Global existence and long-time dynamics for Coulomb-gauge Chern–Simons–Dirac systems

Establish global existence and characterize the long-time behavior of solutions to the two-dimensional Chern–Simons–Dirac equation in Coulomb gauge, including the expected modified-scattering dynamics.

Background

Under the Coulomb gauge condition, the two-dimensional Chern–Simons–Dirac system reduces to a nonlinear Dirac equation with a Hartree-type nonlinearity. The paper notes that local well-posedness has been studied under several gauge choices, but that the global theory and asymptotic behavior remain unresolved. The Fourier multipliers of the Hartree-type terms have the same order-minus-one singularity near the origin as the Coulomb potential, so the system is regarded as scattering-critical and modified scattering is expected. The analysis developed for the Dirac–Hartree equation cannot be directly applied because one of the bilinear components lacks the relevant Dirac null structure.

References

While local well-posedness results have been studied extensively under various gauge choices , not only long-time behavior but also the global existence of solutions to eq:csd-coulomb remains unknown.

— Long-range scattering for 2D Dirac-Hartree equations  (2608.12672 - Lee et al., 13 Aug 2026) in Section “Future Directions,” item (2), “Chern-Simons-Dirac systems”

We do not know whether the assumptions on the decay in the momentum variable in Theorem \ref{Th} are optimal. They arise naturally from our control of the characteristics and of the commuted transport equations, and weakening them does not seem accessible by a straightforward refinement of our estimates.

— Global dynamics of Vlasov--wave systems without the null condition under exponential momentum decay  (2608.27053 - Bigorgne, 27 Aug 2026) in Remark following the Conjecture in Section 1.3, “Main result”