Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces

Abstract

We prove new well-posedness results for energy-critical nonlinear Schrödinger equations in modulation spaces, which are larger than the energy space. First, we remove the ε-derivative loss in L p -smoothing estimates for the linear Schrödinger equation, if p is larger than the Tomas-Stein exponent. Next, we show local well-posedness results for nonlinear Schrödinger equations in modulation spaces containing the scaling critical L 2 -based Sobolev space. The proof is carried out via bilinear refinements and adapted function spaces.

Wellposedness of NLS in Modulation Spaces

“…Using decoupling techniques, Schippa [29] recently proved L p smoothing estimates and extended the range of local wellposedness results for p ∈ {4, 6} and also, inspired by the work [16], gave global results for q = 2, 2 ≤ p < ∞, s > 3/2. Finally we want to mention the preprint [30] in which Schippa very recently considered the energy-critical NLS with initial data in modulation spaces.…”