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\null
\vskip 5mm

\centerline{\big  Existence of weak solutions}

\centerline{\big for a quasilinear version of Benney equations}\vskip 6mm







\centerline{\bf Jo\~ao-Paulo Dias \hbox{\tenrm and} M\'ario Figueira }\medskip

\centerline{CMAF/UL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa -
PORTUGAL.}\vskip 5mm


\vskip 7mm

 \centerline{\bf Abstract}\bigskip

Benney introduced a general strategy for deriving systems of nonlinear partial
 differential equations associated with long- and short-wave solutions. The semi-linear
Benney system was studied recently by Tsutsumi and Hatano. 
Here, we tackle the nonlinear version of it and 
using compensated compactness techniques we prove the global existence of 
weak solutions to the Cauchy problem, in the case that the equation for the
amplitude of the long wave is a quasilinear conservation law with flux 
$f(v) = a v^2 - b v^3$ where $a,b$ are constants with $b>0$.

\bigskip

\noindent AMS classification: 35L70, 35L65.




\vskip 10mm

\noindent {\bf 1. Introduction and main result.}\bigskip

In [1], Benney introduces new models of nonlinear time-dependent
equations to study the interaction between short and long waves. 
The models are important in the theory of
capillary-gravity waves, internal-surface waves and short and long gravity
waves in fluids with finite depth. If $U=U(x,t)$ is a solution to a nonlinear partial
differential equation which has both short and long waves, then geometrical
considerations show that there are four length scales to be taken into account: 
$$ \eqalign{
& a_s = \hbox{\rm amplitude of the short wave,}   \cr
& a_l = \hbox{\rm amplitude of the long wave,}   \cr
& l_s = \hbox{\rm wave length of the short wave,}   \cr
& l_s = \hbox{\rm wave length of the long wave}.   \cr
} $$
Based on these scales, Benney introduces three dimensionless parameters, namely
 $$\eqalign{
&  \varepsilon_s ={{a_s}\over{l_s}} =  \hbox{\rm slope of the short wave,}   \cr
&  \varepsilon_l ={{a_l}\over{l_l}} =  \hbox{\rm slope of the long wave,}   \cr
&  \mu ={{l_s}\over{l_l}} =  \hbox{\rm ratio of short to long wave 
$\ll 1$,}   \cr
}
$$
and derives various models depending on the relative magnitude of the
parameters $\varepsilon_s, \varepsilon_l$ and $\mu$.

Benney treats the general model equation
$$ {{\partial U}\over{\partial t}} + A U = B U,
$$
where $A$ is a linear differential operator involving only spatial derivatives and 
$B$ is a nonlinear differential operator. Noticing that 
$$ {{a_l}\over{a_s}} = {{\varepsilon_l}\over{\varepsilon_s \mu}},
$$
and
restricting attention to the case where the amplitudes of the short and long
waves are 
comparable, 
we see that the long waves are substantially weaker that the short ones. 
In [1, eqs.(3.27), (3.28)], an example of this type of equations is given as: 
%
$$ 
\matrix{
\displaystyle i {{\partial u}\over{\partial t}} + {{\partial^2
u}\over{\partial x^2}} =
    |u|^2 u  + v u,  \hfill    \cr
    \cr
\displaystyle {{\partial v}\over{\partial t}} + {\partial\over{\partial x}}
f(v)
= {\partial\over{\partial x}} |u|^2,  \cr
}
\eqno (1.1)  
$$
where 
$x\in {\bf R},\,\, t \geq 0$, and  
 $f=f(v)$ is a nonlinear polynomial
real function,  $u=u(x, t) \in {\bf C}$ and $v=v(x, t) \in {\bf R}$ represent
the short wave and
the long wave, respectively, and $i=\sqrt{-1}$. 


