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

Michael Christ, James Colliander, Terence Tao

Published 2006-12-15Version 1

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schr\"odinger 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 $H^s$.