Oscillatory integrals and regularity of dispersive equations. (English) Zbl 0738.35022

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

Indiana Univ. Math. J. 40, No. 1, 33-69 (1991).

The paper under review deals with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations of the type \[ \partial_ tu-iP(D)u=0,\quad x\in\mathbb{R}^ n,\;\;t\in\mathbb{R}, \] where \(D=(1/i)(\partial_ 1,\ldots,\partial_ n)\), \(\partial_ j=\partial/\partial x_ j\), and \(P(D)\) is defined via its real symbol, i.e., \[ P(D)f(x)=\int\exp(ix\xi)P(\xi)\hat f(\xi) d\xi. \] The authors study also some applications of these smoothing properties to nonlinear problems and their link with restriction theorem for the Fourier transform and positive convergence results.

Reviewer: R.Manthey (Jena)

MSC:

35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35Q53	KdV equations (Korteweg-de Vries equations)
35Q55	NLS equations (nonlinear Schrödinger equations)
35B65	Smoothness and regularity of solutions to PDEs
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

local and global smoothing properties; nonlinear problems; Fourier transform; convergence; initial value problem for free Schrödinger equation