The (semi-linear) case $f(v) = a v,\, a\in{\bf R}$ was studied
by Tsutsumi and Hatano in the pioneering papers [7,8]. In the present paper
we consider the case
$$ f(v) =  a v^2 - b v^3,
\eqno (1.2) 
$$
$\quad  {\rm where}\quad   a\in{\bf R},\, b>0 \quad
\hbox{\rm are given
constants,}$
with initial data
$$ u(x, 0) = u_0(x),\quad v(x, 0) = v_0(x),\,\,\, x\in{\bf R}.   \eqno (1.3) $$
In order to simplify the presentation and without loss of
generality, we assume that 
$a=b=1$ in
$(1.2)$.
\medskip
Using the general approach known as the vanishing viscosity method 
%[4] 
we start below by studying a regularized version ($\varepsilon > 0$) 
$$  \matrix{
\displaystyle i {{\partial u^\varepsilon}\over{\partial t}} + {{\partial^2
u^\varepsilon}\over{\partial x^2}} =
    |u^\varepsilon|^2 u^\varepsilon  + v^\varepsilon u^\varepsilon  \hfill
\cr
    \cr
\displaystyle {{\partial v^\varepsilon}\over{\partial t}} +
{\partial\over{\partial x}} f(v^\varepsilon)
= {\partial\over{\partial x}} |u^\varepsilon|^2 +
\varepsilon  {{\partial^2 v^\varepsilon}\over{\partial x^2}}  \cr
}
,\quad 
\eqno (1.4)  $$
in the domain ${\bf R}\times{\bf R}_{+}$
and with initial data
$$ 
u(\cdot, 0) = u_0, \qquad v(\cdot, 0) = v_0. 
%\in H^1 ({\bf R}),\quad v_0  \,\,{\rm real}.  
\eqno (1.5)  
$$
Next,  we will used arguments from the
compensated compactness method by Murat and Tartar [4,6] generalized to the
$L^p$-setting [2,3,5].

Our main result is as follows.

\medskip





 \noindent{\bf Theorem 1.} {\it Given $u_0, v_0 \in H^1 ({\bf R})$ with $v_0$ real-valued,  
 there exist functions 
 $$
 u\in L^\infty ({\bf R}_{+} ;
H^1 ({\bf R}) ), 
\qquad
v\in L^\infty ({\bf R}_{+} ;  (L^4 \cap L^2) ({\bf R}) )
$$
 such that
$$ \eqalign{
i \int_0^\infty \int_{\bf R} & u {{\partial \varphi}\over{\partial t}} \,dx
\,dt +
 \int_0^\infty \int_{\bf R} {{\partial u}\over{\partial x}} \,
 {{\partial \varphi}\over{\partial x}} \,dx\,dt + \int_{\bf R} u_0(x)
\varphi (x, 0) \,dx +  \cr
    &+  \int_0^\infty \int_{\bf R} |u|^2u \varphi\, dx\,dt
+  \int_0^\infty \int_{\bf R} v u \varphi \,dx\,dt = 0, \cr
}$$
\medskip

$$  \int_0^\infty \int_{\bf R}  v {{\partial \psi}\over{\partial t}} \,dx
\,dt +
 \int_0^\infty \int_{\bf R}  f(v) {{\partial \psi}\over{\partial x}} \,dx
\,dt +
\int_{\bf R} v_0(x) \psi (x, 0) \,dx -  \int_0^\infty \int_{\bf R}
{\partial\over{\partial x}}
|u|^2 \psi \,dx\,dt = 0, $$
\medskip

\noindent for all functions $\varphi, \psi \in C_0^1 ({\bf R}\times [0,
+\infty[)$ (i.e.~in the class of continuously differentiable functions with compact support), 
with $\varphi$ being complex-valued 
% \in {\bf C}$
and $\psi$ real-valued.
%  \in {\bf R}$. }



\bigskip

To establish our main result we will first prove two lemmas, the first one
being a direct generalization of the conservation laws derived in [7].\bigskip



