In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space .
We also prove a long time existence result; more precisely we prove that for fixed there exists a set , such that any data , evolves up to time into a solution with , . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space , that is in the supercritical scaling regime.
Cite this article
Andrea R. Nahmod, Gigliola Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1687–1759