On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity

Abstract

We prove continuity properties for the flow map associated to the defocusing energy-subcritical power-like nonlinear Schr{\"o}dinger equation, when the power varies. We show local in time continuity in the energy space for any power, and global in time continuity for sufficiently large powers. When the linear dispersive rate is counterbalanced by a time-dependent rescaling, we show a uniform in time continuity of the squared modulus of this rescaled function, in Kantorovich distance, for any power, including long range cases in terms of scattering. The most difficult result addresses the convergence of suitably renormalized solutions to the solution of the logarithmic Schr{\"o}dinger equation, when the power goes to zero, uniformly in time, in Kantorovich distance. The proof relies on estimates for perturbed porous medium equations, involving the harmonic Fokker-Planck operator.