 \noindent{\bf Lemma 2.} {\it If $(u^\varepsilon, v^\varepsilon)\in
 C([0, +\infty[; H^1({\bf R}))^2$ denotes a solution of the Cauchy problem
$(1.4)$, $(1.5)$, then setting for all 
$t\geq 0$
$$ E_1(t) =  {1\over 2} \int_{\bf R}
\left| {{\partial u^\varepsilon}\over{\partial x}}\right|^2 dx  + {1\over
4}  \int_{\bf R}
|u^\varepsilon|^4 dx + {1\over 2}  \int_{\bf R} v^\varepsilon
|u^\varepsilon|^2 dx
- {1\over 6}  \int_{\bf R} (v^\varepsilon)^3 dx + {1\over 8}   \int_{\bf R}
(v^\varepsilon)^4 dx
$$
and
$$E_2(t) =  {1\over 2} \int_{\bf R}
(v^\varepsilon)^2 dx + {\rm Im}
 \int_{\bf R} \left(u^\varepsilon  {{\partial \bar
u^\varepsilon}\over{\partial x}}\right) dx,
$$
it holds: 

$$  \eqalignno{
& {\partial\over{\partial t}} \int_{\bf R} |u^\varepsilon|^2 dx = 0,  &
(1.6)   \cr
  &  &  \cr
  & &  \cr
& {\partial\over{\partial t}} E_1(t) +
{3\over 2} \varepsilon
  \int_{\bf R} \left( v^\varepsilon  {{\partial
v^\varepsilon}\over{\partial x}}\right)^2 dx =
 \varepsilon  \int_{\bf R}  v^\varepsilon
 \left( {{\partial v^\varepsilon}\over{\partial x}}\right)^2 dx -
{\varepsilon\over 2}
 \int_{\bf R}  {{\partial v^\varepsilon}\over{\partial x}}\,
{\partial\over{\partial x}}
|u^\varepsilon|^2 dx,  & (1.7)  \cr
 & &    \cr
 & &  \cr
& {\partial\over{\partial t}} E_2(t) + \varepsilon  \int_{\bf R}
\left({{\partial v^\varepsilon}\over{\partial x}}\right)^2 dx = 0.  & (1.8)
\cr
}
$$
}


\eject
\null
\vskip 5mm

 \noindent{\bf Lemma 3.} {\it Under the hypothesis of Lemma~2 the function
$$ g_\varepsilon (t) =   \int_{\bf R} \left| {{\partial
u^\varepsilon}\over{\partial x}}\right|^2 dx
+  \int_{\bf R} |u^\varepsilon|^4 dx  +  \int_{\bf R} (v^\varepsilon)^4 dx $$
satisfies the following property: 
There exists $\varepsilon_0 >0$ such that for all $\varepsilon \leq
\varepsilon_0$
$$  g_\varepsilon (t) \leq h(t),\quad t\geq 0,$$
where $h$ is a positive continuous function in $[0, +\infty[$ independent of
 $\varepsilon$.
}

\

\rm 

Hence, combining this result with $(1.6)$ and $(1.8)$ we obtain for all $\varepsilon \leq
\varepsilon_0$
$$  \int_{\bf R} (v^\varepsilon)^2 dx + \varepsilon \int_0^t \int_{\bf R}
\left(
{{\partial v^\varepsilon}\over{\partial x}}\right)^2 dx \,d\tau \leq
h_1(t),\quad t\geq 0,  $$
where $h_1$ is a positive continuous function in $[0, +\infty[$ independent of $\varepsilon$.


\bigskip

To conclude this introduction let us point out that 
there is a large number of interesting open problems concerning the class of systems
under consideration in this paper, especially:  

\noindent i) The existence of local-in-time smooth solutions for the system associated with an 
arbitrary flux-function $f$. The finite-time blow-up of these solutions. 


\noindent ii) The uniqueness of weak solutions to the Cauchy problem.


\noindent iii) The existence of stationary waves, and their linearized and nonlinear stability 
under small or large perturbations. 

