Global regularity for the spatially inhomogeneous Landau equation

Establish global regularity for the Landau equation with general spatially inhomogeneous initial data beyond the spatially homogeneous setting.

Background

The paper places the linear stability analysis of traveling Maxwellians within the broader theory of the Landau equation. Although global regularity has recently been established for spatially homogeneous very soft potentials, the corresponding problem for spatially inhomogeneous solutions with general data remains unresolved. The authors note that existing inhomogeneous results generally assume a priori pointwise control of hydrodynamic quantities or address particular types of singularities, rather than proving global regularity for arbitrary large data.

References

Global regularity remains a wide open problem except for spatially homogeneous data.

— Linear stability of traveling Maxwellians for Landau equation with very soft potentials  (2608.18407 - Chaturvedi et al., 19 Aug 2026) in Section 1, Related works, subsection “Local existence and continuation criteria for general data”

Whether the results presented below persist when $\Omega$ is a bounded domain is an interesting open question.

— Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation  (2608.31071 - Golding et al., 31 Aug 2026) in Section 1, subsection “The Landau--Fermi--Dirac Equation”

For very soft potentials, including the Coulomb Landau equation, the corresponding global nonlinear stability problem near vacuum remains, to our knowledge, an important open problem.

— Global forward self-similar solutions of the spatially inhomogeneous Boltzmann equation for cutoff Maxwell molecules  (2609.28285 - Nguyen et al., 23 Sep 2026) in Section 1, paragraph beginning “For very soft potentials”