The Landau equation: Well-posedness and singularities of homoenergetic solutions

We consider the inhomogeneous Landau equation and focus on solutions whose density and temperature evolve in time but remain spatially homogeneous. Such solutions are called homoenergetic. We identify one class of initial data for which homoenergetic solutions exist globally in time, and another class for which they develop finite-time hydrodynamic singularities. The initial data are spatially inhomogeneous, neither close to vacuum nor close to equilibrium; both the well-posedness and the finite-time blow-up results hold for all soft and hard potentials.