Energy Conservation of the DiPerna–Lions Solutions to Vlasov–Maxwell Systems.

In this paper, we provide sufficient conditions for the energy conservation of the DiPerna–Lions solutions given in DiPerna and Lions (Commun Pure Appl Math 42:729–757, 1989), Rein (Commun Math Sci 2:145–158, 2004) to Vlasov–Maxwell systems in R 3 , which requires only macroscopic density ρ ∈ L t 2 L loc 2 for the relativistic case and | ξ | f ∈ L t , x 2 L ξ 1 for the non-relativistic case, improving the results in Sospedra–Alfonso (Commun Math Sci 8:901–908, 2010), Bardos et al. (Q Appl Math 78:193–217, 2020). Next, by mollification in time twice with different parameters and the weak formulations containing t = 0 , we obtain the same results for solutions given in Guo (Commun Math Phys 154:245–263, 1993) with low temporal regularity to the Vlasov–Maxwell systems in bounded domains under some boundary-layer conditions. [ABSTRACT FROM AUTHOR]