In  this talk I will present some results on  the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. I will show that if the data are of size $\eps$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\eps$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma - C'\eps t$. This question comes from motivations from control theory which force us to consider analytic solutions. I will actually start the talk with these motivations.

This is joint work with T. Alazard and C. Zuily.

We consider the question of stabilization for the damped wave equation on tori
$$(\partial_t^2 -\Delta )u +a(x) \partial _t u =0.$$
When the damping coefficient $a(x)$ is continuous the question is quite well understood and the geometric control condition is necessary and sufficient for uniform (hence exponential) decay to hold. When $a(x)$ is only $L^{\infty}$ there are still gaps in the understanding.
Using second microlocalization we completely solve the question for 
Damping coefficients of the form
$$a(x)=\sum_{i=1}^{J} a_j 1_{x\in R_j},$$
Where $R_j$ are cubes.
This is a joint work with P. Gérard