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Large-time dynamics for the one-dimensional Schrödinger equation

Published online by Cambridge University Press:  04 April 2011

Nicolas Burq
Affiliation:
Département de Mathématiques, Université Paris Sud, Bât. 425, 91405 Orsay Cedex et Institut Universitaire de [email protected]

Abstract

Famous results by Rademacher, Kolmogorov and Paley and Zygmund state that random series on the torus enjoy better Lp bounds that the deterministic bounds. We present a natural extension of these harmonic analysis results to a partial-differential-equations setting. Specifically, we consider the one-dimensional nonlinear harmonic oscillator i∂tu + Δu − |x|2u = |u|r−1u, and exhibit examples for which the solutions are better behaved for randomly chosen initial data than would be predicted by the deterministic theory. In particular, on a deterministic point of view, the nonlinear harmonic oscillator equation is well posed in L2(ℝ) if and only if r ≤ 5. However, we shall prove that, for all nonlinearities |u|r−1u, r > 1, not only is the equation well posed for a large set of initial data whose Sobolev regularity is below L2, but also the flows enjoy very nice large-time probabilistic behaviour.

These results are joint work with Laurent Thomann and Nikolay Tzvetkov.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2011

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