinear on $[0,T]\times\mathbb{T}$ with $\mathbb{T}$ is a
periodic domain, possess :

(i) the sharp Kato smoothing property if

$\displaystyle{u_{0}\in H^{s}(\mathbb{T})\Longrightarrow\partial_{x}^{s+1}u\in
L_{loc}^{\infty}(\mathbb{T};L^{2}(0,T)),T>0}$

(ii) the Kato smoothing property if

$\displaystyle{u_{0}\in H^{s}(\mathbb{T})\Longrightarrow u\in
L^{2}(0,T,H^{s+1}(\mathbb{T}))}$

######  Remark 3.3.

It is well known that

(i) the solutions of the Cauchy problem of the KdV equation on a periodic
domain $\mathbb{T}$:

$\displaystyle{u_{t}+uu_{x}+u_{xxx}=0,\quad u(0,x)=u_{0}(x),\quad
x\in\mathbb{T},t\in\mathbb{R}}$

possess neither the sharp Kato smoothing property nor the Kato smoothing
property.
(ii) but for the solutions of the Cauchy problem of the KdV-Burgers (KdVB)
equation on a periodic domain $\mathbb{T}$:

$\displaystyle{u_{t}+uu_{x}+u_{xxx}-u_{xx}=0,\quad u(0,x)=u_{0}(x),\quad
x\in\mathbb{T},t\in\mathbb{R}}$

possess the sharp Kato smoothing property and the Kato smoothing property
We present (ii) in following theorem:

Let $x\in[0,L],L>0$ then $\displaystyle{\frac{x}{L}\in[0,1]}$ and let
$\displaystyle{e_{n}(x)=e^{2in\pi x},n=0,\pm 1,\pm 2,.....}$ with $x\in[0,1]$
Then $\displaystyle{\\{e_{n}(x)\\}_{-\infty}^{\infty}}$ forms an orthonormal
basis in the space $L^{2}(0;1)$.We may define the Sobolev space
$\displaystyle{H_{p}^{s}:=H_{p}^{s}(0,1)}$ of order $s$ ($s\in\mathbb{R}$ ) as
the space of all real periodic functions of period $1$.

$\displaystyle{v(x)=\sum_{n=-\infty}^{\infty}v_{n}e_{n}(x)}$

such that

$\displaystyle{||v||_{s}:=\sum_{n=-\infty}^{\infty}(1+|n|^{2s})^{\frac{s}{2}}|v_{n}|^{2}<\infty}$
(3.3)

$\displaystyle{||v||_{s}}$ is a Hilbert norm for $H_{p}^{s}$.

For any $\displaystyle{s\in\mathbb{R},v\in H_{p}^{s}}$ with

$\displaystyle{v(x)=\sum_{k=-\infty}^{\infty}v_{k}e_{k}(x)}$

we define the operator $D_{x}^{s}$ by

$\displaystyle{D_{x}^{s}v(x)=\sum_{k=-\infty}^{\infty}(1+k^{2})^{s/2}v_{k}e_{k}(x)}$

such that:

$\displaystyle{||v||_{s}=||D_{x}^{s}v||_{L^{2}(0,1)}}$

######  Theorem 3.4.

Consider the Cauchy problem of the KdV-Burgers equation posed on $(0;L),L>0$
with periodic boundary conditions,

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+uu_{x}+u_{xxx}-u_{xx}=0,\quad
x\in(0,L)}\\\ \\\ u(0,x)=u_{0}(x)\\\ \\\
u(t;0)=u(t;L);u_{x}(t;L)=u_{x}(t;L);u_{xx}(t;0)=u_{xx}(t;L)\\\
\end{array}\right.\\\ $ (3.4)

which is well-known to be globally well-posed in the space $H_{p}^{s}$ for
$s\geq 0$: Let $s\geq 0$ and $T>0$ be given. Then

(i) For any $u_{0}\in H_{p}^{s}$ the Cauchy problem of the KdV-Burgers
equation admits a unique mild solution

$\displaystyle{u\in C([0,T],H_{p}^{s})}$

which possesses the Kato smoothing property:

$\displaystyle{u_{0}\in H_{p}^{s}\Longrightarrow u\in L^{2}(0,T;H_{p}^{s+1})}$

It also possesses the sharp Kato smoothing property.

(ii) For any $u_{0}\in H_{p}^{s}$ the corresponding solution $u$ of Cauchy
problem of the KdV-Burgers equation belongs to the space
$\displaystyle{C([0,T];H_{p}^{s})\cap L^{2}([0,T];H_{p}^{s+1})}$ and satisfies

$\displaystyle{sup_{0\leq x\leq
L}||\partial_{x}^{s+1}u(.,x)||_{L^{2}(0,T)}\leq\omega_{s,T}||u_{0}||_{s}}$

where
$\displaystyle{\omega_{s;T}:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}}$ is a
nondecreasing continuous function.

By contrast, the theory pertaining to the (1.1) for the forced KDV equation
has remained less developed. The following result was given by [[5]] in the
early 1970’s:

######  Theorem 3.5.

For given $T>0$ and $s\geq 3$, if

(i) $\displaystyle{u_{0}\in H^{s}(\mathbb{R})}$,

(ii) $\displaystyle{f\in C(-T,T;H^{s}(\mathbb{R}))}$,

(iii) $\displaystyle{f_{t}\in C(-T,T;L^{2}(\mathbb{R}))}$,

then

The initial-value problem (1.1) has a unique solution $\displaystyle{u\in
C(-T,T;H^{s}(\mathbb{R}))\cap C(-T,T;L^{2}(\mathbb{R}))}$.

In addition, the solution $u$ depends continuously in
$C(-T,T;H^{s}(\mathbb{R})$ on $u_{0}$ in $H^{s}(\mathbb{R})$ and $f$ in
$\displaystyle{u\in C(-T,T;H^{s}(\mathbb{R}))\cap
C^{1}(-T,T;L^{2}(\mathbb{R}))}$.

######  Remark 3.6.

This result was strengthened by [[39]] where he showed that the conclusion of
this theorem holds without assumption (iii).

$\rhd$ It frequently arises in numerical simulation where unbounded domains
are hard to model and the relative simplicity of imposing periodic boundary
conditions is very attractive.

Let $H^{s}(\mathbb{S})$ denote the real Sobolev space of order $s$ ($s\geq 0$)
on the unit length circle in the plane. $H^{s}(\mathbb{S})$ may be
characterized as the space of real $1$-periodic functions $v$ whose Fourier
series

$\displaystyle{v(x)\sim\sum_{-\infty}^{\infty}v_{k}e^{2i\pi kx}}$

is such that

$\displaystyle{||v||_{s}=\\{\sum_{-\infty}^{\infty}(1+|k|^{2s}|v_{k}|^{2}\\}^{\frac{1}{2}}}$

$\displaystyle{||v||_{s}}$ defines a Hilbert space norm on the linear space
$H^{s}(\mathbb{S})$ .

Let $D^{s}$ represent the fractional derivative of order $s$, so if $v$ has
the Fourier series as above then

$\displaystyle{D^{s}v(x)\sim\sum_{-\infty}^{\infty}v_{k}|k|^{s}e^{2i\pi kx}}$

By using energy estimates we have the following theorem

######  Theorem 3.7.

For $s\geq 2$ and $T>0$, if $u_{0}\in H^{s}(\mathbb{S})$ and $f\in
L^{1}(-T,T;H^{s}(\mathbb{S}))$, then the initial-value problem :

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+uu_{x}+u_{xxx}=f\quad
t\in\mathbb{R},x\in\mathbb{S}}\\\ \\\ u(0,x)=u_{0}(x)\\\ \end{array}\right.$
(3.5)

has a unique solution $u\in C(-T,T;H^{s}(\mathbb{S}))$. More over, the
solution $u$ depends continuously on $u_{0}$ in $H^{s}(\mathbb{S})$ and $\in
L^{1}(-T,T;H^{s}(\mathbb{S}))$

##  4  Aimar’s results on K.S equation (1982)

$\rhd_{1}$ The equation

$\displaystyle{u_{t}+\alpha u_{xx}+\beta u_{xxxx}+\gamma uu_{x}=0}$ (KS)

arises in interesting physical situations, for example as a model for long
waves on a viscous fluid flowing down an inclined plane [[37]] and to derive
drift waves in a plasma [[6]]. Equation  (KS) was derived also independently
by Kuramoto [[22]] and Kuramoto-Tsuzuki [[23]], [[24]] as a model for phase
turbulence in reaction-diffusion systems and by Sivashinsky [[33]] as a model
for plane flame propagation, describing the combined influence of diffusion
and thermal conduction of the gas on the stability of a plane flame front.

$\rhd_{2}$ The Michelson system arises as an equation for the derivative of a
traveling wave or a steady state solution of the one-dimensional Kuramoto-
Sivashinsky equation (see [[2]], [[3]], [[21]], [[28]], [[32]], [[33]] and the
references therein).

