#### Vol. 1, No. 3, 2019

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Almost sure local well-posedness for the supercritical quintic NLS

### Justin T. Brereton

Vol. 1 (2019), No. 3, 427–453
##### Abstract

This paper studies the quintic nonlinear Schrödinger equation on ${ℝ}^{d}$ with randomized initial data below the critical regularity ${H}^{\left(d-1\right)∕2}$ for $d\ge 3$. The main result is a proof of almost sure local well-posedness given a Wiener randomization of the data in ${H}^{s}$ for $s\in \left(\frac{1}{2}\left(d-2\right),\frac{1}{2}\left(d-1\right)\right)$. 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.

##### Keywords
almost sure well-posedness, supercritical, dispersive, PDEs, NLS equation
Primary: 35K55
Secondary: 35R60