On global solutions to the Vlasov-Poisson system with radiation damping

Abstract

In this paper, the dynamics of three dimensional Vlasov-Poisson system with radiation damping is investigated. We prove global existence of a classical as well as weak solution that propagates boundedness of velocityspace support or velocity-space moment of order two respectively. This kind of solutions possess finite mass but need not necessarily have finite kinetic energy. Moreover, uniqueness of the classical solution is also shown.

Estimate of largest velocities for Vlasov–Poisson plasma with radiation damping

“…Well‐posedness and asymptotic behavior for the system () has been well understood(see previous studies 2–5 ). Specifically, for $$$ {f}_0^{\pm}\in {C}_c^{\infty}\left({\mathbb{R}}^6\right) $$$, Kunze and Rendall proved the global existence and uniqueness of classical solutions and obtained the large time asymptotic behavior of the charge density $$$ \rho \left(t,x\right) $$$, the electrostatic field $$$ E\left(t,x\right) $$$ and the damping term $$$ {D}^{\left[2\right]}(t) $$$ 2 .…”

“…Well-posedness and asymptotic behavior for the system (1.1) has been well understood(see previous studies [2][3][4][5] ). Specifically, for 𝑓 ± 0 ∈ C ∞ c (R 6 ), Kunze and Rendall proved the global existence and uniqueness of classical solutions and obtained the large time asymptotic behavior of the charge density 𝜌(t, x), the electrostatic field E(t, x) and the damping term D [2] (t).…”

“…2 When 𝑓 ± 0 ∈ L p (R 6 )(p > 3) with (1 + |v| 2 )𝑓 ± 0 ∈ L 1 (R 6 ), Chen et al 3 proved the global existence of a weak solution and also obtained the large time asymptotic behaviors of its macroscopic quantities. When 𝑓 ± 0 ∈ C 1 0 ⋂ L 1 (R 6 ) with bounded velocity-spatial support, Xiao and Zhang 4 proved global existence and uniqueness of a classical solution with possibly infinite energy. And the propagation for velocity and velocity-spatial moments of order greater than 2 has been studied in Xiao and Zhang.…”

“…Below, we first recall some notations and known results. 2,4 The kinetic energy 𝜀 kin (t) and potential energy 𝜀 pot (t) are, respectively, defined by…”

“…When $$$ {f}_0^{\pm}\in {L}^p\left({\mathbb{R}}^6\right)\left(p>3\right) $$$ with $$$ \left(1+{\left|v\right|}^2\right){f}_0^{\pm}\in {L}^1\left({\mathbb{R}}^6\right) $$$, Chen et al 3 proved the global existence of a weak solution and also obtained the large time asymptotic behaviors of its macroscopic quantities. When $$$ {f}_0^{\pm}\in {C}_0^1\bigcap {L}^1\left({\mathbb{R}}^6\right) $$$ with bounded velocity‐spatial support, Xiao and Zhang 4 proved global existence and uniqueness of a classical solution with possibly infinite energy. And the propagation for velocity and velocity‐spatial moments of order greater than 2 has been studied in Xiao and Zhang 5 …”