$\displaystyle{u_{t}=-u_{xxxx}-u_{xx}-uu_{x}}$ (4.1)

This one-dimensional Kuramoto-Sivashinsky (K-S) equation has been rigorous
studied of view mathematic (existence and uniqueness of the solution on some
Sobolev spaces) by Aimar in its PhD (1982) under the following form:

######  Theorem 4.1.

(Aimar 1982)

Let

$\displaystyle{u_{t}=-u_{xxxx}-u_{xx}-\frac{1}{2}[u^{\prime}(x)]^{2};(t,x)\in[0,T]\times[0,L]}$
(4.2)

be the one-dimensional Kuramoto-Sivashinsky equation with

$u_{x}(t,0)=u_{x}(t,L)=0$ and $u_{xxx}(t,0)=u_{xxx}(t,L)=0$ as limit
conditions and $T,L$ are parameters.

Let $\mathbb{H}^{s}:=\mathbb{H}^{s}(0,L)$ be the classical Sobolev space where
$\mathbb{H}^{0}:=\mathbb{L}^{2}$.

Let
$\displaystyle{\mathbb{V}=\\{v\in\mathbb{H}^{1};v_{xxx}\in\mathbb{L}^{2}\\}}$

If we choose $u(0,x)$ be in $\mathbb{H}^{1}$

Then there exists an uniqueness solution of the problem (4.2) such that

i)
$\displaystyle{u\in\mathbb{L}^{2}(0,T;\mathbb{V})\cap\mathbb{L}^{\infty}(0,T;\mathbb{H}^{2})}$

ii) $\displaystyle{u_{t}\in\mathbb{L}^{2}([0,T]\times[0,L])}$

######  Remark 4.2.

(i) As $\displaystyle{u\in\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$ then
$\displaystyle{u:[0,T]\longrightarrow\mathbb{L}^{2}}$ is continuous.

(ii) If $\displaystyle{u\in\mathbb{L}^{\infty}(0,T;\mathbb{H}^{2})}$ then
$\displaystyle{u:[0,T]\longrightarrow\mathbb{H}^{1}}$ is continuous and the
choose of $u(0,x)$ in $\mathbb{H}^{1}$ is justified.

(iii) If we replace the limit conditions of above theorem by:

$\left\\{\begin{array}[c]{l}\displaystyle{u(t,0)=u(t,L)}\\\ \\\
\displaystyle{u_{x}(t,0)=u_{x}(t,L)}\\\ \\\
\displaystyle{u_{xx}(t,0)=u_{xx}(t,L)}\\\ \\\
\displaystyle{u_{xxx}(t,0)=u_{xxx}(t,L)}\\\ \end{array}\right.$

Then we have existence and uniqueness of the solution of kuramoto-Sivashinsky
with these limit conditions.

(iv) In (4.2) the second-order term acts as an energy source and has a
destabilizing effect, the nonlinear term transfers energy from low to high
wave numbers, while the fourth-order term removes the energy on small scales.
The Kuramoto-Sivashinsky equation is known to exhibit spatio-temporal chaos.

By adding a periodic second member $f$ in (4.2) and taking $L=1$ and the
following conditions

$\left\\{\begin{array}[c]{l}\displaystyle{u(t,0)=u(t,1)}\\\ \\\
\displaystyle{u_{x}(t,0)=u_{x}(t,1)}\\\ \\\
\displaystyle{u_{xx}(t,0)=u_{xx}(t,1)}\\\ \\\
\displaystyle{u_{xxx}(t,0)=u_{xxx}(t,1)}\\\ \end{array}\right.$

($\rhd$ Now, we consider numerical solutions of Kuramoto-Shivasinsky equation,
we will give a semi-implicit numerical method with a result of stability.
Namely, the numerical approximation is taken on regular grid by using an
implicit finite differences for linear operator and semi-implicit finite
differences for non linear operator.

Let $h=\frac{1}{N}$, $\displaystyle{\Omega=]0,1[}$ and
$\displaystyle{\Omega_{h}=\\{x_{p};x_{p}=(p-1)h,p=1,...,N+1\\}}$

Let $\sigma_{h}(x_{p})$ be the interval of center $x_{p}$ and of diameter $h$.

Let $w_{h}^{p}$ be the characteristic function of $\sigma_{h}(x_{p})$ defined
by:

$\left\\{\begin{array}[c]{l}\displaystyle{w_{h}^{p}(x)=1\quad if\quad
x\in[x_{p}-\frac{1}{2},x_{p}+\frac{1}{2}[}\\\ \\\
\displaystyle{w_{h}^{p}(x)=0\quad if\quad
x/\\!\\!\\!\\!\\!\in[x_{p}-\frac{1}{2},x_{p}+\frac{1}{2}[}\\\
\end{array}\right.$

We put
$\displaystyle{\mathbb{V}_{h}=span\\{w_{h}^{1},w_{h}^{2},...,w_{h}^{N+1}\\}}$
and we define $u_{h}(t)$ in $\mathbb{V}_{h}$ by
$\displaystyle{u_{h}(t)=\sum_{p=1}^{N+1}u_{h}^{p}(t)w_{h}^{p}}$

The usual scalar product on $\mathbb{V}_{h}$ is given by

$\displaystyle{<u_{h}(t),u_{h}(t)>=h\sum_{p=1}^{N}u_{h}^{p}(t)v_{h}^{p}(t)}$

with associated norm:

$\displaystyle{||u_{h}(t)||_{h}^{2}=h\sum_{p=1}^{N}|u_{h}^{p}(t)|^{2}}$

$||.||_{h}$ is an approximation of the usual norm in $\mathbb{L}^{2}(\Omega)$.

For $u_{h}(t)$ in $\mathbb{V}_{h}$, we define following classical operators:

$\left\\{\begin{array}[c]{l}\displaystyle{\delta_{h}^{+}u_{h}^{p}(t)=\frac{u_{h}^{p+1}(t)-u_{h}^{p}(t)}{h}}\\\
\\\
\displaystyle{\delta_{h}^{-}u_{h}^{p}(t)=\frac{u_{h}^{p}(t)-u_{h}^{p-1}(t)}{h}}\\\
\\\
\displaystyle{\Delta_{h}u_{h}^{p}(t)=\frac{u_{h}^{p+1}(t)-2u_{h}^{p}(t)+u_{h}^{p-1}(t)}{h^{2}}}\\\
\\\
\displaystyle{\Delta_{h}^{2}u_{h}^{p}(t)=\frac{u_{h}^{p+2}(t)-4u_{h}^{p+1}(t)+6u_{h}^{p}(t)-4u_{h}^{p-1}(t)+u_{h}^{p-2}(t)}{h^{4}}}\\\
\end{array}\right.$

We turn now to consider the nonlinear term of Kuramoto-Shivasinsky equation,
we choose the following semi-implicit finite differences :

$\bullet$
$\displaystyle{[D_{h}u_{h}^{p}(t)]^{2}=\frac{1}{3}[[\delta_{h}^{+}u_{h}^{p}(t)]^{2}+[\delta_{h}^{-}u_{h}^{p}(t)]^{2}+\delta_{h}^{+}u_{h}^{p}(t)\delta_{h}^{+}u_{h}^{p}(t)]}$

We observe that
$\displaystyle{||\delta_{h}^{+}u_{h}(t)||_{h}^{2}=||\delta_{h}^{-}u_{h}(t)||_{h}^{2}}$
that we denote by $\displaystyle{||\delta_{h}^{1}u_{h}(t)||_{h}^{2}}$

and

$\displaystyle{||\delta_{h}^{+}\Delta_{h}u_{h}(t)||_{h}^{2}=||\Delta_{h}\delta_{h}^{-}u_{h}(t)||_{h}^{2}}$
that we denote by
$\displaystyle{||\delta_{h}^{1}\Delta_{h}u_{h}(t)||_{h}^{2}}$

Now we define two norms on $\mathbb{V}_{h}$ as follows:

$\displaystyle{||u_{h}(t)||_{h,1}^{2}=||u_{h}(t)||_{h}^{2}+||\delta_{h}^{1}u_{h}(t)||_{h}^{2}}$
this norm is an approximation of usual norm of $\mathbb{H}^{1}(\Omega)$

and

$\displaystyle{||u_{h}(t)||_{h,1}^{2}=||u_{h}(t)||_{h}^{2}+||\Delta_{h}u_{h}(t)||_{h}^{2}}$
this norm is an approximation of usual norm of $\mathbb{H}^{2}(\Omega)$

Then the approximed problem can be written as follows:

$\left\\{\begin{array}[c]{l}\displaystyle{\frac{d}{dt}u_{h}^{p}+\Delta_{h}^{2}u_{h}^{p}(t)+\Delta_{h}u_{h}^{p}(t)+\frac{1}{2}[\delta_{h}u_{h}^{p}(t)]^{2}=f_{h}^{p}(t)}\\\
\\\ with\quad boundary\quad conditions\\\ \\\
\displaystyle{u_{h}^{N+1}(t)=u_{h}^{1}(t)}\\\ \\\
\displaystyle{u_{h}^{N+2}(t)=u_{h}^{2}(t)}\\\ \\\
\displaystyle{u_{h}^{0}(t)=u_{h}^{N}(t)}\\\ \\\
\displaystyle{u_{h}^{-1}(t)=u_{h}^{N-1}(t)}\\\ \\\ with\quad initial\quad
conditions\\\ \\\ \displaystyle{u_{h}^{p}(0)=u_{0,h}^{p};p=1,2,....,N}\\\ \\\
\end{array}\right.$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{P}_{d})}$

Let $f_{h}^{*},u_{0,h}^{*}$ are a perturbation of second member and the
initial condition of the equation respectively.

