The Landau Equation in a Periodic Box

Gevrey regularity for solution of the spatially homogeneous landau equation

“…Recently, a lot of progress has been made on the study of the Sobolev regularizing property, cf. [6,11,13,18,19] and references therein, which shows that in some sense the Landau equation can be regarded as a non-linear and non-local analog of the hypo-elliptic Fokker-Planck equation. That means the weak solution, which constructed under rather weak hypothesis on the initial datum, will become smooth or, even more, rapidly decreasing in v at infinity.…”

Asymptotic behaviour of the Landau equation with Coulomb potential

“…On the other hand, in a perturbative regime, Guo [19] proved well-posedness in the high-order Sobolev space with fast decay in velocity…”

Hypoelliptic estimates for some linear diffusive kinetic equations

“…Details about the full Landau equation may be found for example in the works by Y. Guo [10], C. Mouhot and L. Neumann [20], or C. Villani [25], and we may only recall here that the Landau equation reads as the evolution equation of the density of particles…”

“…Following the standard procedure described in [10] or [20], we linearize the Landau equation around M by posing…”