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Solving the 4NLS with white noise initial data

Part of: Stochastic analysis Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 November 2020

Tadahiro Oh ,
Nikolay Tzvetkov  and
Yuzhao Wang
Tadahiro Oh
Affiliation:
School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom; E-mail: hiro.oh@ed.ac.uk
Nikolay Tzvetkov
Affiliation:
CY Cergy Paris University, Cergy-Pontoise, F-95000, UMR 8088 du CNRS, France; E-mail: nikolay.tzvetkov@cyu.fr
Yuzhao Wang
Affiliation:
School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom; E-mail: hiro.oh@ed.ac.uk School of Mathematics, School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom; E-mail: y.wang.14@bham.ac.uk

Abstract

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We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

Keywords

fourth-order nonlinear Schrrödinger equationbiharmonic nonlinear Schrrödinger equationwhite noiseinvariant measurerandom-resonant/nonlinear decompositionrandom Fourier restriction norm space

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Differential Equations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e48
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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