A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Christ, Michael; Colliander, James; Tao, Terence

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Authors

Christ, Michael

Colliander, James

Tao, Terence

Description

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$ norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in $H^s$.
22 pages