ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHR(sic)DINGER EQUATION WITH LARGE INITIAL DATA

Abstract

We consider the fourth-order nonlinear Schr(sic)dinger equation (4NLS) (i partial derivative(t) + epsilon Delta + Delta(2))u = c(1)u(m) + c(2)(partial derivative u)u(m-1) + c(3)(partial derivative u)(2)u(m-2) and establish the conditional almost sure global well-posedness for random initial data in H-s(R-d) for s is an element of (s(c) - 1/2, s(c)], when d >= 3 and m >= 5, where s(c) := d/2 - 2/(m - 1) is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new M -norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for d = 2, m >= 4 and d >= 3, 3 <= m < 5.