\vskip 10mm


\noindent {\bf 2. The Cauchy problem for the regularized model.}\bigskip

We start with a sketch of the proof of Lemma 2. The conservation law
$(1.6)$ is standard for the Schr\"odinder equation [7, eq.(43)]. The equation $(1.7)$ is a variant of
the conservation of energy established in [7, eq.(44)]. The difference is in computing the following term
(we drop the superscript $\varepsilon$ and assume ``enough'' regularity for $u$ and $v$
without loss of generality):
$$\eqalign{
{1\over 2}  \int_{\bf R} |u|^2 {{\partial v}\over{\partial t}} dx 
& = {1\over 2} \left[
 \int_{\bf R} |u|^2 {\partial\over{\partial x}} |u|^2 dx -
  \int_{\bf R} |u|^2 {\partial\over{\partial x}} f(v) dx + \varepsilon
\int_{\bf R} |u|^2
{{\partial^2 v}\over{\partial x^2}} dx \right]    \cr
  \cr
& =  {1\over 2} \left[  \int_{\bf R} {\partial\over{\partial x}} |u|^2 f(v)
dx \,+\,
  \varepsilon  \int_{\bf R} |u|^2 {{\partial^2 v}\over{\partial x^2}} dx
\right]   \cr
  \cr
& =  {1\over 2} \left[  \int_{\bf R} {{\partial v}\over{\partial t}} f(v) dx +
 \int_{\bf R} \left({\partial\over{\partial x}} f(v)\right) f(v) dx +
 \varepsilon  \int_{\bf R} {{\partial^2 v}\over{\partial x^2}} \,(|u|^2 -
f(v)) dx\right]  \cr
  \cr
}
$$  
  thus 
  $$\eqalign{
& {1\over 2}  \int_{\bf R} |u|^2 {{\partial v}\over{\partial t}} dx    \cr
%
& =  {1\over 2} \left[{\partial\over{\partial t}}
  \int_{\bf R} F(v) dx +  \varepsilon  \int_{\bf R} {{\partial
v}\over{\partial x}}
\,(2v - 3 v^2)  {{\partial v}\over{\partial x}} dx \right]
 -  {1\over 2} \varepsilon  \int_{\bf R} {{\partial
v}\over{\partial x}}
\, {\partial\over{\partial x}} |u|^2 dx   \cr
  \cr
& =  {1\over 2} \left[{\partial\over{\partial t}}  \int_{\bf R} F(v) dx - 3
\varepsilon  \int_{\bf R} v^2
\left( {{\partial v}\over{\partial x}}\right)^2 dx + 2  \varepsilon
\int_{\bf R} v
\left( {{\partial v}\over{\partial x}}\right)^2 dx \right]
 -  {1\over 2} \varepsilon  \int_{\bf R} {{\partial v}\over{\partial x}}
{\partial\over{\partial x}} |u|^2 dx,   \cr
}
$$
where we have set $\displaystyle F(v) = \int_0^v f(\xi)d\xi$.
Finally, we obtain $(1.8)$ by the same calculations as in [7, eq.(45)] with the modification
$$\eqalign{
&{\rm Im}{\partial\over{\partial t}}  \int_{\bf R} u  {{\partial \bar
u}\over{\partial x}} dx =
   -  \int_{\bf R} v {\partial\over{\partial x}} |u|^2 dx =   \cr
&= -  \int_{\bf R} v
\left( {{\partial v}\over{\partial t}} +  {\partial\over{\partial x}} f(v)
-  \varepsilon
 {{\partial^2 v}\over{\partial x^2}} \right) dx
  = -  {1\over 2} {\partial\over{\partial t}}  \int_{\bf R} v^2 dx -
\varepsilon
 \int_{\bf R} \left( {{\partial v}\over{\partial x}}\right)^2 dx, \cr
}
$$
which completes the proof of Lemma 2. \qed