If we put $\displaystyle{w_{h}=u_{h}-u_{h}^{*}}$ and
$\displaystyle{g_{h}=f_{h}-f_{h}^{*}}$, we get the following result of the
stability:

######  Theorem 4.3.

(Aimar 1982)

If $\displaystyle{||u_{0,h}(t)||_{h,2}}$,
$\displaystyle{||u_{0,h}^{*}(t)||_{h,2}}$,
$\displaystyle{\int_{0}^{T}||f_{h}(t)||_{h,2}dt}$ and
$\displaystyle{\int_{0}^{T}||f_{h}^{*}(t)||_{h,2}dt}$ are uniformly bounded.
Then

(i) $u_{h}$ and $u_{h}^{*}$ are defined on $[0,T]$

(ii) $\displaystyle{\forall\epsilon>0\quad\exists\eta>0}$ such that

$\displaystyle{\forall w_{0,h}}$ and $\displaystyle{\forall g_{h}}$ satisfying
$\displaystyle{max(||w_{0,h}||_{h,1}^{2},\int_{0}^{T}||g_{h}(\tau)||_{h,1}^{2}d\tau)<\eta}$

implies

$\displaystyle{||w_{h}(t)||_{h,1}^{2}\leq\epsilon;\forall t\in[0,T],\forall
h\in]0,h_{0}[,h_{0}>0}$

The essential points of the proof

Under the following assumptions

(a) $\displaystyle{||u_{0,h}(t)||_{h,2}\leq C}$ ;
$C$ is independent of $h$ (similar for $u_{0,h}^{*}$)

(b) $\displaystyle{\int_{0}^{T}||f_{h}^{*}(t)||_{h,2}dt\leq C}$;
$C$ is independent of $h$ (similar for $f^{*}$)

we have

($\alpha$) $\displaystyle{\forall
t\in(0,T),T<+\infty;||D_{h}^{2}u_{h}(t)||^{2}\leq C}$;
$C$ is independent of $h$

From ($\alpha$), we deduce the following inequalities

($\alpha_{1}$)
$\displaystyle{||[D_{h}u_{h}(t)]^{2}-[D_{h}u_{h}^{*}(t)]^{2}||^{2}\leq
C||\delta_{h}^{1}(u_{h}(t)-u_{h}^{*}(t))||_{h}^{2}}$;
$C$ is independent of $h$.

($\alpha_{2}$)
$\displaystyle{{}_{{}_{{}_{{}_{{}_{{}_{p\in\\{1,...,N\\}}}}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!||\delta_{h}^{1}u_{h}^{p}(t)||\leq
C_{1}}$;
$C_{1}$ is independent of $h$.

($\alpha_{3}$) $\displaystyle{[\delta_{h}^{1}u_{h}^{p}(t)]^{2}\leq
C_{2}+||D_{h}^{2}u_{h}(t)||^{2}\quad\forall p\in\\{1,...,N\\}}$;
$C_{2}$ is independent of $h$.

Transition phenomen to under harmonic (Aimar 1982)

Evolution of flame front (Aimar 1982)

######  Remark 4.4.

We can apply to  (4.2) an another numerical method by using a lower order term
$\lambda v_{xx}$, with $\lambda$ chosen in order to counteract the effect of
$-v_{xxxx}$.

Eq.  (4.2) is approximed on regular grid, using centered finite differences:

$\displaystyle{\frac{v_{j}^{n+1}-v_{j}^{n}}{\delta
t}=-v_{j}^{n}\frac{v_{j+1}^{n}-v_{j-1}^{n}}{2\delta
x}-\frac{v_{j-1}^{n}-2v_{j}^{n}+v_{j+1}^{n}}{\delta
x^{2}}-\frac{v_{j-2}^{n}-4v_{j-1}^{n}+6v_{j}^{n}-4v_{j+1}^{n}+v_{j+2}^{n}}{\delta
x^{4}}}$

$\displaystyle{-\lambda\frac{v_{j-1}^{n}-2v_{j}^{n}+v_{j+1}^{n}}{\delta
x^{2}}-\lambda\frac{v_{j-1}^{n+1}-2v_{j}^{n+1}+v_{j+1}^{n+1}}{\delta x^{2}}}$

where $\lambda$ has to be chosen such that the above algorithm is stable.

We see that while this is certainly not the method of choice to solve (4.2),
it sufficiently accurate to represent the statistics of the solution.

##  5 Presentation of KdV-KS equation (1996)

The above theorem was generalized by Aimar-Intissar (1996) to the Korteweg-de
Vries-Kuramoto-Sivashinsky equation (KdV-KS):

$\displaystyle{u_{t}=-u_{xxxx}-u_{xxx}-Ru_{xx}-uu_{x}}$ (KdV-KS)

where $R>0$ as the “anti-diffussion” parameter corresponding to the Reynolds
number.

We give bellow the theorem of existence and uniqueness of the solution of KdV-
KS equation and its proof ($\mathbb{F}$-Method).

This last equation is an approximate equation for surface waves two
dimentional incompressible viscous fluid down an inclined plane under the
assumption of small amplitude and long wave. It has a mixed form of Korteweg-
de Vries equation (KdV) for surface waves of water.

$\displaystyle{u_{t}=-u_{xxx}-uu_{x}}$ (KdV)

and Kuramoto-Sivashinsky equation  (4.1) for thermodiffusive instability of
premixed flame fronts and wave propagation in reaction diffusion systems.

Let $L=2\pi$ and $\mathfrak{H}:=\mathbb{L}^{2}(0,2\pi)$, if we consider the
operator $\mathbb{A}u=u_{xxxx}+u_{xxx}+u_{xx}$ with domain
$\displaystyle{D(\mathbb{A})=\\{u\in\mathbb{H}_{loc}^{4}(\mathbb{R});u(x+2\pi)=u(x)\,\\\
,\forall\,\,x\in\mathbb{R}\\}}$ then we see that it is nonselfadjoint
unbounded linear operator with compact resolvent and its real and imaginary
parts are linear operators associated to Kuramoto-Shivashinsky equation and
Korteweg-de Vries equation respectively.

The derivative of above equation give  (4.1). Hence equation  (4.1) has been
studied extensively in the effort to understand the complicated dynamics in
nonlinear partial differential equations (PDEs).

Physically, the two forms  (4.1) and (4.2) of the K-S equation model different
things; equation  (4.1) models small disturbances in liquid films falling down
an inclined or vertical plane and propagation of concentration waves in
chemical reactions , where equation ( (4.2) models instabilities in laminar
flame fronts  [[32]], [[33]].

From equation (4.2) Aimar-Michelson-Sivashinsky was observed that the mean
value of the solution
$\displaystyle{v(t):=\frac{1}{2L}\int_{0}^{L}[u(t,x)]dx}$ satisfies the drift
equation

$\displaystyle{\frac{d}{dt}v(t)=-\frac{1}{2L}\int_{0}^{L}[u_{x}(t,x)]^{2}dx}$
(5.1)

and that solutions took the form

$u(t,x)=c_{0}^{2}+\phi(t,x)$ where $c_{0}$ is a constant independent of the
initial condition,

and $\displaystyle{\psi(t):=\frac{1}{2L}\int_{0}^{L}[\phi(t,x)]dx}$ is close
to zero.

Therefore $\displaystyle{\frac{d}{dt}\psi(t)\sim-c_{0}^{2}}$ and hence we
substitute a solution of the form
$\displaystyle{u(t,x)=-c^{2}t+\phi(x);c\in\mathbb{R}}$, we get

$\displaystyle{-c^{2}+\phi_{xxxx}+\phi_{xx}+\frac{1}{2}[\phi_{x}]^{2}=0}$
(5.2)

The energy $E(t)$ of solutions of  ( (KdV-KS)) is defined by

$\displaystyle{E(t):=\frac{1}{2L}\int_{0}^{L}[u(t,x)]^{2}dx}$ (5.3)

and its higher order energy $\hat{E}(t)$ is defined by

$\displaystyle{\hat{E}(t):=\frac{1}{2L}\int_{0}^{L}[u_{xx}(t,x)]^{2}dx}$ (5.4)

The stability of  ( (KdV-KS)) significantly depends on the anti-diffusion
parameter $R>0$.
It was observed in numerical simulations of the KS equation that an orbit will
shadow an unstable periodic orbit for a time before diverging from it as some
observed dynamics in low-dimensional systems such as the Lorenz equations.

##  6 On travelling wave solution of the Korteweg-de Vries-Kuramoto-
Sivashinsky equation

In this way it is useful to study the simple behaviours of  (4.1), such as the
travelling wave solutions of the form $\tilde{u}=u(x-\omega t)$, and the
steady-state solutions.

