Analyticity and infinite breakdown of regularity in mass-subcritical Hartree scattering

TL;DR

This paper investigates the regularity and well-posedness of the scattering problem for the defocusing mass-subcritical Hartree NLS, revealing an infinite loss of regularity between weighted spaces and L^2.

Contribution

It demonstrates the analytic well-posedness in weighted spaces and the failure in L^2, extending previous work on finite regularity breakdown in mass-subcritical NLS.

Findings

Well-posedness in weighted spaces $\\Sigma$ and $\\mathcal{F}H^1$

Failure of well-posedness in $L^2$

Infinite regularity loss between weighted and unweighted spaces

Abstract

We study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $iu_t + \Delta u = F(u) = (|x|^{-\gamma}*|u|^2)u$ on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < \gamma < 2$. We show that the scattering problem associated to this equation is analytically well-posed in the weighted spaces $\Sigma = H^1\cap\mathcal{F}H^1$ and $\mathcal{F}H^1$. Furthermore, we show that the same problem fails to be analytically well-posed for data in $L^2$. This constitutes an infinite loss of regularity between the scattering problems in weighted spaces and in $L^2$. This further develops an earlier investigation initiated by the author in which a finite breakdown of regularity was proved for the $L^2$ scattering problem for the mass-subcritical NLS with power nonlinearity $F(u) = |u|^pu$.