The spatially inhomogeneous Vlasov-Nordstr\"{o}m-Fokker-Planck system in the intrinsic weak diffusion regime

TL;DR

This paper proves the global existence, asymptotic behavior, and dynamic stability of solutions to the spatially inhomogeneous Vlasov-Nordström-Fokker-Planck system in the weak diffusion regime, addressing challenges from relativistic velocity singularities.

Contribution

It establishes the first rigorous results on global solutions and stability for the inhomogeneous relativistic Vlasov-Nordström-Fokker-Planck system without friction.

Findings

Existence of unique global classical solutions.

Characterization of asymptotic behavior using weighted energy methods.

Verification of dynamic stability in self-similar solution framework.

Abstract

The spatially homogeneous Vlasov-Nordstr\"{o}m-Fokker-Planck system is known to exhibit nontrivial large time behavior, naturally leading to weak diffusion of the Fokker-Planck operator. This weak diffusion, combined with the singularity of relativistic velocity, present a significant challenge in analysis for the spatially inhomogeneous counterpart. In this paper, we demonstrate that the Cauchy problem for the spatially inhomogeneous Vlasov-Nordstr\"{o}m-Fokker-Planck system, without friction, maintains dynamically stable relative to the corresponding spatially homogeneous system. Our results are twofold: (1) we establish the existence of a unique global classical solution and characterize the asymptotic behavior of the spatially inhomogeneous system using a refined weighted energy method; (2) we directly verify the dynamic stability of the spatially inhomogeneous system in the framework of self-similar solutions.