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

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

2006-12-15

arxiv.org/abs/math/0612457v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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 $H^s$.

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