SCATTERING AND NONSCATTERING OF THE HARTREE-TYPE NONLINEAR DIRAC SYSTEM AT CRITICAL REGULARITY

We consider Cauchy problem of Hartree-type nonlinear Dirac equation with potentials given by V_b(x) = 1 e^-b|x|/4π |x| (b≥0). In previous works regarding a global well-posedness of Dirac equation with these potentials, a standard argument is to utilize null form estimates in order to prove global well-posedness for H^s-data, s > 0. However, the null structure inside the equations is not enough to attain critical regularity. We impose an extra regularity assumption with respect to the angular variable. First, we prove the global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for b > 0 for small L²_x-data with additional angular regularity. We also show that only a small amount of angular regularity is required to obtain the global existence of solutions. Second, we obtain nonscattering result for a certain class of solutions with the Coulomb potential V0(x). 4π | x| (b≥0). In previous works regarding a global well-posedness of Dirac equation with these potentials, a standard argument is to utilize null form estimates in order to prove global well-posedness for H^s-data, s > 0. However, the null structure inside the equations is not enough to attain critical regularity. We impose an extra regularity assumption with respect to the angular variable. First, we prove the global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for b > 0 for small L²_x-data with additional angular regularity. We also show that only a small amount of angular regularity is required to obtain the global existence of solutions. Second, we obtain nonscattering result for a certain class of solutions with the Coulomb potential V₀(x).