Strong ill-posedness for fractional Hartree and cubic NLS Equations

2021/01/11 by Bhimani, Divyang G., Haque, Saikatul · 2 citations
#35A01 (secondary) #35Q55 #42B35 (primary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.03991

Abstract

We consider fractional Hartree and cubic nonlinear Schrödinger equations on Euclidean space \mathbb R^d and on torus \mathbb T^d. 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.

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