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\null
\vskip -10mm


\centerline{\Teo \'Equations aux D\'eriv\'ees Partielles} \vskip 10mm


\centerline{\big  Existence of local strong solutions for a quasilinear
Benney system}\vskip 6mm



\centerline{ $\hbox{\bf Jo\~ao-Paulo Dias}^{(1)}$, $\hbox{\bf M\'ario
Figueira}^{(1)}$
\hbox{\rm and} $\hbox{\bf Filipe Oliveira}^{(2)}$ } \bigskip
\noindent $(1)$ CMAF/UL and FCUL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa
- PORTUGAL
\medskip

\noindent $(2)$ Dep. Matem\'atica, FCT/UNL, Monte da Caparica - PORTUGAL

\vskip 5mm

\noindent \hbox to 8cm{\hrulefill}\bigskip


 \noindent {\bf Abstract}\medskip


We prove in this Note the existence and uniqueness of a strong local
solution to the Cauchy
problem for the quasilinear Benney system (1).


\vskip 10mm

\centerline{\big Existence d'une solution locale forte}\medskip
\centerline{\big pour un syst\`eme de Benney quasilin\'eaire} \bigskip

 \noindent {\bf R\'esum\'e}\medskip

Nous prouvons dans cette Note l'existence et unicit\'e d'une solution
locale forte du
probl\`eme de Cauchy pour le syst\`eme de Benney quasilin\'eaire (1). \bigskip

\noindent \hbox to 8cm{\hrulefill}\bigskip


\vskip 6mm

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

We consider the system introduced by Benney in [1] to study the interaction
between short
and long waves, for example gravity waves in fluids :
$$ \left\{\matrix{
 i u_t + u_{xx} =
    |u|^2 u  + v u  \hfill & & (a)   \cr
                 & x\in{\bf R}, t\geq 0,\quad &   \cr
 v_t + [f(v)]_x
=  |u|_x^2    \hfill &  & (b) \cr
}\right.
 \eqno (1)  $$
where $f$ is a polynomial real function, $u$ and $v$ (real) represent the
short and
the long wave, respectively.\bigskip

\noindent \hbox to 8cm{\hrulefill}

\noindent{ \halign{# & # \cr
{\it Adresses e-mail} : & $\hbox{\rm dias@ptmat.fc.ul.pt (J.P. Dias)}$   \cr
                                     & $\hbox{\rm figueira@ptmat.fc.ul.pt
(M. Figueira)}$   \cr
                                     & $\hbox{\rm fso@fct.unl.pt (F.
Oliveira)}$   \cr}
\vfill
\eject
\null



In [2] the existence of weak solutions for $(1)$ was proved for
$ f(v) =  a v^2 - b v^3,\ $ with $a$ and $b$ real constants, $b > 0$, in
the following sense:
\medskip





 \noindent{\bf Theorem 1.1} {\it Given $u_0, v_0 \in H^1 ({\bf R})$ with
$v_0$ real-valued, there
exists 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
$$ \eqalign{
i \int_0^\infty \int_{\bf R}  u {{\partial \varphi}\over{\partial t}} \,dx
\,dt &+
 \int_0^\infty \int_{\bf R} {{\partial u}\over{\partial x}} \,
 {{\partial \varphi}\over{\partial x}} \,dx\,dt +    \cr
  & \int_{\bf R} u_0(x) \varphi (x, 0) \,dx
    +  \int_0^\infty \int_{\bf R} |u|^2u \varphi\, dx\,dt
+  \int_0^\infty \int_{\bf R} v u \varphi \,dx\,dt = 0, \cr
}$$
\medskip

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

\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 and $\psi$ real-valued.}

\bigskip

 This result was obtained for this particular system by application of the
vanishing viscosity method and we could not extend the necessary estimates
to the Burger's
case ($a=1, b=0$) or to more general cases. Here we will prove the
existence of (local) strong
solutions to
$(1)$ for general $f$, extending previous results in [6, 7] for $f$ linear :

\bigskip



 \noindent{\bf Theorem 1.2} {\it Let $(u_0, v_0) \in  H^3({\bf R})\times
H^2({\bf R})$
and
$f\in C^3({\bf R})$. Then there exists a unique strong solution $(u, v)$ of
the
Cauchy problem associated to $(1)$, with
$$ (u, v) \in C^j([0, T]; H^{3-2j}({\bf R})) \times C^j([0, T];
H^{2-j}({\bf R})),\, j=0, 1. $$
Here, the life-span $T > 0$ depends exclusively on $f$ and on the initial
data $(u_0, v_0)$.}