\vskip 6mm

 \noindent{\bf Proof of Lemma 3.}  \bigskip

\noindent The right-hand side in $(1.7)$ can be estimated as follows. 
We have (after
dropping out the superscript $\varepsilon$)
$$ \eqalign{
&  \varepsilon  \int_{\bf R} v \left( {{\partial v}\over{\partial
x}}\right)^2 dx -
{\varepsilon\over 2}  \int_{\bf R} {{\partial v}\over{\partial x}}
 {\partial\over{\partial x}} |u|^2 dx    \cr
   \cr
& \leq {\varepsilon\over 2}  \int_{\bf R} \left( {{\partial
v}\over{\partial x}}\right)^2 dx +
{\varepsilon\over 2}  \int_{\bf R} \left(v {{\partial v}\over{\partial
x}}\right)^2 dx +
\varepsilon  \int_{\bf R} \left| {{\partial v}\over{\partial x}}\right| \,|u|\,
\left|{{\partial u}\over{\partial x}}\right| dx,
}
$$
and therefore by $(1.6), (1.7)$ and $(1.8)$ :
$$ g(t) = \left(\left|{{\partial u}\over{\partial x}}\right|_{L^2}^2 +
|u|_{L^4}^4 +
|v|_{L^4}^4 \right) (t) \leq c + c \, \varepsilon \int_0^t
\left|{{\partial u}\over{\partial x}}\right|_{L^2}^{3/2}
\left| {{\partial v}\over{\partial x}}\right|_{L^2} d \tau.
 $$
Here we have used that 
$$
 |u|_{L^\infty} \leq c |u|_{L^2}^{1/2}
\left|{{\partial u}\over{\partial x}}\right|_{L^2}^{1/2}
$$
and  $c>0$ denote a constant
independent of $\varepsilon$,
$$  |v|_{L^2}^2 (t) + \varepsilon \int_0^t
\left| {{\partial v}\over{\partial x}}\right|_{L^2}^2 d \tau \leq c  + c
\left|{{\partial u}\over{\partial x}}\right|_{L^2} (t). $$
We derive
$$ g(t) \leq c +  c \, \varepsilon \int_0^t g(\tau)^{3/4}
\left| {{\partial v}\over{\partial x}}\right|_{L^2} d \tau = \theta (t)
\eqno (1.9) $$
and
$$ \varepsilon  \int_{\bf R} \left| {{\partial v}\over{\partial
x}}\right|_{L^2}^2 d \tau \leq
c + c \,g(t)^{1/2} \leq c + c\, \theta(t)^{1/2}.  \eqno (1.10)  $$

We see that 
$$ \theta'(t) \leq c\, \varepsilon^{1/2} \theta(t)^{3/4} \left(
\varepsilon^{1/2}
\left| {{\partial v}\over{\partial x}}\right|_{L^2} (t) \right)  $$
$$  \theta'(t)\, \theta^{-3/4}(t) \leq c  \varepsilon^{1/2} \left(
\varepsilon^{1/2}
\left| {{\partial v}\over{\partial x}}\right|_{L^2} (t) \right).  \eqno
(1.11)  $$
Let us now multiply $(1.11)$ by $e^t$ and integrate by parts over the
interval $[0, t]$. After 
applying Cauchy-Schwarz inequality we obtain
$$ - \int_0^t   \theta^{1/4}(\tau)\, e^\tau d\tau + \left[
\theta^{1/4}(\tau) e^\tau \right]_0^t \leq
c \, \varepsilon^{1/2} \left(\int_0^t e^{2\tau} d \tau\right)^{1/2}
 \left(\int_0^t  \varepsilon \left| {{\partial v}\over{\partial
x}}\right|_{L^2}^2 d \tau \right)^{1/2} $$
and so, by $(1.10)$,
$$  \theta^{1/4}(t)\, e^t \leq c + c \int_0^t  \theta^{1/4}(\tau)\, e^\tau
d\tau + c\,\varepsilon^{1/2}
\,e^t\,  \theta^{1/4}(t) + c\, \varepsilon^{1/2} e^t.$$

Hence, for $ \varepsilon \leq  \varepsilon_0$ such that $ c\,
\varepsilon_0^{1/2} = 1/2$, it follows that 
$$  \theta^{1/4}(t)\, e^t \leq c + c \int_0^t  \theta^{1/4}(\tau) d\tau +
c\, \varepsilon_0^{1/2} e^t  $$
and the conclusion follows by Gronwall's inequality and $(1.9)$. \qed

\bigskip

\noindent We are in a position to establish the global existence for the approximating
problem $(1.4), (1.5)$ : \bigskip


\noindent{\bf Proposition 4.} {\it For each $\varepsilon > 0$ the Cauchy
problem  $(1.4), (1.5)$
admits a unique solution $(u, v) \in C([0, +\infty[; H^1({\bf R}))^2$. Moreover, 
for all $T > 0$
$$
v\in L^2(0, T;  H^2({\bf R})),\,
\qquad 
 \displaystyle  {{\partial v}\over{\partial t}} \in
 L^2(0, T;  L^2({\bf R})).
 $$
}
\bigskip


