Scattering and non-scattering of the Hartree-type nonlinear Dirac system at critical regularity

Abstract

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by V b (x) = 1 4π e −b|x| |x| (b ≥ 0). In previous works, a standard argument is to utilise 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 the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global wellposedness and scattering of Dirac equations with Hartree-type nonlinearity for b > 0 for small L 2x -data with additional angular regularity. We also show that only small amount of angular regularity is required to obtain global existence of solutions. Secondly, we obtain non-scattering result for a certain class of solutions with the Coulomb potential b = 0.

Global Existence for The Massive Dirac Equations with small initial datum on Tori

“…In [18] The results are as follows: first, these two kinds of systems have local well-posedness in square integrable fractional Sobolev space, where the order of Sobolev space is greater than a constant; second, for any square integrable Sobolev space of order less than the constant, the flow maps determined by these two kinds of systems, if them exist, are not differentiable at the origin of order 3. In the same case of 2 elements, [7] considers the same Yukawa potential as [6]. It is concluded that the Dirac equation is unique and exists globally with small initial data in a square integrable function space, and the solution scatterers.…”

“…Mathematicians have also made a lot of explorations on the equation of variable including time. The scattering and non-scattering problems of hartree-type nonlinear Dirac systems with critical regularity were studied by Cho, Hong and Lee in [6]. The nonlinear term of this equation is not a function but a functional, and is the convolution of a function called the Yukawa potential (under some conditions it is called the Coulomb potential) with the quadratic form of the solution.…”

“…Through the process, the authors observe that only a small amount of angular regularity is required to obtain global well-posedness. In addition, [6] studies a class of solutions with given Coulomb potential, which are not scattering. In addition to the Dirac type equations with 3-spatial variables, the case of 2 elements has also been studied.…”

Charge conjugation approach to scattering for the Hartree type Dirac equations with chirality

“…If γ 5 is replaced by the identity (in this case φ is a scalar field), then (1.1) and (1.3) have been extensively studied [5,12,1,3,4,20,21,22,13,6,7,9]. Our work is motivated from these works and is concerned with the relation between charge conjugation and chirality (see (1.7) below).…”

“…This shows that the chirality is not conserved and a smallness on both data ψ 0,R and ψ 0,L rather than a partial smallness is necessary for the global well-posedness unlike the charge conjugation as stated in Theorem 1.1. x space unless the Majorana condition appears. In fact, one may consider scattering states satisfying coercivity condition guaranteeing the full time decay t − 3 2 (see [11]) as in [9,7], to which any solution of (1.1) dose not converge in L 2…”