Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. (English) Zbl 0928.35158

Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis

Invent. Math. 134, No. 3, 489-545 (1998).

The authors treat smoothing and local existence theory of generalized nonlinear Schrödinger equations where nonlinearities up to first order derivatives are allowed, and where the linear part may be nonelliptic. Those equations may model e.g. problems in water wave theory. One of their main results concerns the removal of size restriction for the initial data (which have to lie in certain Sobolev spaces) of local solutions; these solutions inherit a local smoothing effect in space time. The method consists in introducing the space derivatives of wave function as new variables, writing the original equation as a system, and using results of earlier papers of the same authors. The system has the advantage that the crucial term vanishes for \(t=0\) if a solution of the original equation is inserted.

Reviewer: H.Lange (Köln)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

smoothing; nonlinear Schrödinger equations; generalized nonlinear Schrödinger equations; local smoothing effect in space time