\noindent {\bf Proof.} First we prove the local-in-time existence and
uniqueness of the solution.
To this end we consider the product space ${\bf B}_R^T\times{\rm B}_R^T$, where
$$ {\bf B}_R^T = \{ w \in X_T = C([0, T]; H^1({\bf R})) \,: \, \| w \|_
{L^\infty(0, T; H^1)} \leq R \}, $$
and ${\rm B}_R^T$ denotes the corresponding real-valued version for the second factor, 
and pairs of functions $(\tilde u, \tilde v)$. 
We search for the map
$$ (\tilde u, \tilde v)
\longrightarrow \Phi (\tilde u, \tilde v) = (u, v)   \eqno (2.1)  $$
where $(u, v) \in X_T \times X_T$ is the solution of the linear problem (we
put $ \varepsilon = 1$ for
simplicity)
$$  \matrix{
\displaystyle i {{\partial u}\over{\partial t}} + {{\partial^2
u}\over{\partial x^2}} =
    |\tilde u|^2 \tilde u  + \tilde v \tilde u  \hfill    \cr
    \cr
\displaystyle {{\partial v}\over{\partial t}} + {\partial\over{\partial x}}
f(\tilde v)
= {\partial\over{\partial x}} |\tilde u|^2 + {{\partial^2 v}\over{\partial
x^2}}  \cr
   \cr
   u(0) = u_0,\quad v(0)=v_0  \hfill \cr
}
 \eqno (2.2)  $$


Recall that, in the usual semi-group framework,
$$ u(t) = e^{i\Delta t} u_0 - i \int_0^t e^{i \Delta (t-s)} \left[ |\tilde
u|^2 \tilde u +
\tilde v \tilde u \right] (s) ds $$
and 
$$  {\partial\over{\partial x}} |\tilde u|^2 -  {\partial\over{\partial x}}
f(\tilde v) \in
C([0, T]; L^2({\bf R})).$$
It is now easy to prove that there exists $T > 0$ and $R > 0$ such that the
map $\Phi$ defined in $(2.1)$
is a strict contraction in the complete metric space  ${\bf
B}_R^T\times{\rm B}_R^T$ and so,
by the Banach fixed point theorem, we obtain a unique local in time
solution $(u, v)$ of
the Cauchy problem $(1.4), (1.5)$ in the space $X_T \times X_T$.

\noindent To conclude that there is a unique global solution $(u, v)$
in the space\break
$C([0, +\infty[; H^1({\bf R}))^2$ we can apply $(1.6)$, Lemma 3 (in $[0,
T]$) and, to
estimate
  $ \left|{{\partial v}\over{\partial x}}\right|_{L^2} (t)$, the formula
(in the semi-group framework)
$$ v(t) = e^{\Delta t} v_0 +  \int_0^t e^{ \Delta (t-s)} \left[
{\partial\over{\partial x}} |u|^2 -
 {\partial\over{\partial x}} f(v) \right] (s) ds. $$
Since we have
$$  \left| {\partial\over{\partial x}} e^{\Delta t} \rho \right|_{L^2} \leq
{c\over{t^{3/4}}} \,\,|\rho|_{L^1},\quad t > 0, $$
we deduce
$$ \eqalign{
 \left|{{\partial v}\over{\partial x}}\right|_{L^2} (t) & \leq
 \left|{{\partial v_0}\over{\partial x}}\right|_{L^2}  + c \int_0^t
{1\over{(t-s)^{3/4}}} \left[\,
\left| u {{\partial \bar u}\over{\partial x}}\right|_{L^1} +
\left| v^2 {{\partial v}\over{\partial x}}\right|_{L^1} + \left| v
{{\partial v}\over{\partial x}}\right|_{L^1}
\right] (s) ds \leq    \cr
   \cr
  & \leq  \left|{{\partial v_0}\over{\partial x}}\right|_{L^2} +
 c \int_0^t {1\over{(t-s)^{3/4}}} ds +  c \int_0^t {1\over{(t-s)^{3/4}}}
 \left|{{\partial v}\over{\partial x}}\right|_{L^2} (s) ds.   \cr
}$$
Hence, using Gronwall's inequality, we arrive at the desired estimate.
Since, for each $T > 0$,
$ {\partial\over{\partial x}} [\,|u|^2 - f(v)] \in L^2(0, T; L^2({\bf
R}))$, we conclude
that $v\in L^2(0, T; H^2({\bf R}))$ and
${{\partial v}\over{\partial t}} \in  L^2(0, T; L^2({\bf R}))$. \qed

