Local well-posedness for the Cauchy problem of the Zakharov type system
A talk in the Oberseminar Analysis series by
Isao Kato
| Abstract: | In this talk, we consider the Cauchy problem of the degenerated Zakharov
type system. The system has no dispersion in some direction in the usual
Zakharov system. The linear part of the degenerated Zakharov system is
more complicated than that of the Zakharov system, so it is difficult to
apply directly the local well-posedness result by
Ginibre-Tsutsumi-Velo(1997). There are few well-posedness results for
this system. The latest result is given by Barros-Linares(2015) for the
three dimensional case by using the linear estimates. To obtain the
local well-posedness result with lower regularity initial data, we apply
the Fourier restriction norm method. We treat the non-resonant part and
the resonant part more carefully than in the case of the Zakharov system
because of the lack of dispersion, then we obtain the system is locally
well-posed in some anisotropic Sobolev space with lower regularity. |