We begin by substituting $\tilde{u}=u(\xi)$ where $\xi=x-\omega t$. Then
equation  (4.1) transforms to

$\displaystyle{-\omega u_{\xi}+u_{\xi\xi\xi\xi}+u_{\xi\xi}+uu_{\xi}=0}$ (6.1)

This can be rewritten as a one-dimensional system by the change of variables
$x_{1}=\tilde{u}$, $\displaystyle{x_{2}=\tilde{u}_{\xi}}$,
$\displaystyle{x_{2}=\tilde{u}_{\xi\xi}}$ and
$\displaystyle{x_{4}=\tilde{u}_{\xi\xi\xi}}$

Then

$\left\\{\begin{array}[c]{l}\displaystyle{\frac{d}{d\xi}x_{1}=x_{2}}\\\ \\\
\displaystyle{\frac{d}{d\xi}x_{2}=x_{3}}\\\ \\\
\displaystyle{\frac{d}{d\xi}x_{3}=x_{4}}\\\ \\\
\displaystyle{\frac{d}{d\xi}x_{4}=\omega x_{2}-x_{1}x_{2}-x_{3}}\\\
\end{array}\right.$ (6.2)

Integrating the last equation once gives

$\displaystyle{\frac{d}{d\xi}x_{3}=\omega x_{1}-x_{2}-\frac{1}{2}x_{1}^{2}}$
\+ constant

We remove the constant by taking $x_{1},x_{2},x_{3},x_{4}=0$ to be a valid
solution for all $\omega$. Now under the transformation
$x=x_{1}-\omega,y=x_{2},z=x_{3},t=\xi$ we obtain the system

$\left\\{\begin{array}[c]{l}\displaystyle{{}^{{}^{\bullet}}\\!\\!\\!x=y}\\\
\\\ \displaystyle{{}^{{}^{\bullet}}\\!\\!\\!y=z}\\\ \\\
\displaystyle{{}^{{}^{\bullet}}\\!\\!\\!z=\frac{\omega^{2}}{2}-\frac{1}{2}x^{2}-y}\\\
\end{array}\right.$ (6.3)

where the dot denotes always differentiation with respect to time. Thus we
have derived the Michelson system with $\omega=c\sqrt{2}$.

We will consider equation  (4.1) on periodic boundary conditions,
$u(t,x)=u(t,x+L)$ (with $L$ as a parameter). There is a rigorous proof of an
inertial manifold [[31]], [[36]] for equation  (4.1) which makes this equation
effectively equivalent to a finite dimensional system for long time behavior.

Estimates of the dimension of the latter are discussed in [[36]]. This,
together with the fact that it is a scalar equation, makes it a paradigm for
the study of rich spatiotemporal dynamics in one dimensional PDEs, and thus
the K-S equation has been well-studied numerically. In particular in [[2]],
[[3]].
reported a rich and diverse variety in the nature of the solutions to the K-S
equation on periodic boundary conditions, as the control parameter $L$ is
varied.

##  7 Existence and Uniquness of KdV-KS equation on domain
$[0,T]\times[0,2\pi],T>0$

$\mathbb{F}$-Method

Integrating KdV-KS equation once, we deduce

$\displaystyle{u_{t}+u_{xxxx}+u_{xxx}+Ru_{xx}+\frac{1}{2}u_{x}^{2}=c_{0}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{1})}$

where $c_{0}$ is the integration constant.

If we introduce $f(t,x)$ in place of $c_{0}$ and
$\displaystyle{u(t,x)=e^{\lambda t}v(t,x)}$ where $\lambda\in\mathbb{R}$ we
get :

$\displaystyle{v_{t}+v_{xxxx}+v_{xxx}+Rv_{xx}+\frac{1}{2}e^{\lambda
t}v_{x}^{2}=e^{-\lambda t}f}$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{2})}$

Let $\displaystyle{\tilde{f}=e^{-\lambda t}f}$, then can study existence and
uniqueness of the solution of the following problem:

$\left\\{\begin{array}[c]{l}\displaystyle{v_{t}+v_{xxxx}+v_{xxx}+Rv_{xx}+\frac{1}{2}e^{\lambda
t}v_{x}^{2}=\tilde{f};\quad(t,x)\in]0,\infty[\times\mathbb{R}}\\\ \\\
\displaystyle{v(t,x+2\pi)=v(t,x)}\\\ \\\ \displaystyle{v(0,x)=v_{0}}\\\
\end{array}\right.$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{P})}$

Let
$\displaystyle{\mathfrak{H}:=\mathbb{L}^{2}(0,T)=\\{f:(0,T)\longrightarrow\mathbb{R};\int_{0}^{T}|f(t)|^{2}dt<\infty,T>0\\}}$
and $H^{s}(0,T)$ be the classical Sobolev space of order $s$.

Let $\mathbb{X}$ be a Banach space with norm
$\displaystyle{||.||_{\mathbb{X}}}$ and

\-
$\displaystyle{\mathbb{L}^{p}(0,T;\mathbb{X})=\\{f:(0,T)\longrightarrow\mathbb{X};\int_{0}^{T}||f(t)||_{\mathbb{X}}^{p}dt<\infty;p\in\mathbb{N}\\}}$

\-
$\displaystyle{\mathbb{L}^{\infty}(0,T;\mathbb{X})=\\{f:(0,T)\longrightarrow\mathbb{X};\quad_{{}_{{}_{{}_{t\in]0,T[}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Sup.ess||f(t)||_{\mathbb{X}}<\infty\\}}$

The main result of these results is the following theorem:

######  Theorem 7.1.

Let $\displaystyle{\tilde{f}\in\mathbb{L}^{2}(0,T;\mathbb{H}^{1})}$ and
$\displaystyle{v_{0}\in\mathbb{H}^{1}}$

Then there exist an uniqueness solution of the problem $\mathbb{P}$ satisfying

i)
$\displaystyle{v\in\mathbb{L}^{2}(0,T;\mathbb{H}^{3})\cap\mathbb{L}^{\infty}(0,T;\mathbb{H}^{2})}$

ii) $\displaystyle{v_{t}\in\mathbb{L}^{2}(\Omega)}$ where
$\Omega=]0,T[\times]0,2\pi[$

The proof of this theorem consists of serval lemmas. We begin by

######  Lemma 7.2.

(a priori inequalities of $\mathbb{L}^{2}$ type for solutions of KdV-KS
equation)

Let $\displaystyle{\tilde{f}\in\mathbb{L}^{2}(0,T;\mathbb{H}^{1})}$

i) If $\displaystyle{v_{0}\in\mathbb{H}^{1}}$ and $v$ is solution of
$\mathbb{P}$ then we have

$\alpha$)
$\displaystyle{||v_{x}(t)||^{2}+\int_{0}^{t}||v_{xxx}(s)||^{2}ds+(2\lambda-R^{2}-1)\int_{0}^{t}||v_{x}(s)||^{2}ds\leq
C(v_{0},\tilde{f})\quad\forall t\in]0,T[}$

$\beta$)
$\displaystyle{||v(t)||^{2}+2\int_{0}^{t}||v_{xx}(s)||^{2}ds+(2\lambda-\frac{3}{4})\int_{0}^{t}||v(s)||^{2}ds\leq
C(v_{0},\tilde{f})\quad\forall t\in]0,T[}$

where $C(v_{0},\tilde{f})$ is constant.

ii) If $\displaystyle{v_{0}\in\mathbb{H}^{2}}$ and $v$ is solution of
$\mathbb{P}$ then we have

$\displaystyle{\int_{0}^{t}||v_{t}(s)||^{2}ds+||v_{xx}(t)||^{2}+\lambda||v(t)||^{2}\leq
C(v_{0},\tilde{f})\quad\forall t\in]0,T[}$

where $C(v_{0},\tilde{f})$ is constant.

iii) If $\displaystyle{v_{0}\in\mathbb{H}^{3}}$ and $v$ is solution of
$\mathbb{P}$ then we have

$\displaystyle{\int_{0}^{t}||v_{tx}(s)||^{2}ds+||v_{xxx}(t)||^{2}+\lambda||v_{x}(t)||^{2}\leq
C(v_{0},\tilde{f})\quad\forall t\in]0,T[}$

where $C(v_{0},\tilde{f})$ is constant.

Proof

We work within the framework of the Hilbert space
$\mathcal{H}=\mathbb{L}^{2}(0,2\pi)$.