\vskip 6mm



 \noindent{\bf 3. Proof of Theorem 1.}  \bigskip

Let $(u_ \varepsilon, v_ \varepsilon)$ (for $\varepsilon \leq
\varepsilon_0$) be the approximating
solutions in $C([0, +\infty[; H^1({\bf R}))^2$ for the problem $(1.4),
(1.5)$. By Lemma 3 we have that 
$$ \{u_ \varepsilon \}_\varepsilon \,\,\hbox{\rm bounded in}\,\, L^\infty
({\bf R}_+ ; H^1({\bf R})),\qquad
 \left\{{{\partial u_\varepsilon}\over{\partial t}} \right\}_\varepsilon
\,\,\hbox{\rm bounded in}\,\,
 L^\infty ({\bf R}_+ ; H^{-1}({\bf R})). $$
Recall that, for each $R > 0$, the injection $H^1 (I_R) \hookrightarrow L^2 (I_R)$ is
compact, where
$I_R = \{ x\in{\bf R} : |x| \leq R \}$. 
So, for each $T > 0$, the
injection of the space
$$ W_R^T = \left\{ w\in L^2(0, T ; H^1 (I_R))\,\, :\,\,  {{\partial
w}\over{\partial t}} \in
  L^2(0, T ; H^{-1} (I_R) \right\} $$
in  $L^2(0, T ; L^2 (I_R))$ is compact, by a well known result. 

By applying a standard diagonalization procedure, there exists $u\in  L^\infty ({\bf R}_+ ;
H^1({\bf R}))$ and a
subsequence of $ \{u_ \varepsilon \}_\varepsilon$ still denoted by $ \{u_
\varepsilon \}_\varepsilon$
such that
$$ u_ \varepsilon \build\rightharpoonup_{\varepsilon\rightarrow 0}^{} u
\quad {\rm in}\quad
 L^\infty ({\bf R}_+ ; H^1({\bf R}))\,\,{\rm weak *}\quad {\rm and}\,\,
\hbox{\rm a.e. in}\quad  {\bf R}_+\times{\bf R}. \eqno (3.1) $$
We can also deduce that there exist $v\in  L^\infty ({\bf R}_+ ; (L^4 \cap
L^2) ({\bf R}))$ and
$w\in  L^\infty ({\bf R}_+ ; L^1 ({\bf R}))$ such that

$$\eqalign{
  &  v_ \varepsilon \build\rightharpoonup_{\varepsilon\rightarrow 0}^{} v
\quad {\rm in}\quad
 L^\infty ({\bf R}_+ ; (L^4 \cap L^2) ({\bf R}))\,\,{\rm weak *},   \cr
   \cr
  &  f(v_ \varepsilon) \build\rightharpoonup_{\varepsilon\rightarrow 0}^{}
w \quad {\rm in}\quad
 L^\infty ({\bf R}_+ ; L^{4/3} ({\bf R}))\,\,{\rm weak *}.   \cr
}\eqno (3.2)
$$

