On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity

Abstract

We study the nonlinear Schrödinger equation with initial data in ${\mathcal{Z}}_p^s({\mathbb{R}}^d)={\dot{H}}^s({\mathbb{R}}^d)\cap L^p({\mathbb{R}}^d)$, where $0<s<\min \{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.