The inner product and norm are given respectively by,

$\displaystyle{<u,v>=\int_{0}^{2\pi}u(x)v(x)dx}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{3})}$

$\displaystyle{||u||=\sqrt{\int_{0}^{2\pi}|u(x)|^{2}dx}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{4})}$

i) let $\displaystyle{v_{0}\in\mathbb{H}^{1}}$ and $v$ be solution of
$\mathbb{P}$, to proof a priori inequality $\alpha$) multiplying
($\mathbb{F}_{2}$) by $-v_{xx}$ and integrating on $[0,2\pi]$, then it easy to
see from the periodic conditions that

$\displaystyle{\int_{0}^{2\pi}v_{xxx}v_{xx}dx=\frac{1}{2}\int_{0}^{2\pi}[v_{xx}^{2}]_{x}dx=\frac{1}{2}[v_{xx}^{2}]_{0}^{2\pi}=0}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{5})}$

and

$\displaystyle{\int_{0}^{2\pi}v_{x}^{2}v_{xx}dx=-\frac{1}{3}\int_{0}^{2\pi}[v_{x}^{3}]_{x}dx=-\frac{1}{3}[v_{x}^{3}]_{0}^{2\pi}=0}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{6})}$

Now by using Green formula and periodic conditions, we get

$\displaystyle{\frac{1}{2}\frac{d}{dt}||v_{x}||^{2}+||v_{xxx}||^{2}+\lambda||v_{x}||^{2}=-R\int_{0}^{2\pi}v_{x}v_{xx}dx+\int_{0}^{2\pi}\tilde{f}_{x}v_{x}dx}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{7})}$

we use the Cauchy-Schwarz inequality on the terms of second member of
($\mathbb{F}_{7}$) to obtain

$\displaystyle{-R\int_{0}^{2\pi}v_{x}v_{xx}dx|\leq\frac{1}{2}||v_{xxx}||^{2}+\frac{1}{2}R^{2}||v_{x}||^{2}}$

and

$\displaystyle{|\int_{0}^{2\pi}\tilde{f}_{x}v_{x}dx|\leq\frac{1}{2}||v_{x}||^{2}+\frac{1}{2}||\tilde{f}_{x}||^{2}}$

This implies the following a priori inequality

$\displaystyle{\frac{d}{dt}||v_{x}||^{2}+||v_{xxx}||^{2}+(2\lambda-R^{2}-1)||v_{x}||^{2}\leq||\tilde{f}_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{8})}$

integrating ($\mathbb{F}_{8}$) of $0$ to $t$ to get

$\displaystyle{||v_{x}(t)||^{2}+\int_{0}^{t}||v_{xxx}(s)||^{2}ds+(2\lambda-R^{2}-1)\int_{0}^{t}||v_{x}(s)||^{2}ds\leq
C(v_{0},\tilde{f})}$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{9})}$

This finish the proof of $\alpha$) of i) of this lemma.

######  Remark 7.3.

If we choose $\lambda\geq\frac{R^{2}+1}{2}$ we have

a) $\displaystyle{v_{x}\in\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$

b) $\displaystyle{v_{xxx}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$

c) $\displaystyle{v_{t}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$

d) $\displaystyle{v_{xx}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$

e) $\displaystyle{v_{x}\in\mathbb{L}^{2}(0,T;\mathbb{H}^{2})}$ and
$\displaystyle{v_{x}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{\infty})}$

To proof a priori inequality $\beta$), multiplying ($\mathbb{F}_{2}$) by $v$
and integrating on $[0,2\pi]$, then it easy to see from the periodic
conditions that

$\displaystyle{\int_{0}^{2\pi}v_{xxx}vdx=[v_{xx}v]_{0}^{2\pi}-\int_{0}^{2\pi}v_{xx}v_{x}dx=[v_{xx}v]_{0}^{2\pi}-\frac{1}{2}[v_{x}^{2}]_{0}^{2\pi}=0}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{9})}$

Now by using Green formula and once more periodic conditions, we get

$\displaystyle{\frac{1}{2}\frac{d}{dt}||v||^{2}+||v_{xx}||^{2}+\lambda||v||^{2}=R||v_{x}||^{2}-\frac{1}{2}e^{\lambda
t}\int_{0}^{2\pi}v_{x}^{2}vdx+\int_{0}^{2\pi}\tilde{f}_{x}vdx}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{10})}$

As

$\displaystyle{|e^{\lambda t}\int_{0}^{2\pi}v_{x}^{2}vdx|\leq e^{\lambda
t}||v_{x}||_{\infty}||v_{x}||||v||\leq\frac{1}{2}e^{2\lambda
t}||v_{x}||_{\infty}^{2}||v_{x}||^{2}+\frac{1}{2}||v||^{2}}$

and

$\displaystyle{|\int_{0}^{2\pi}\tilde{f}vdx|\leq\frac{1}{2}||v||^{2}+\frac{1}{2}||\tilde{f}||^{2}}$

we deduce the following a priori inequality

$\displaystyle{\frac{d}{dt}||v||^{2}+2||v_{xx}||^{2}+(2\lambda-\frac{3}{2})||v||^{2}\leq||v_{x}||^{2}[2R+\frac{1}{2}||v_{x}||_{\infty}^{2}]+||\tilde{f}_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{11})}$

integrating ($\mathbb{F}_{11}$) of $0$ to $t$ and as (see above remark)
$\displaystyle{v_{x}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{\infty})\cap\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$
we deduce that

$\displaystyle{||v(t)||^{2}+2\int_{0}^{t}||v_{xx}(s)||^{2}ds+(2\lambda-\frac{3}{2})\int_{0}^{t}||v(s)||^{2}ds\leq
C(v_{0},\tilde{f})}$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{12})}$

This finish the proof of the property $\beta$).

######  Remark 7.4.

If we choose $\lambda\geq\frac{3}{4}$ we deduce that

a) $\displaystyle{u\in\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$

b) $\displaystyle{u\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$

ii) Let $\displaystyle{v_{0}\in\mathbb{H}^{2}}$ and $v$ be solution of
$\mathbb{P}$, to proof a priori inequality ii), multiplying ($\mathbb{F}_{2}$)
by $v_{t}$ and integrating on $[0,2\pi]$ and by using Green formula with once
more periodic conditions, we get

$\displaystyle{||v_{t}||^{2}+\frac{1}{2}\frac{d}{dt}||v_{xx}||^{2}+\frac{\lambda}{2}\frac{d}{dt}||v||^{2}=-\int_{0}^{2\pi}v_{xxx}v_{t}dx-R\int_{0}^{2\pi}v_{xx}v_{t}dx-}$
$\displaystyle{\frac{1}{2}e^{\lambda
t}\int_{0}^{2\pi}v_{x}^{2}v_{t}dx+\int_{0}^{2\pi}\tilde{f}_{x}v_{t}dx}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{13})}$

Now by using the following inequalities

1)
$\displaystyle{|\int_{0}^{2\pi}v_{xxx}v_{t}dx|\leq\frac{1}{8}||v_{t}||^{2}+2||v_{xxx}||^{2}}$

2)
$\displaystyle{|R\int_{0}^{2\pi}v_{xx}v_{t}dx|\leq\frac{1}{8}||v_{t}||^{2}+2R^{2}||v_{xx}||^{2}}$

3) $\displaystyle{|e^{\lambda t}\int_{0}^{2\pi}v_{x}^{2}v_{t}dx|\leq
e^{\lambda t}||v_{x}||_{\infty}||v_{x}||||v_{t}||\leq e^{2\lambda
t}||v_{x}||_{\infty}^{2}||v_{x}||^{2}+\frac{1}{4}||v_{t}||^{2}}$

4)
$\displaystyle{\int_{0}^{2\pi}\tilde{f}_{x}v_{t}dx|\leq\frac{1}{8}||v_{t}||^{2}+2||\tilde{f}||^{2}}$

we get the following inequality

$\displaystyle{||v_{t}||^{2}+\frac{d}{dt}||v_{xx}||^{2}+\lambda\frac{d}{dt}||v||^{2}\leq
4||v_{xxx}||^{2}+4R^{2}||v_{xx}||^{2}+}$

$\displaystyle{e^{\lambda
t}||v_{x}||_{\infty}^{2}||v_{x}||^{2}+4||\tilde{f}_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{14})}$

integrating ($\mathbb{F}_{14}$) of $0$ to $t$

As
$\displaystyle{u_{xx}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2}),u_{xxx}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$
and
$\displaystyle{v_{x}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{\infty})\cap\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$
we deduce that

$\displaystyle{\int_{0}^{t}||v_{t}(s)||^{2}ds+||v_{xx}(t)||^{2}+\lambda||v(t)||^{2}\leq
C(v_{0},\tilde{f})}$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{15})}$

This finish the proof of the property ii).

######  Remark 7.5.

If we choose $\lambda\geq 0$ we deduce that

a) $\displaystyle{u_{t}\in\mathbb{L}^{2}(0,T;\mathbb{L}^{2})}$

b) $\displaystyle{u_{xx}\in\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}$

iii) let $\displaystyle{v_{0}\in\mathbb{H}^{3}}$ and $v$ be solution of
$\mathbb{P}$, to proof a priori inequality ii), multiplying ($\mathbb{F}_{2}$)
by $v_{txx}$ and integrating on $[0,2\pi]$ and by using Green formula with
once more periodic conditions, we get

$\displaystyle{||v_{tx}||^{2}+\frac{1}{2}\frac{d}{dt}||v_{xxx}||^{2}+\frac{\lambda}{2}\frac{d}{dt}||v_{x}||^{2}=-\int_{0}^{2\pi}v_{xxxx}v_{tx}dx-R\int_{0}^{2\pi}v_{xxx}v_{tx}dx-}$
$\displaystyle{e^{\lambda
t}\int_{0}^{2\pi}v_{x}v_{xx}v_{tx}dx+\int_{0}^{2\pi}\tilde{f}_{x}v_{tx}dx}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{16})}$

Now by using the following inequalities

1)
$\displaystyle{|\int_{0}^{2\pi}v_{xxxx}v_{tx}dx|\leq\frac{1}{8}||v_{tx}||^{2}+2||v_{xxxx}||^{2}}$

2)
$\displaystyle{|R\int_{0}^{2\pi}v_{xxx}v_{tx}dx|\leq\frac{1}{8}||v_{tx}||^{2}+2R^{2}||v_{xxx}||^{2}}$

