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- W2484001570 abstract "7.1. Sketch of the ProblemFor many nonlinear problems, one uses the following steps. Firstly, a convenient approximation is chosen. This approximation scheme should satisfy the dual criteria that the modified problem is “easily” solvable, and that it retain the expected a priori bounds. Then one uses compactness to pass to the limit (in the sense of distributions) in the modified problem. What follows appeared originally in DiPerna–Lions [3]; we also use the approach in Kruse [14].Let compatible data ƒ0≥0 , E0 , B0 be prescribed at t=0 . Let δϵ be the standard Friedrichs mollifier. We will consider two approximation schemes. Firstly, consider the modified system(MVM)ƒt+v⋅∇xƒ+(E+v×B)⋅∇vƒ=0,(7.1)∂tE=c∇×B−jϵ∇⋅E=ρ∂tB=−c∇×E∇⋅B=0.Below the function ν(x) represents a given neutralizing background density, and ρ=4π∫ƒdv−ν(x) ; j=4π∫vƒdv ; jϵ=δϵ*j . By the work of Horst [11], an initial–value problem for this system possesses global smooth solutions for fixed ϵ>0 . Let ( ƒn , En , Bn ) be such an approximate solution corresponding to δn . (We abuse notation with the mollifier)." @default.
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- W2484001570 date "1996-01-01" @default.
- W2484001570 modified "2023-10-16" @default.
- W2484001570 title "7. Velocity Averages: Weak Solutions to the Vlasov–Maxwell System" @default.
- W2484001570 doi "https://doi.org/10.1137/1.9781611971477.ch7" @default.
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