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.