2022/07/05 by Bertrand Lods, Lods, Bertrand, Mustapha Mokhtar‐Kharroubi +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2207.01897
openalex publication_date 2022/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C0-semigroups (V(t))t ≥0 in L1(\mathbbTd× ℝd) governing conservative linear kinetic equations on the torus with general scattering kernel k(v,v') and degenerate (i.e. not bounded away from zero) collision frequency σ(v)=∫ℝd k(v',v)m(d v'), (with m(d v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N0 is the maximal integer s ≥0 such that (1)/(σ(⋅))∫ℝdk(⋅,v)σ-s(v)m(d v) ∈ L∞(ℝd) then, for initial datum f such that d s∫_\mathbbTd× ℝd|f(x,v)|σ^-N0(v)d x m(d v) <∞ it holds ‖V(t)f-\varrhofΨ‖L1=\dfracεf(t)(1+t)^N0-1, \varrhof:= ∫ℝdf(x,v)m(d v) where Ψ is the unique invariant density of (V(t))t ≥0 and limt→∞εf(t)=0. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of V(t) and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp ``subgeometric'' convergence rate for Markov semigroups associated to general transition kernels.