Well-posedness of the self-consistent physical-frequency BGK model

Establish well-posedness, including uniqueness, for the self-consistent BGK equation with collision frequency equal to the local density, beyond the global existence theory proved for slab domains with inflow or Maxwell boundary conditions.

Background

The paper studies the BGK equation in a slab with the physical collision frequency νf=ρf\nu_f=\rho_f, where the collision rate depends on the unknown density and the local Maxwellian depends on the solution’s mass, momentum, and temperature. It proves global renormalized existence for large data with finite mass, kinetic energy, and entropy under inflow or Maxwell boundary conditions.

The cited open issue concerns the broader well-posedness problem for the original self-consistent model. The paper addresses only its global existence component; uniqueness and more general collision frequencies are explicitly left unresolved and outside the scope of the result.

References

More recently, Section 1.1 explicitly identifies well-posedness for the original self-consistent model with $\lambda(\rho)=\rho$ as an open problem. Theorem \ref{thm1} below addresses the global existence part of this problem in a slab with inflow or Maxwell reflection; uniqueness and more general collision frequencies are outside the scope of the result.

— Global large-data solutions for the BGK equation with physical collision frequency in a slab  (2610.08658 - Choi et al., 6 Oct 2026) in Section 1, subsection “Previous results and the large-data problem”