Critical well-posedness and scattering results for fractional Hartree-type equations. (English) Zbl 1463.35500

Herr, Sebastian; Yang, Changhun

Differ. Integral Equ. 31, No. 9-10, 701-714 (2018).

Summary: Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.

MSC:

35R11	Fractional partial differential equations
35B33	Critical exponents in context of PDEs
35P25	Scattering theory for PDEs
35Q40	PDEs in connection with quantum mechanics
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

radial function; small initial data; Fourier localization operator