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

Demirbas, Seckin

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 471-485

Résumé

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.

Détail

DOI : 10.4153/CMB-2015-025-7

Mots-clés : 35Q55, NLS, fractional Schrodinger equation, almost sure global wellposedness