ON THE PROBABILISTIC WELL-POSEDNESS OF THE NONLINEAR SCHRODINGER EQUATIONS WITH NON-ALGEBRAIC NONLINEARITIES

We consider the Cauchy problem for the nonlinear Schrodinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on R-d , d = 5, 6, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on R-d , d = 5, 6, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.