The Landau Equation for Maxwellian Molecules and the Brownian Motion on Son (r)

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition. 1. Statement of the problem and existing results The spatially homogeneous Landau equation for Maxwellian molecules is a common model in plasma physics. It can be obtained as a certain limit of the spatially homogeneous Boltzmann equation for N dimensional particles subject to pairwise interaction, when the collisions become grazing and when the interaction forces between particles at distance r are order 1/r2N+1 (see Villani [24] and Guérin [15]). The Landau equation reads as a nonlocal Fokker-Planck equation. Given an initial condition (f(0, v), v ∈ RN ), the solution is denoted by (f(t, v), t ≥ 0, v ∈ RN ), N ≥ 2, and satisfies ∂tf(t, v) = Lf(t, v), (1.1) where Lf(t, v) = ∇ · ∫ RN dv∗ a(v − v∗) (f(t, v∗)∇f(t, v)− f(t, v)∇f(t, v∗)) . (1.2) Here, a is an N ×N nonnegative and symmetric matrix that depends on the collisions between binary particles. It is given by (up to a multiplicative constant) a(v) = |v|IdN − v ⊗ v, where IdN denotes the identity matrix of size N , and v ⊗ v = vv>, v> denoting the transpose of v, v being seen as a column vector in RN . The unknown function f(t, v) represents the density of particles of velocity v ∈ RN at time t ≥ 0 in a gas. It is assumed to be independent of the position of the particles (spatially homogeneous case). The density f(t, v) being given, the nonlocal operator L can be seen as a standard linear Fokker-Planck operator, with diffusion matrix a(t, v) = ∫ RN a(v − v∗)f(t, v∗)dv∗ Date: June 2014. 1991 Mathematics Subject Classification. 60H30, 60H40, 60H10.