Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity · arXivDeskmath/0703640Mar 21, 2007
Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity
Stéphane Vento
Abstract
We establish the local well-posedness of the generalized Benjamin-Ono equation ∂tu+H∂x2u±uk∂xu=0
in
,
s>1/2−1/k for
and without smallness assumption on the initial data. The condition
s>1/2−1/k is known to be sharp since the solution map
is not of class
Ck+1 on
for
s<1/2−1/k . On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill-posedness in
,
.