Asymptotic Properties of the Solutions to the Vlasov–Maxwell System in the Exterior of a Light Cone

Journal of Functional Analysis

2020

Self Cite

In this article, we present a vector field method for the study of solutions to massless relativistic transport equations. Compared to [10], we remove the Lorentz boosts of the commutation vector fields, which makes our method more suitable for the study of systems coupling a massless Vlasov equation with an equation which has a different speed of propagation. We also believe that our approach can be adapted more easily to the context of curved backgrounds.In a second time, we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless Vlasov-Poisson system. No compact support assumption is required on the data and, in particular, the initial decay in the velocity variable is optimal. In order to compensate the slow decay rate of the particle density near the light cone, we take advantage of the null structure of the equations.

Section: Main Difficultiesmentioning

confidence: 89%

“…In [3] or [5], we only used the inequality v 0 − x i r v i v 0 t+r z. These two lemmas directly implies that…”

Section: Structure Of the Papermentioning

confidence: 99%

Section: Decay Estimates For Velocity Averagesmentioning

confidence: 99%

“…• if the initial data do not vanish for small velocities, the massless relativistic Vlasov-Poisson could fail to admit a local C 1 solution (see Section 8 of [3]).…”

Section: The Problem Caused By the Small Velocitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A vector field method for massless relativistic transport equations and applications

Journal of Functional Analysis

2020

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Sharp Asymptotics for the Solutions of the Three-Dimensional Massless Vlasov–Maxwell System with Small Data

2020

Ann. Henri Poincaré

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This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially optimal. Consistently with Proposition 8.1 of Bigorgne (Asymptotic properties of small data solutions of the Vlasov–Maxwell system in high dimensions. arXiv:1712.09698, 2017), the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.

Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter

Arch Rational Mech Anal

Fajman

Joudioux

et al. 2021

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We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein–Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.