Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space (Q6611113)

The authors study a nonlocal evolution equation on the hyperbolic space \(H^N\). A model for particle transport governed by a nonlocal interaction kernel defined on the tangent bundle and invariant under the geodesic flow is presented. The authors study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, it is proved that the solution of the rescaled nonlocal problem converges to that of the local transport equation. Then, a large class of interaction kernels that satisfy those conditions is constructed.\N\NAlso a nonlocal and nonlinear convection-diffusion equation on \(H^N\) governed by two kernels, one for each of the diffusion and convection parts, is considered. In this case, the authors are able to prove that the solution converges to the solution of a local problem as the kernels get concentrated. The fundamental tool used in this paper is a a compactness result on manifolds inspired by the work of \textit{J. Bourgain} et al. [in: Optimal control and partial differential equations. In honour of Professor Alain Bensoussan's 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha. 439--455 (2001; Zbl 1103.46310)].