\bigskip

The main difficulty here is the derivative-loss in the right-hand side of
equation $(1-a)$. This
cannot be handled easily by the Schr\"odinger kernel, due to its limited
smoo-\break
 thing properties. The method employed in [6, 7] for $f$ linear, based in the
inhomogeneous smoothing effect of the Schr\"odinger group, can not be
easily implemented for
$f$ nonlinear. We will address this problem by introducing some auxiliary
functions and
rewriting system $(1)$ without derivative loss. A similar technique was
introduced in [5] to
solve the fully nonlinear wave equation and employed in [4], in the context
of the
Zakharov-Rubenchik system.
 \medskip Another interesting open problem is the study of the probable
blow-up of the local
smooth solutions.



\vskip 6mm

\noindent {\bf 2. An equivalent system.}\bigskip

Let us take $(u, v)$ a solution of $(1)$. By setting $F = u_t$, we obtain
from $(1-a)$
$$ i F + u_{xx} - u = |u|^2 u + u (v - 1), $$
and
$$ u = (\Delta - 1)^{-1} ( |u|^2 u + u (v - 1) - i F), \eqno (2) $$
with  $\quad \Delta = \displaystyle {{\partial^2}\over{\partial x^2}}.$
  Also, differentiating $(1-a)$ with respect to $t$ leads to
$$ i F_t + F_{xx} = 2  |u|^2 F + u^2 \bar F + F v + u v_t, $$
and from $(1-b)$,
$$ i F_t + F_{xx} = 2  |u|^2 F + u^2 \bar F + F v + u |u|_x^2 - u v_x
f'(v).   \eqno (3)  $$
These computations are our motivation to consider the following Cauchy problem:
$$ \left\{\matrix{
i F_t + F_{xx} = 2  |u|^2 F + u^2 \bar F + F v + u |\tilde u|_x^2 - u v_x
f'(v) &\quad (a)   \cr
   \cr
v_t + [f(v)]_x = |\tilde u|_x^2       \hfill   & \quad (b)  \cr
   \cr
F(x, 0) = F_0(x) \in H^1({\bf R}),\,\, v(x, 0) = v_0(x) \in H^2({\bf R})
\hfill   &   \cr
}
\right.
\eqno (4)
$$

where $u$ and $\tilde u$ are given in terms of $F$ by
$$ u(x, t) = u_0 + \int_0^t F(x, s) ds\quad {\rm and} \quad
\tilde u (x, t) =  (\Delta - 1)^{-1} ( |u|^2 u + u (v - 1) - i F).   \eqno
(5) $$
Note that in this system derivative losses do not occur. Indeed, the
regularization
of $(\Delta - 1)^{-1}$ puts $\tilde u$ in $H^3$ and therefore the
right-hand side
of $(4-a)$ is in $H^1$, like $F$.\medskip

\noindent We will prove the following lemma:\bigskip



\noindent{\bf Lemma 2.1} {\it Let $(F_0, v_0)\in H^1({\bf R})\times
H^2({\bf R})$ and
$f\in C^3({\bf R})$. Then there exists $T > 0$ and a unique strong solution
$(F, v)$
of the Cauchy problem $(4-a,b)$, with
$$ (F, v) \in C^j([0, T]; H^{1-2j}({\bf R})) \times C^j([0, T];
H^{2-j}({\bf R})),\, j=0, 1. $$
Here, the life-span $T > 0$ depends exclusively on $f$ and on the initial
data $(F_0, v_0)$.}\medskip

\noindent This lemma will be proved in the next section, using the general
theory of Kato
for quasilinear equations ([3]).  \bigskip

