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Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

Published online by Cambridge University Press:  20 November 2018

Seckin Demirbas
Seckin Demirbas*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A. e-mail: demirba2@illinois.edu
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Abstract

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In a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in ${{H}^{s}}$ for $s>1-\alpha /2$ and globally well-posed for $s>10\alpha -1/12$. In this paper we define an invariant probability measure $\mu$ on ${{H}^{s}}$ for $s<\alpha -1/2$, so that for any $\text{ }\!\!\varepsilon\!\!\text{ }>0$ there is a set $\Omega \subset {{H}^{s}}$ such that $\mu \left( {{\Omega }^{c}} \right)<\text{ }\!\!\varepsilon\!\!\text{ }$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for $\frac{1-\alpha }{2}<\alpha -\frac{1}{2},i.e.,\alpha >\frac{2}{3}$ in an almost sure sense.

Keywords

35Q55NLSfractional Schrodinger equationalmost sure global wellposedness
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 471 - 485
Copyright
Copyright © Canadian Mathematical Society 2015

References

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