\noindent Moreover we have, with $p=3$,
$$ | f'(\xi) | \,\leq \, c (1 + | \xi|^{p-1} ),\quad \xi\in{\bf R}, $$
and for each real convex $C^2$ entropy function $\eta$ with compact support
we deduce from $(1.4)$
with $q\in C^2({\bf R})$ such that $q' = \eta' f'$ ($q$ is the
corresponding entropy flux)
$$\eqalign{
{\partial\over{\partial t}} \eta (v^\varepsilon) +  {\partial\over{\partial
x}}
q(v^\varepsilon) & = \eta' (v^\varepsilon)\, |u^\varepsilon|_x^2 \,+\,
\varepsilon \,\eta'  (v^\varepsilon) \,
{{\partial^2 v^\varepsilon}\over{\partial x^2}}   \cr
   \cr
   & = \varepsilon \,{{\partial^2}\over{\partial x^2}} (  \eta
(v^\varepsilon)) \,-\, \varepsilon \,
\eta''(v^\varepsilon) \left({{\partial v^\varepsilon}\over{\partial
x}}\right)^2.   \cr
}\eqno (3.3)
$$
For each open bounded set $\Omega$ in ${\bf R}\times ]0, +\infty[$, $ \{v_
\varepsilon \}_\varepsilon$
is bounded in $L^p(\Omega)$ with $p=3$, and, by Lemma 3, $
\left\{ \varepsilon \,\eta''(v^\varepsilon) \left({{\partial
v^\varepsilon}\over{\partial x}}\right)^2
\right\}_\varepsilon$ is bounded in $L^1(\Omega)$ (since $\eta''$ is
continuous with compact support)
and so is bounded in the space of measures with finite mass ${\cal M}
(\Omega)$. Hence, by $(3.3)$
$$ {\partial\over{\partial t}} \eta (v^\varepsilon) +
{\partial\over{\partial x}}
q(v^\varepsilon) \,\in\, [\, \hbox{\rm compact set of}\,\,
H^{-1}(\Omega)\,]\,\,+\,\,
  [\, \hbox{\rm bounded set of}\,\, {\cal M} (\Omega)\,]. $$
We can now apply the Corollary 3.1 of Theorem 3.2 in [5] and, by a suitable
diagonalization
procedure, we can deduce $ f(v_ \varepsilon)
\build\longrightarrow_{\varepsilon\rightarrow 0}^{} f(v) $
in ${\cal D}' ({\bf R}\times ]0, +\infty[)$. Hence, by $(3.2)$, we have
 $ f(v_ \varepsilon) \build\longrightarrow_{\varepsilon\rightarrow 0}^{} f(v) $
in  $L^\infty ({\bf R}_+ ; L^{4/3}({\bf R}))\,\, {\rm weak*}$. 
Finally, in view of $(1.4), (1.5)$ and passing
to the limit $\varepsilon\rightarrow 0$, we obtain the desired result. \qed

 
\vskip 15mm

 

\noindent {\bf Acknowledgements:} This research was partially supported by
FCT under program
 POCTI (Portugal/FEDER-EU).

\vskip 6mm



 \centerline{REFERENCES}\bigskip

\item{[1]} {\teo D.J. Benney,} {\it A general theory for interactions
between short and long waves,}
Studies in Appl. Math., 56 (1977), 81-94.
\bigskip

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and kinetic relations:scalar
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\bigskip

\item{[3]} {\teo C. Kondo {\rm and} P.G. LeFloch,} {\it Zero
diffusion-dispersion limits for hyperbolic
conservation laws,} SIAM J. Math. Anal., 33 (2002), 1320-1329.
\bigskip
 

\item{[4]} {\teo F. Murat,} {\it Compacit\'e par compensation,} Ann. Scuola
Norm. Sup. Pisa, 5 (1978), 489-507.
\bigskip


\item{[5]} {\teo M.E. Schonbek,} {\it Convergence of solutions to nonlinear
dispersive equations,}
 Comm. Partial Diff. Eq., 7 (1982), 959-1000.
\bigskip


\item{[6]} {\teo L. Tartar,} {\it Compensated compactness and applications
to partial
differential equations,} Research Notes in Math., 39, pg.136-212, Pitman, 1979.
\bigskip


\item{[7]} {\teo M. Tsutsumi {\rm and} S. Hatano,} {\it Well-posedness of
the Cauchy problem for the long wave - short wave resonance equations,}
Nonlinear Anal., TMA, 22 (1994), 155-171.
\bigskip

\item{[8]} {\teo  M. Tsutsumi {\rm and} S. Hatano,}  {\it Well-posedness of
the Cauchy problem for Benney's first equations of  long wave short wave
interactions,}
 Funkcialaj Ekvacioj, 37 (1994), 289-316.
 \bigskip



\vskip 15mm

\noindent \hbox to 3cm{\hrulefill} \medskip

\noindent Authors e-mail addresses:\smallskip

\noindent J.P.Dias: dias@ptmat.fc.ul.pt

\noindent M.Figueira: figueira@ptmat.fc.ul.pt




















\end