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Keywords = Vlasov–Fokker–Planck equation with local alignment forces

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22 pages, 325 KB  
Article
Global Solutions to the Vlasov–Fokker–Planck Equation with Local Alignment Forces Under Specular Reflection Boundary Condition
by Yanming Chang and Yingzhe Fan
Axioms 2025, 14(10), 760; https://doi.org/10.3390/axioms14100760 (registering DOI) - 11 Oct 2025
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Abstract
In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space Lk1LTLv2. A key difficulty is [...] Read more.
In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space Lk1LTLv2. A key difficulty is that the macroscopic averaged velocity u does not directly possess a dissipative structure in the equation. To overcome this, we rely on the dissipation ub from the linear part, combined with the dissipation of the macroscopic component b derived from the associated macroscopic equation. Moreover, since no direct energy functional is available for u, we fully exploit the dissipative mechanisms of both ub and b when handling the estimates for the nonlinear terms. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Related Topics)
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