Almost sure local well-posedness for the supercritical quintic NLS

This paper studies the quintic nonlinear Schrödinger equation on $ℝ^{d}$ with randomized initial data below the critical regularity $H^{(d - 1) ∕ 2}$ for $d \geq 3$ . The main result is a proof of almost sure local well-posedness given a Wiener randomization of the data in $H^{s}$ for $s \in (\frac{1}{2} (d - 2), \frac{1}{2} (d - 1))$ . The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.

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Keywords

almost sure well-posedness, supercritical, dispersive, PDEs, NLS equation

Mathematical Subject Classification 2010

Primary: 35K55

Secondary: 35R60

Milestones

Received: 9 April 2018

Accepted: 19 June 2018

Published: 8 August 2018

Authors

Justin T. Brereton
Department of Mathematics University of California Berkeley, CA United States

Vol. 1, No. 3, 2019

Justin T. Brereton

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors