Strong ill-posedness for fractional Hartree and cubic NLS equations

Abstract

We consider fractional Hartree and cubic nonlinear Schrödinger equations on Euclidean space Rd and on torus Td. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in Fourier amalgam spaces with negative regularity. In particular, these spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show that this can be even worse by exhibiting norm inflation with an infinite loss of regularity. To establish these phenomena, we employ a Fourier analytic approach and introduce new resonant sets corresponding to the fractional dispersion (−Δ)α/2. In particular, when dispersion index α is large enough, we obtain norm inflation above scaling critical regularity in some of these spaces. It turns out that our approach could treat both equations (Hartree and power-type NLS) in a unified manner. The method should also work for a broader range of nonlinear equations with Hartree-type nonlinearity.