Theoretical analysis demonstrates spectral gap existence and irreducibility in collisional transport equations with general vector fields, suggesting robust long-time asymptotic stability.
Motivated by kinetic theory, this paper investigates the spectral theory and long-time asymptotics of semigroups ( W(t)) t≥ 0 associated with integro-differential equations ∂ f/∂ t +a.∂ f/∂ x+b.∂ f/∂ v+σ f = Kf\ in Lᵖ(Ω × Rᵈ) (1≤ p<+∞) under non incoming boundary condition. We consider a broad classes of vector fields ( a, b) satisfying appropriate transversality conditions, together with a general class of scattering operators K. The analysis focuses on the existence of a spectral gap for ( W(t)) t≥ 0 and on its irreducibility. A systematic construction is provided, and several open questions are highlighted.
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Mustapha Mokhtar‐Kharroubi (2026) studied this question.
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