We now explain why Lemma $2.1$ implies our main Theorem $1.2$:\medskip

\noindent  If $(F, v)$ is a solution of $(4)$, by differentiating $(5)$
with respect to $t$ we obtain
$$ u_t = F. $$
Replacing in $(1-a)$ yields by $(4-b)$
$$ \eqalign{
 (i u_t + u_{xx})_t & = 2 |u|^2 F + u^2 \bar F + F v + u |\tilde u|_x^2 - u
v_x f'(v)   \cr
   & =  2 |u|^2 u_t + u^2 \bar u_t + u_t v + u v_t   \cr
}
$$
Hence $(i u_t + u_{xx} - |u|^2 u - u v )_t = 0$ and we get
$$ i u_t + u_{xx} - |u|^2 u - u v = \phi_0(x), $$
where $\phi_0(x) = i F_0(x) + u_0''(x) - |u_0(x)|^2 u_0(x) - u_0(x) v_0(x).
$ By setting
$$ F_0(x) = i ( u_0''(x)  - |u_0(x)|^2 u_0(x) - u_0(x) v_0(x) ), \eqno (6)   $$
we obtain $\phi_0 = 0$ and $(u, v)$ satisfies $(1-a)$. Furthermore, from
$(1-a)$,
$$ u = (\Delta - 1)^{-1} (  |u|^2 u + u (v-1) - i u_t).  \eqno (7) $$
Therefore $u = \tilde u$ and $(u, v)$ satisfies $(1-b)$.
Note that $u_t = F \in C([0, T]; H^1({\bf R}))$. Also $ u(., t) = u_0(.) +
\int_0^t F(., s) ds
\in C([0, T]; H^1 ({\bf R}))$, but from $(7)$ we have in fact
$$ u\in C([0, T]; H^3 ({\bf R})). $$


\vskip 6mm

\noindent {\bf 3. Proof of Lemma 2.1.}\bigskip

In order to apply a variant of theorem 6 in [3] we need to set the Cauchy
problem $(4)$ in the
framework of real spaces. We introduce the new variables
$$ F_1 = \Re F,\,\, F_2 = \Im F,\,\, u_1 = \Re u,\,\, u_2 = \Im u $$
and with $U = (F_1, F_2, v),\, F_{10} = \Re F_0,\, F_{20} = \Im F_0$
$\,(4)$ can be written as follows :
$$ \left\{\matrix{
\displaystyle {\partial\over{\partial t}} U + A (U) U = g(t, U)   \hfill    \cr
   \cr
(F_1(x, 0), F_2(x, 0), v(x, 0)) = ( F_{10} (x),  F_{20} (x), v_0(x)) \in
( H^1 ({\bf R}))^2 \times  H^2 ({\bf R})   \cr
}
\right.
\eqno (8)
$$
where
$$ A(U) = \left[\matrix{
0  &   \Delta   &   0   \cr
-\Delta  & 0  &  0  \cr
0  &  0  & \displaystyle f'(v){\partial\over{\partial x}}   \cr
}
\right]
 $$
and

$$ g(t, U) = \left[\matrix{
2 |u^2| F_2 - (u_1^2 - u_2^2) F_2 + 2 u_1 u_2 F_1 + F_2 v +
u_2 |\tilde u|_x^2 - u_2 v_x f'(v)   \cr
  \cr
- 2 |u^2| F_1 -  (u_1^2 - u_2^2) F_1 - 2 u_1 u_2 F_2 - F_1 v
- u_1 |\tilde u|_x^2 + u_1 v_x f'(v) \cr
   \cr
|\tilde u|_x^2   \cr
}\right]
$$
which is a non-local source term.

