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

Abstract

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

“…It can be shown, as for the Vlasov-Maxwell system [9], that this structure is conserved under commutation with complete lifts. What is crucial in a subsequent step is to assure that this null structure can be exploited at all levels of regularity, which is not straightforward to validate.…”

“…The vector field method for transport equations has in the meantime been applied successfully to the Vlasov-Nordström system [16] and the massive Einstein-Vlasov system in [17]. In a series of papers, [6][7][8][9], the method has also been extended to the Vlasov-Maxwell system in various contexts, in particular, without the need of any compact support assumptions.…”

“…The commutation formula obtained here is more geometric than the one used in [17]. In the spirit of [9] for the Vlasov-Maxwell system (see in particular Subsection 2.5), we express the error terms using Lie derivatives of the metric instead of derivatives of its Cartesian components. We recall the notations…”

“…The main part of this section then consists in deriving such estimates for |K | N − 3. For this, we follow an improvement of the strategy used in [18] (see Subsection 4.5.7), which was used in [9,Section 7] in the context of the Vlasov-Maxwell system. Contrary to the method of [18], this improvement will allow us to exploit all the null structures of the system.…”

“…Remark 14.4. Contrary to [18], we kept, as in [9], the v-derivatives in the statement of Lemma 14.2 in order to take advantage of the good behavior of radial component of ∇ v F. If we had already transformed the v-derivatives, we would be left with terms such as x j r (t − r )∂ x j F from (∇ v F) r (see Lemma 3.9). We would then have to deal with factors such as t 3 |x| 3 during the treatment of the homogeneous part F hom (apply three boost to x k |x| ) since we will have to commute at least three times the equation T g (F hom ) + A • F hom = 0.…”