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

TL;DR

This paper investigates the convergence of non-local convection-diffusion equations on hyperbolic space to local equations as the interaction kernels become concentrated, providing new insights into non-local to local transition on curved manifolds.

Contribution

It establishes the convergence of non-local models to local PDEs on hyperbolic space and constructs a broad class of kernels satisfying the necessary conditions.

Findings

Non-local models converge to local equations under kernel concentration.

Constructed a large class of admissible interaction kernels.

Developed a compactness tool on manifolds inspired by Bourgain-Brezis-Mironescu.

Abstract

We study a non-local evolution equation on the hyperbolic space $\mathbb{H}^N$. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We 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, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions. We also consider a non-local, non-linear convection-diffusion equation on $\mathbb{H}^N$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu.