Massimiliano Berti,Jean-Marc Delort
Almost global existence of solutions for 
capillarity-gravity water waves equations 
with periodic spatial boundary conditions
(1998K, PDF)

ABSTRACT.  The goal of this monograph is to prove that any solution of the Cauchy 
problem for the capillarity-gravity water waves equations, in one space dimension, 
with periodic, even in space, initial data of small size , is almost 
globally defined in time on Sobolev spaces, i.e. it exists on a time interval of 
length of magnitude −N for any N, as soon as the initial data are smooth 
enough, and the gravity-capillarity parameters are taken outside an exceptional 
subset of zero measure. In contrast to the many results known for 
these equations on the real line, with decaying Cauchy data, one cannot 
make use of dispersive properties of the linear flow. Instead, our method is 
based on a normal forms procedure, in order to eliminate those contributions 
to the Sobolev energy that are of lower degree of homogeneity in the 
solution. 
Since the water waves equations are a quasi-linear system, usual normal 
forms approaches would face the well known problem of losses of derivatives 
in the unbounded transformations. In this monograph, to overcome such 
a difficulty, after a paralinearization of the capillarity-gravity water waves 
equations, necessary to obtain energy estimates, and thus local existence 
of the solutions, we first perform several paradifferential reductions of the 
equations to obtain a diagonal system with constant coefficients symbols, up 
to smoothing remainders. Then we may start with a normal form procedure 
where the small divisors are compensated by the previous paradifferential 
regularization. The reversible structure of the water waves equations, and 
the fact that we look for solutions even in x, guarantees a key cancellation 
which prevents the growth of the Sobolev norms of the solutions.