3) $\displaystyle{|e^{\lambda t}\int_{0}^{2\pi}v_{x}v_{xx}v_{tx}dx|\leq
e^{\lambda t}||v_{x}||_{\infty}||v_{xx}||||v_{tx}||\leq 2e^{2\lambda
t}||v_{x}||_{\infty}^{2}||v_{xx}||^{2}+\frac{1}{8}||v_{tx}||^{2}}$

4)
$\displaystyle{\int_{0}^{2\pi}\tilde{f}_{x}v_{tx}dx|\leq\frac{1}{8}||v_{tx}||^{2}+2||\tilde{f}_{x}||^{2}}$

we get the following inequality

$\displaystyle{||v_{tx}||^{2}+\frac{d}{dt}||v_{xxx}||^{2}+\lambda\frac{d}{dt}||v_{x}||^{2}\leq
4||v_{xxxx}||^{2}+4R^{2}||v_{xxx}||^{2}+}$
$\displaystyle{4e^{\lambda
t}||v_{x}||_{\infty}^{2}||v_{xx}||^{2}+4||\tilde{f}_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{17})}$

Now, we observe that

$\displaystyle{v_{xxxx}=-v_{t}-v_{xxx}-Rv_{xx}-\lambda v-\frac{1}{2}e^{\lambda
t}v_{x}^{2}+\tilde{f}}$

we deduce

$\displaystyle{||v_{xxxx}||^{2}\leq
6[||v_{t}||^{2}+||v_{xxx}||^{2}+R^{2}||v_{xx}||^{2}+\lambda^{2}||v||^{2}]+}$
$\displaystyle{6[\frac{1}{4}e^{2\lambda
t}||v_{x}^{2}||^{2}+||\tilde{f}||^{2}]}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{18})}$

$\displaystyle{||v_{x}^{2}||^{2}\leq||v_{x}||_{\infty}^{2}||v_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{19})}$

integrating ($\mathbb{F}_{17}$) of $0$ to $t$ then we obtain

$\displaystyle{\int_{0}^{t}||v_{tx}(s)||^{2}ds+||v_{xxx}(t)||^{2}+\lambda||v_{x}(t)||^{2}\leq||v_{0_{{}_{xxx}}}||^{2}+\lambda||v_{0_{{}_{x}}}||^{2}+4\int_{0}^{t}||v_{xxxx}(s)||^{2}ds+}$
$\displaystyle{4R^{2}\int_{0}^{t}||v_{xxx}(s)||^{2}ds+4\int_{0}^{t}||\tilde{f}_{x}(s)||^{2}ds+4\int_{0}^{t}e^{2\lambda
s}||v_{x}(s)||_{\infty}^{2}||v_{xx}(s)||^{2}ds}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{20})}$

From the inequalities established in i) and ii) and the asymptions on $v_{0}$
and $f$ we know that all terms of second member of ($\mathbb{F}_{20}$) are
bounded excepted the terms $\displaystyle{\int_{0}^{t}||v_{xxxx}(s)||^{2}ds}$
and $\displaystyle{\int_{0}^{t}e^{2\lambda
s}||v_{x}(s)||_{\infty}^{2}||v_{xx}(s)||^{2}ds}$ they are bounded by:

$\displaystyle{\int_{0}^{t}e^{2\lambda
s}||v_{x}(s)||_{\infty}^{2}||v_{xx}(s)||^{2}ds\leq e^{2\lambda
t}||v_{xx}||_{\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})}||v_{x}||_{\mathbb{L}^{2}(0,T;\mathbb{L}^{\infty})}}$

and

$\displaystyle{\int_{0}^{t}||v_{xxxx}(s)||^{2}ds\leq
6[\int_{0}^{t}||v_{t}(s)||^{2}ds+\int_{0}^{t}||v_{xxx}(s)||^{2}ds+R^{2}\int_{0}^{t}||v_{xx}(s)||^{2}ds]+}$
$\displaystyle{6[\lambda^{2}\int_{0}^{t}||v(s)||^{2}ds]+\frac{1}{4}\int_{0}^{t}e^{2\lambda
s}||v_{x}^{2}(s)||^{2}ds+\int_{0}^{t}||\tilde{f}(s)||^{2}ds]}$

As $v_{x}$ is bounded in $\mathbb{L}^{2}(0,T;\mathbb{L}^{\infty})$ and
$v_{xx}$ is bounded in $\mathbb{L}^{\infty}(0,T;\mathbb{L}^{2})$ we deduce
that the term $\displaystyle{\int_{0}^{t}e^{2\lambda
s}||v_{x}(s)||_{\infty}^{2}||v_{xx}(s)||^{2}ds}$ is bounded for all
$t\in]0,T[$

Now from ($F_{19}$), i) and ii) we deduce also that the term
$\displaystyle{\int_{0}^{t}||v_{xxxx}(s)||^{2}ds}$ is bounded for all
$t\in]0,T[$

Then

$\displaystyle{\int_{0}^{t}||v_{tx}(s)||^{2}ds+||v_{xxx}(t)||^{2}+\lambda||v_{x}(t)||^{2}\leq
C(v_{0},\tilde{f})}$ $\forall t\in]0,T[$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{21})}$

This finish the proof of the property iii).

######  Lemma 7.6.

If $v$ is solution of the problem $\mathbb{P}$, then we have uniqueness of the
solution.

Proof

We consider two solutions $u$ and $v$ of the problem $\mathbb{P}$ and we put
$u=v-w$ then $u$ is solution of the following problem

$\left\\{\begin{array}[c]{l}\displaystyle{u_{t}+u_{xxxx}+u_{xxx}+Ru_{xx}+\frac{1}{2}e^{\lambda
t}u_{x}^{2}=0;\quad(t,x)\in]0,\infty[\times\mathbb{R}}\\\ \\\
\displaystyle{u(t,x+2\pi)=u(t,x)}\\\ \\\ \displaystyle{u(0,x)=0;\quad\forall
x\in\mathbb{R}}\\\ \end{array}\right.$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{P}_{0})}$

Multiplying equation of the problem ($\mathbb{P}_{0}$) in $L^{2}(0,2\pi)$ by
$-u_{xx}$, by following the procedure used in the above lemma we obtain

$\displaystyle{\frac{1}{2}\frac{d}{dt}||u_{x}||^{2}+||u_{xxx}||^{2}+\lambda||u_{x}||^{2}=-R\int_{0}^{2\pi}u_{x}u_{xxx}dx\quad+}$

$\displaystyle{\frac{1}{2}e^{\lambda
t}\int_{0}^{2\pi}u_{x}(v_{x}+w_{x})u_{xx}dx}$$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{22})}$

Now we use the following inequalities

1) $\displaystyle{e^{\lambda
t}\int_{0}^{2\pi}u_{x}(v_{x}+w_{x})u_{xx}dx\leq\frac{1}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}+\frac{1}{2}||u_{xx}||^{2}}$

2)$\displaystyle{|R\int_{0}^{2\pi}u_{x}u_{xxx}dx|\leq\frac{1}{2}||u_{xxx}||^{2}+\frac{1}{2}R^{2}||u_{x}||^{2}}$

to deduce

$\displaystyle{\frac{d}{dt}||u_{x}||^{2}+||u_{xxx}||^{2}-\frac{1}{2}||u_{xx}||^{2}+(2\lambda-R^{2})||u_{x}||^{2}\leq}$

$\displaystyle{\frac{1}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{23})}$

Multiplying equation of problem ($\mathbb{P}_{0}$) in $L^{2}(0,2\pi)$ by $u$
then we get

$\displaystyle{\frac{d}{dt}||u||^{2}+||u_{xx}||^{2}+\lambda||u||^{2}=-R\int_{0}^{2\pi}u_{xx}u_{x}dx-\frac{1}{2}e^{\lambda
t}\int_{0}^{2\pi}u_{x}(v_{x}+w_{x})udx}$

As

$\displaystyle{e^{\lambda
t}\int_{0}^{2\pi}u_{x}(v_{x}+w_{x})u_{xx}dx\leq\frac{1}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}+\frac{1}{2}||u_{xx}||^{2}}$

and

$\displaystyle{|R\int_{0}^{2\pi}u_{xx}udx|\leq\frac{1}{2}||u_{xx}||^{2}+\frac{1}{2}R^{2}||u||^{2}}$

then we have

$\displaystyle{\frac{d}{dt}||u||^{2}+||u_{xx}||^{2}+(2\lambda-1-R^{2})||u||^{2}\leq
e^{2\lambda t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{24})}$

By adding ($\mathbb{F}_{23}$) and ($\mathbb{F}_{24}$) we obtain

$\displaystyle{\frac{d}{dt}||u||_{H^{1}}^{2}+||u_{xxx}||^{2}+\frac{1}{2}||u_{xx}||^{2}+(2\lambda-R^{2})||u_{x}||^{2}+(2\lambda-1-R^{2})||u||^{2}\leq}$

