2021/04/12 by Sanchit Chaturvedi, Chaturvedi, Sanchit, Jonathan Luk +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2104.05692
openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the Vlasov-Poisson-Landau system with Coulomb potential in the weakly collisional regime on a 3-torus, i.e. \beginaligned ∂t F(t,x,v) + vi ∂xi F(t,x,v) + Ei(t,x) ∂vi F(t,x,v) = νQ(F,F)(t,x,v),
E(t,x) = ∇ Δ-1 (∫\mathbb R3 F(t,x,v) d v - (1)/((2π)3)∫\mathbb T3 ∫\mathbb R3 F(t,x,v) d v d x), \endaligned with ν≪ 1. We prove that for ε>0 sufficiently small (but independent of ν), initial data which are O(εν1/3)-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as t→ ∞. The solutions exhibit uniform-in-ν Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov-Poisson-Fokker-Planck equation with the same threshold. However, unlike in the Fokker-Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo's weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.