Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space
type: Item
instance of: scholarly article
title: Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space
zbMATH Open document ID: 1548.45004
author: Maria del Mar Gonzalez
author: Liviu I. Ignat
author: Dragoş Manea
author: Sergiu Moroianu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2024-09-26T00:00:00Z
review text: 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)].
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45L05
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 7919109
zbMATH Keywords: nonlocal convection-diffusion
zbMATH Keywords: dissipative kernels
zbMATH Keywords: hyperbolic space
zbMATH Keywords: convergence of nonlocal equations to local equations
zbMATH Keywords: Riemannian geodesic flow
MaRDI profile type: Publication
cites work: Q3162662
cites work: Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces
cites work: Nonlinear Schrödinger equation on real hyperbolic spaces
cites work: The wave equation on hyperbolic spaces
cites work: Fourier Analysis and Nonlinear Partial Differential Equations
cites work: A nonlocal diffusion problem on manifolds
cites work: The Fujita exponent for the Cauchy problem in the hyperbolic space
cites work: The Nonlinear Schrödinger Equation on Hyperbolic Space
cites work: Some constructions for the fractional Laplacian on noncompact manifolds
cites work: Variational Methods for Nonlocal Fractional Problems
cites work: Handbook of Peridynamic Modeling
cites work: Q3159214
cites work: Q4307544
cites work: Applied functional analysis. Functional analysis, Sobolev spaces and elliptic differential equations
cites work: Nonlocal diffusion and applications
cites work: Q4217277
cites work: Q5472808
cites work: Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups
cites work: Nonlocal Modeling, Analysis, and Computation
cites work: Nonlocal convection-diffusion volume-constrained problems and jump processes
cites work: A new approach for a nonlocal, nonlinear conservation law
cites work: Sobolev-Räume, Einbettungssätze und Normungleichungen auf offenen Mannigfaltigkeiten
cites work: Q5189317
cites work: The Heat Kernel on Hyperbolic Space
cites work: Q3331943
cites work: Long-time behavior for a nonlocal convection diffusion equation
cites work: A Compactness Tool for the Analysis of Nonlocal Evolution Equations
cites work: A nonlocal convection-diffusion equation
cites work: Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length
cites work: Fractional Sobolev norms and BV functions on manifolds
cites work: The incompressible Navier-Stokes equations on non-compact manifolds
cites work: Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation
cites work: Q5182015
cites work: A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization
cites work: Q3995594
DOI: 10.1016/J.NA.2024.113618
description: scientific article; zbMATH DE number 7919109