Decay estimates for the $3D$ relativistic and non-relativistic Vlasov-Poisson systems

TL;DR

This paper establishes global regularity and decay estimates for the 3D relativistic and non-relativistic Vlasov-Poisson systems using a Fourier method, demonstrating scattering to linear solutions and providing a basis for studying more complex systems.

Contribution

Introduces a Fourier-based approach for proving global regularity and decay in 3D Vlasov-Poisson systems, applicable to both relativistic and non-relativistic cases, and offers a comparison framework for more complex models.

Findings

Proved global regularity for both relativistic and non-relativistic cases.

Established sharp decay estimates and scattering behavior.

Provided a systematic Fourier method for future complex system analysis.

Abstract

We study the small data global regularity problem of the $3D$ Vlasov-Poisson system for both the relativistic case and the non-relativistic case. The main goal of this paper is twofold. (i) Based on a Fourier method, which works systematically for both the relativistic case and the non-relativistic case, we give a short proof for the global regularity and the sharp decay estimate for the $3D$ Vlasov-Poisson system. Moreover, we show that the nonlinear solution scatters to a linear solution in both cases. The result of sharp decay estimates for the non-relativistic case is not new, see Hwang-Rendall-Vel\'azquez and Smulevici. (ii) The Fourier method presented in this paper serves as a good comparison for the study of more complicated $3D$ relativistic Vlasov-Nordstr\"om system and $3D$ relativistic Vlasov-Maxwell system.