Three new results on continuation criteria for the 3D relativistic Vlasov–Maxwell system

Abstract

In this paper, we consider sufficient conditions, called continuation criteria, for global existence and uniqueness of classical solutions to the three-dimensional relativistic Vlasov-Maxwell system. In the compact momentum support setting, we prove that p 18 5r −1+β 0 1 2 .

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

“…Although global existence in the whole Minkowski spacetime for classical solutions to the VM system with large data still remains an open problem, several continuation critera are known (see for instance the recent results of [13], [11] and [14]). For the weak solutions, the problem was solved in [5] and revisited in [15].…”

“…It then remains to add the previous two inequalities and to notice, using (14), that 4T [G] 00 = |α| 2 + |α| 2 + 2|ρ| 2 + 2|σ| 2 and 2T [G] L0 = |α| 2 + |ρ| 2 + |σ| 2 .…”

Higher regularity of the “tangential” fields in the relativistic Vlasov–Maxwell system

“…There has been quite some activity concerning the relativistic Vlasov-Maxwell over the years, but nonetheless the question whether (for instance smooth) initial data will yield a global in time solution still remains open. See [1] and [7] for a general introduction and overview, [4] for a summary of results up to approximately 2015 and [5] for some newer and further refined criteria concerning unrestricted global existence.…”

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

“…Here we introduce a small number of results from the literature on the Vlasov-Maxwell system by Degond [9], DiPerna-Lions [11], Glassey-Strauss [25,27], Glassey-Schaeffer [20][21][22][23][24] , Horst [35], Guo [30][31][32][33], Rein [46], Bouchut-Golse-Pallard [2], Klainerman-Staffilani [4], and Strain [47]. Recent results include the work on the Strichartz estimates [39] and continuation criteria [27,37,38,43]. Regarding the rigorous derivation of the Vlasov equations, see the work of Dobrushin [12].…”

“…Here, the initial parameter δ 0 is the same constant as the one used in the definition of I 0 at (8), C is defined as (43), and K is defined as (16).…”