Dynamics of Partial Differential Equations
Volume 15 (2018)
On the energy-critical fractional Schrödinger equation in the radial case
Pages: 265 – 282
We consider the Cauchy problem for the energy-critical nonlinear Schrödinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in the focusing case with energy below the ground state. The main feature of the present work is the nonlocality of the operator. This does not allow us to use standard computations for the rigidity part of the theorem. Instead we develop a commutator argument which has its own interest for problems with nonlocal operators.
nonlinear wave equation, nonlinear Schrödinger equation
2010 Mathematics Subject Classification
This work was supported by Australian Research Council Discovery Project (Grant No. DP170101060). L. Zhao is supported by grant NSFC of China no. 11771415. Y. Wang is supported by NSFC of China, no. 11771140. Y. Sire is supported by a Simons fellowship.
Received 27 November 2017