Local well-posedness of Dirac equations with nonlinearity derived from honeycomb structure in 2 dimensions

Abstract

The aim of this paper is to show local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlinearity derived from honeycomb structure in H s for s > 7 8 and s > 3 8 , respectively. We also provide the smoothness failure of flows of Dirac equations.

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

“…In addition to the Dirac type equations with 3-spatial variables, the case of 2 elements has also been studied. In [18], Lee discussed two kinds of systems: one is massless honeycomb potential Dirac equation; The other is the Hartree Dirac equation. The potential of the former is a diagonal matrix of order 2, whose elements are complex quadratic forms of solutions.…”

“…The nonlinear term of the latter is the convolution of the Coulomb potential with the square of the norm of the solution. 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].…”