$\displaystyle{\frac{3}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{25})}$

by taking $\displaystyle{\lambda\geq\frac{R^{2}+1}{2}}$, we deduce that

$\displaystyle{\frac{d}{dt}||u||_{H^{1}}^{2}\leq\frac{3}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u_{x}||^{2}}$

and consequently

$\displaystyle{\frac{d}{dt}||u||_{H^{1}}^{2}\leq\frac{3}{2}e^{2\lambda
t}||v_{x}+w_{x}||_{\infty}^{2}||u||_{H^{1}}^{2}}$$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{26})}$

Now, we use the Gronwall lemma to deduce that
$\displaystyle{||u||_{H^{1}}^{2}=0}$ and this implies that $u=0$, hence we
have uniqueness of the solution of the problem $\mathbb{P}$

Now, we use Galerkin procedure for the proof of the existence of solution of
KdV-KS equation based on its variational formulation

Let $\displaystyle{\mathcal{V}_{m}=span\\{u_{1},u_{2},...,u_{m}\\}}$ be space
spanned by the $m$ first eigenvectors of the operator $\Delta$ defined by
$\Delta v=-v_{xx}$ with domain:

$\displaystyle{D(\Delta)=\\{v\in\mathbb{H}_{loc}^{2};v(x+2\pi)=v(x)\quad\forall
x\in\mathbb{R}\\}}$

Defining an approximation of the solution of problem $\mathbb{P}$ by

$\displaystyle{v^{m}=\sum_{j=1}^{m}\alpha_{j}^{m}u_{j}}$ where
$\alpha_{j}^{m}$ satisfy the conditions of the following system

$\left\\{\begin{array}[c]{l}\displaystyle{<v_{t}^{m},u_{j}>+<v_{xxxx}^{m},u_{j}>+<v_{xxx}^{m},u_{j}>+R<v_{xx}^{m},u_{j}>+\lambda<v^{m},u_{j}>+}\\\
\\\ \displaystyle{\frac{1}{2}e^{\lambda
t}<(v_{x}^{m})^{2},u_{j}>=<\tilde{f},u_{j}>;j=1,...,m}\\\ \\\
\displaystyle{v^{m}(0)=v_{0}^{m}\in\mathbb{H}^{2}(0,2\pi)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\color[rgb]{1,0,0}(\mathbb{F}_{27})}}\\\
\\\ \displaystyle{v_{0}^{m}\longrightarrow v_{0}\quad
in\quad\mathbb{H}^{1}(0,2\pi)}\\\ \\\ \displaystyle{\exists
c>0;||v_{0}^{m}||_{\mathbb{H}^{2}}\leq c||v_{0}||_{\mathbb{H}^{1}}}\\\
\end{array}\right.$

As

$\displaystyle{<v_{xxx}^{m},u_{j}>=<\sum_{j=1}^{m}\alpha_{j}^{m}(t)\frac{d^{3}}{dx^{3}}u_{j},u_{j}>=\sum_{k=1}^{m}\alpha_{j}^{m}(t)\int_{0}^{2\pi}\frac{d^{3}}{dx^{3}}(u_{k})u_{j}dx}$

$\displaystyle{-\sum_{k=1}^{m}\lambda_{k}\alpha_{j}^{m}(t)\int_{0}^{2\pi}\frac{d}{dx}(u_{k})u_{j}dx}$

where $\lambda_{k}$ is an eigenvalue of operator $\Delta$ associated to
eigenvectors $u_{k}$.

Then if we put
$\displaystyle{\gamma_{kj}=\int_{0}^{2\pi}\frac{d}{dx}u_{k}u_{j}dx}$, we can
write

$\displaystyle{<v_{xxx}^{m},u_{j}>=\sum_{k=1}^{m}-\lambda_{k}\gamma_{kj}\alpha_{k}^{m}(t)}$

Now, for nonlinear term, we can write

$\displaystyle{<(v_{x}^{m})^{2},u_{j}>=\int_{0}^{2\pi}[\sum_{i=1}^{m}\alpha_{i}^{m}(t)\frac{d}{dx}u_{i}]^{2}u_{j}dx=\int_{0}^{2\pi}[\sum_{i=1}^{m}\sum_{l=1}^{m}c_{il}\alpha_{i}^{m}(t)\alpha_{l}^{m}(t)\frac{d}{dx}u_{i}\frac{d}{dx}u_{l}]u_{j}dx}$
$\displaystyle{=\sum_{i=1}^{m}\sum_{l=1}^{m}\gamma_{ilj}\alpha_{i}^{m}(t)\alpha_{l}^{m}(t)}$
where $\displaystyle{\gamma_{ilj}=\frac{d}{dx}u_{i}\frac{d}{dx}u_{l}]u_{j}dx}$

The coefficients $\alpha_{i}^{m}$ satisfy the following system:

$\left\\{\begin{array}[c]{l}\displaystyle{\frac{d}{dt}\alpha_{j}^{m}+(\lambda+\lambda_{j}^{2}-R\lambda_{j})\alpha_{j}^{m}-\sum_{k=1}^{m}\lambda_{k}\gamma_{kj}\alpha_{k}^{m}}\\\
\\\ \displaystyle{+\quad\frac{1}{2}e^{\lambda
t}\sum_{i=1}^{m}\sum_{l=1}^{m}\gamma_{ilj}\alpha_{i}^{m}(t)\alpha_{l}^{m}(t)=\tilde{f}_{j}(t);j=1,...,m}\quad\quad\quad\quad\quad\quad\quad{\color[rgb]{1,0,0}(\mathbb{F}_{28})}\\\
\\\ \displaystyle{\alpha_{j}^{m}(0)=<v_{0}^{m},u_{j}>}\\\ \end{array}\right.$

where $\displaystyle{\tilde{f}_{j}(t)=<\tilde{f},u_{j}>}$

The non linear system ($\mathbb{F}_{28}$) is loc-Lipschitzian on
$\mathcal{V}_{m}$ hence its solution exists and it is maximal with continuous
derivative on $[0,t_{m}[$ ($0\leq t_{m}<+\infty$)

Under the two following conditions:

(i) $\displaystyle{\tilde{f}\in\mathbb{L}^{2}(0,T,\mathbb{H}^{1})}$

(ii) $\displaystyle{v_{0}^{m}\in\mathbb{H}^{2};\exists
c>0,||v_{0}^{m}||_{\mathbb{H}^{2}}\leq c||v_{0}||_{\mathbb{H}^{1}}}$

we present some properties of $v^{m}$:

1) $\displaystyle{v^{m},v_{x}^{m},v_{t}^{m}}$ satisfy a priori inequalities of
the first lemma of this annex and from (ii) we deduce that
$\displaystyle{||v_{0}^{m}||\leq C(v_{0},\tilde{f})}$ uniformly and $t_{m}=T$

2) For $\displaystyle{\lambda\geq sup(\frac{R^{2}+1}{2},\frac{3}{4})}$ and
$m\longrightarrow+\infty$ we have

$\alpha$) $v^{m}$ is in a bounded set of
$\mathbb{L}^{\infty}(0,T,\mathbb{H}^{3})$

$\beta$) $v^{m}$ is in a bounded set of $\mathbb{L}^{2}(0,T,\mathbb{H}^{4})$

$\gamma$) $v_{t}^{m}$ is in a bounded set of
$\mathbb{L}^{\infty}(0,T,\mathbb{H}^{3})$

$\delta$) $v_{tx}^{m}$ is in a bounded set of
$\mathbb{L}^{\infty}(0,T,\mathbb{H}^{3})$

.$\hfill{}{\color[rgb]{1,0,0}(\clubsuit)}$

These properties allow us to pass to the limit if we found one compactly
property to pass to the limit in the nonlinear term.
Considering the Sobolev space
$\displaystyle{\mathbb{H}^{r,s}(Q)=\mathbb{L}^{2}(0,T;\mathbb{H}^{r})\cap\mathbb{H}^{s}(0,T;\mathbb{L}^{2})}$
where
$Q=[0,T]\times[0,2\pi]$.

From the above properties we deduce that $v_{x}^{m}$ is in a bounded set of
$\mathbb{H}^{3,1}(Q)$ hence it is in a bounded set of $\mathbb{H}^{1}(Q)$. Now
by the regularity results of Sobolev that the injection of $\mathbb{H}^{1}(Q)$
in $\mathbb{L}^{4}(Q)$ is compact allow us to extract a weakly convergent
subsequence which we continue to denote by $v^{m}$ such that

a) $v^{m}\longrightarrow v$ in $\mathbb{L}^{2}(0,T;\mathbb{H}^{4})$ weakly.

b) $v_{t}^{m}\longrightarrow v_{t}$ in $\mathbb{L}^{2}(Q)$ weakly.

c) $v^{m}\longrightarrow v$ in $\mathbb{L}^{\infty}(0,T;\mathbb{H}^{3})$
weakly*.

d) $v_{x}^{m}\longrightarrow v$ in $\mathbb{L}^{4}(Q)$ strongly.