\noindent Now we set $X = (H^{-1} ({\bf R}))^2 \times L^2({\bf R})$,
$ Y = (H^1 ({\bf R}))^2 \times H^2({\bf R})$ and introduce $S : Y
\longrightarrow X$ defined by
  $ S = (1 - \Delta) I$, which is an isomorphism. Moreover $A: U = (F_1,
F_2, v) \in W
\longrightarrow G(X, 1,\beta)$, where $W$ is an open ball in $Y$ centered
at the origin and
with radius $R$ and $G(X, 1, \beta)$ denotes the set of all linear
operators $D$ in $X$ such
that $-D$ generates a $C_0$ - semigroup
$\left\{e^{-tD}\right\}$ with
$$ \left\| e^{-tD } \right\| \leq e^{\beta t},\,\, t\in [0, +\infty[, $$
$$ \beta = {1\over 2} \sup_{x\in{\bf R}} | f''(v(x)) \,v_x(x) | \leq c \,R
\, \alpha (R), \quad
     $$
where $c> 0$ is a numerical constant and $\alpha (R)$ is a continuous
function (cf.[3], \S8).
It is easy to see that $g$ verifies, for fixed $T > 0$,
$$ \| g(t, U)\|_Y \leq \lambda,\quad t\in [0, T],\,\, U\in W. $$
Now, with $B_0(v) \in {\cal L} (L^2({\bf R})),\, v $ in a ball $W_1$ in
$H^2({\bf R})$,
$ B_0 (v) $ defined by $(8.7)$ in [3]
$$ B_0(v) = - [f''(v) v_{xx} + f'''(v) v_x^2 ] {\partial\over{\partial x}}
(1 - \Delta)^{-1}
- 2 f'(v) v_x {{\partial^2}\over{\partial x^2}} (1 - \Delta)^{-1}, $$
we introduce an operator $B(U) \in  {\cal L} (X),\, U = (F_1, F_2, v) \in W$, defined by
$$ \left[\matrix{
  0  &  0  &  0   \cr
0  &  0  &   0   \cr
0  &  0
&  B_0(v)  \cr
}\right]
$$
In [3], \S 8, Kato proved that for $v\in W_1$ we have
$$ (1 - \Delta) \left( f'(v) {\partial\over{\partial x}} \right) (1 -
\Delta)^{-1} =
f'(v)  {\partial\over{\partial x}} + B_0(v). $$
Hence, we easily derive for $U\in W$
$$ S A(U) S^{-1} = A(U) + B(U). $$
Now, for each pair $(U, U^*),\, U = (F_1, F_2, v)$ and $ U^* = (F_1^*,
F_2^*, v^*)$ in $W$ we will prove that
$$ \| g(t, U) - g(t, U^*) \|_{L^1(0, T' ; X)} \leq c(T') \sup_{0\leq t \leq T'}
\| U(t) - U^*(t) \|_X   \eqno (9)
$$
for $T'\in [0, T]$ where $c(T')$ is a continuous increasing function such
that $c(0)=0$.

\noindent Let us point out that if $h\in L^2({\bf R})$ and $w\in H^1 ({\bf
R})$ we easily derive
$$ \| h w \|_{H^{-1}} \leq \| h \|_{H^{-1}} \|w \|_{H^1}. $$
Hence, for example, we get, with an obvious notation,
$$ \| F_1 u_1 (u_1^* - u_1) \|_{H^{-1}} \leq  \| F_1 \|_{H^1} \| u_1
\|_{H^1} \|u_1^* - u_1
\|_{H^{-1}} $$ and, for $t\leq T'$
$$ \eqalign{
\biggl\| f'(v) v_x  & \left(\int_0^t F_2 d\tau - \int_0^t F_2^* d\tau
\right) \biggr\|_{H^{-1}}
\leq \| f'(v) v_x \|_{H^1} \int_0^t \| F - F^* \|_{H^{-1}} d\tau   \cr
   \cr
   & \leq c(T') \sup_{0\leq t \leq T'} \| U(t) - U^* (t) \|_X   \cr
} $$
where $c(T')$ is a continuous increasing function such that $c(0) = 0$.
Now, Lemma 2.1 is an
easy consequence of Theorem 6 in [3], where the local condition $(7.7)$ is
replaced
by $(9)$ which is sufficient for the proof of this theorem.


 \vskip 5mm


\noindent {\bf Acknowledgements:} This research was partially supported by
FCT under program
 POCI 2010 (Portugal/FEDER-EU).

\vskip 5mm



 \centerline{REFERENCES}\bigskip

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

\bigskip

\item{[2]} J.P. Dias and M. Figueira, Existence of weak solutions for a
quasilinear version
of Benney equations, J. Hyp. Diff. Eq., to appear.

\bigskip


\item{[3]}  T. Kato, Quasi-linear equations of evolution, with applications
to partial
differential equations, Lecture Notes in Math.,Springer, 448 (1975), 25-70.
\bigskip

\item{[4]} F. Oliveira, Stability of the Solitons for the one-dimensional
 Zakharov-Rubenchik Equation, Physica D, 175 (2003), 220-240.
\bigskip


\item{[5]} Y. Shibata and Y. Tsutsumi, Local existence of solutions for the
initial
boundary problem of fully nonlinear wave equation, Nonlinear Analysis TMA,
11 (1987), 335-365.

\bigskip


\item{[6]} M. Tsutsumi and S. Hatano, Well-posedness of
the Cauchy problem for the long wave - short wave resonance equations,
Nonlinear Anal., TMA, 22 (1994), 155-171.
\bigskip

\item{[7]} M. Tsutsumi and S. Hatano, Well-posedness of
the Cauchy problem for Benney's first equations of  long wave short wave
interactions,
 Funkcialaj Ekvacioj, 37 (1994), 289-316.
 \bigskip





\end