Then we can to pass to limit in first equation of $\mathbb{F}_{27}$ to get

$\displaystyle{<v_{t},u_{j}>+<v_{xxxx},u_{j}>+<v_{xxx},u_{j}>+R<v_{xx},u_{j}>+\lambda<v,u_{j}>}$

$\displaystyle{+\frac{1}{2}e^{\lambda
t}<v_{x}^{2},u_{j}>=<\tilde{f},u_{j}>,\quad\forall j}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{29})}$

and since the system of eigenvectors of $\Delta$ is dense in $\mathbb{H}^{3}$
we deduce that

$\displaystyle{<v_{t},u>+<v_{xxxx},u>+<v_{xxx},u>+R<v_{xx},u>+\lambda<v,u>}$

$\displaystyle{+\frac{1}{2}e^{\lambda
t}<v_{x}^{2},u>=<\tilde{f},u>,\quad\forall u\in\mathbb{H}^{3}}$
$\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{30})}$

As $v$ is continuous of $[0,T]$ into $\mathbb{H}^{1}$ and from the above
property c) with the fact that $v_{0}^{m}$ is uniformly bounded, we deduce
that $v(0,x)=v_{0}\in\mathbb{H}^{1}$ thus finish the proof of existence of
solution of KdV-KS equation.

######  Remark 7.7.

1) To pass to the limit in the nonlinear term, we can observed (as $v_{t}^{m}$
is in bounded set of $\mathbb{L}^{2}(0,T;\mathbb{L}^{2})$) that $v^{m}$ is in
bounded set of $\mathbb{H}^{0,1}(Q)$ hence $v_{x}^{m}$ is in bounded set of
$\mathbb{H}^{-1,1}(Q)$. Let
$\mathbb{D}=\\{(x,y);y=ax+b,a\in\mathbb{R},b\in\mathbb{R}\\}$ be the line such
that $(3,0)\in\mathbb{D}$ and $(-1,1)\in\mathbb{D}$. The point
$(x,x)\in\mathbb{D}$ is $(\frac{3}{5},\frac{3}{5})$, from this geometric
interpretation we deduce that the space of interpolation between
$\mathbb{H}^{3,0}(Q)$ and $\mathbb{H}^{-1,1}(Q)$ is
$\mathbb{H}^{\frac{3}{5},\frac{3}{5}}(Q)$ hence $v_{x}^{m}$ is in bounded set
of $\mathbb{H}^{\frac{3}{5},\frac{3}{5}}(Q)$. Now as the injection of
$\mathbb{H}^{\frac{3}{5},\frac{3}{5}}(Q)$ in
$\mathbb{H}^{\frac{1}{2},\frac{1}{2}}(Q)$ is compact and the injection of
$\mathbb{H}^{\frac{1}{2},\frac{1}{2}}(Q)$ in $\mathbb{L}^{4}(Q)$ is continuous
then the injection of $\mathbb{H}^{\frac{3}{5},\frac{3}{5}}(Q)$ in
$\mathbb{L}^{4}(Q)$ is also compact and this complete our discussion on
existence of solutions of KdV-KS equation.

2) An other generalisation of KS equation is given by:

$\left\\{\begin{array}[c]{l}\displaystyle{u_{t}+\nu u_{xxxx}+\delta
u_{xxx}+u_{xx}+uu_{x}=0}\\\ \\\ u(t,x)=u(t,x+2\pi)\\\ \\\ u(0,x)=u_{0}(x)\\\
\end{array}\right.$ $\hfill{}{\color[rgb]{1,0,0}(\mathbb{F}_{31})}$
where $\displaystyle{\nu=(\frac{2\pi}{L})^{2}}$ is a positive parameter that
decreases as the system size $L$ increases and the parameter $\delta$ measures
dispersive effects.
It is well known that the equation ($\mathbb{F}_{31}$) is of the active-
dissipative type and instabilities are present depending on the value of $\nu$
and if $\nu>1$ the zero solution, representing a flat film, is unique.
However, when $\nu<1$ the zero solution is linearly unstable and bifurcates
into nonlinear states including steady states, travelling waves and solutions
exhibiting spatiotemporal chaos. Some of these solutions are stable, and
others are unstable. It is established that sufficiently large values of
$\delta$ act to regularise the dynamics (even chaotic ones) into nonlinear
travelling wave pulses - see Kawahara in [[15]] and Kawahara-Toh in [[16]],
[[17]]. However, in a regime of moderate values of $\delta$ travelling waves
or pulses appear to be randomly interacting with each other giving rise to
what is widely known as weak/dissipative turbulence (in the “Manneville sense”
[[26]]) for a weak interaction theory between pulses that are sufficiently
separated.

Time evolution of the equation ($\mathbb{F}_{31}$) for $\nu=0.01\,\,(L=20\pi)$

(a) shows the chaotic behaviour in the absence of dispersion
(b) shows the weak/dissipative turbulent behaviour for small values of
$\delta$
(c) shows the chaotic regularisation with relatively large values of
dispersion.

##  8 A study of the nonhomogeneous boundary value problem for KdV-KS equation
in quarter plane: (2016)

In 2016, Jing Li, Bing-Yu Zhang and Zhixiong Zhang in  [[25]] was studied the
local well-posedness and the global well-posedness of the non homogeneous
initial boundary value problem of KdV-KS equation in quarter plane.
They was considered the following problem:

$\left\\{\begin{array}[]{c}\displaystyle{u_{t}+u_{xxx}+\delta
u_{xxx}+u_{xx}+uu_{x}=0,\quad(t,x)\in(0,T)\times\mathbb{R}^{+}}\\\ \\\
u(0,x)=\phi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
x\in\mathbb{R}^{+}\\\ \\\ u(t,0)=h_{1}(t)\quad
u_{x}(t,0)=h_{2}(t)\quad\quad\quad\quad t\in(0,T),\\\ \end{array}\right.\quad$
(8.1)

where $T>0$, $\displaystyle{\delta\in\mathbb{R}}$ and $\displaystyle{\phi\in
H^{s}(\mathbb{R}^{+})}$.

They was conjectured that the well-posedness will fail when $s<-2.$.

Some their main results was based on $s$-compatibility conditions.

######  Definition 8.1.

Let $T>0$ and $s0$ be given.

$\displaystyle{(\phi,h_{1},h_{2})\in H^{s}(\mathbb{R}^{+})\times
H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ is
said to be $s$-compatible if

(i) When $\displaystyle{s-4[\frac{s}{4}]\leq\frac{1}{2}}$, then
$\displaystyle{\phi_{k}(0)=h_{1}^{(k)}(0),\quad\phi_{k}^{{}^{\prime}}(0)=h_{2}^{(k)}(0),\quad\quad
k=0,1,......,[\frac{s}{4}]-1}$.

(ii) When $\displaystyle{\frac{1}{2}<s-4[\frac{s}{4}]\leq\frac{3}{2}}$, then
$\displaystyle{\phi_{k}(0)=h_{1}^{(k)}(0),\quad\phi_{k}^{{}^{\prime}}(0)=h_{2}^{(k)}(0),\quad\quad
k=0,1,......,[\frac{s}{4}]-1}$:
$\displaystyle{\phi_{k}(0)=h_{1}^{(k)}(0),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
k=[\frac{s}{4}]}$.

(iii) When $\displaystyle{s-4[\frac{s}{4}]>\frac{3}{2}}$, then
$\displaystyle{\phi_{k}(0)=h_{1}^{(k)}(0),\quad\phi_{k}^{{}^{\prime}}(0)=h_{2}^{(k)}(0),\quad\quad
k=0,1,......,[\frac{s}{4}]}$.
Here,
$\left\\{\begin{array}[]{c}\displaystyle{\phi_{0}(x)=\phi(x)},\\\ \\\
\displaystyle{\quad\quad\phi_{k}(x)=-\phi_{k-1}^{{}^{\prime\prime\prime\prime}}(x)-\delta\phi_{k-1}^{{}^{\prime\prime\prime}}(x)-\phi_{k-1}^{{}^{\prime\prime}}(x)-\sum_{j=0}^{j-1}C_{k-1}^{j}\phi_{j}(x)\phi_{k-j-1}^{{}^{\prime}}(x)}\\\
\end{array}\right.$
where $\displaystyle{C_{n}^{p}=\frac{n!}{p!(n-p)!}}$ and $[\bullet]$ is the
floor function.

Their main result is stated as follows.

######  Theorem 8.2.

([[25]])
(i) If $s\geq 0$ and $\displaystyle{(\phi,h_{1},h_{2})\in
H^{s}(\mathbb{R}^{+})\times H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times
H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ is $s$-compatible, then equation (7.1)
admits a unique solution $\displaystyle{u\in
C([0,T];H^{s}(\mathbb{R}^{+})\bigcap L^{2}(0,T;H^{s+2}(\mathbb{R}^{+})}$ with
$\displaystyle{u_{x}\in C([0,+\infty);H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$.
Moreover, the corresponding solution map from the space of initial and
boundary data to the solution space is continuous.
(ii) If $-2<s<0$, $\displaystyle{(\phi\in\in H^{s}(\mathbb{R}^{+})}$ ,
$\displaystyle{(h_{1},h_{2})\times H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times
H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ and
$\displaystyle{(t^{\frac{|s|}{4}+\epsilon}h_{1},t^{\frac{|s|}{4}+\epsilon}h_{2})\in
H^{\frac{3}{8}}(0,T)\times H^{\frac{1}{8}}(0,T)}$ then equation (7.1) admits a
unique solution $\displaystyle{u\in C([0,T];H^{s}(\mathbb{R}^{+})}$.
Moreover, the corresponding solution map from the space of initial and
boundary data to the solution space is continuous.

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