Uniqueness of the Compactly Supported Weak Solutions of the Relativistic Vlasov-Darwin System
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Uniqueness of the Compactly Supported Weak Solutions of the Relativistic Vlasov-Darwin System Reinel Sospedra-Alfonso · Martial Agueh
Received: 7 September 2011 / Accepted: 13 August 2012 / Published online: 29 August 2012 © Springer Science+Business Media B.V. 2012
Abstract We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system. Our proof extends the method used by Loeper in (J. Math. Pures Appl., 86:68–79, 2006) to obtain uniqueness results for the Vlasov-Poisson system. Keywords Vlasov-Darwin system · Weak solutions · Uniqueness · Optimal transportation Mathematics Subject Classification 82B40 · 82C40 · 45G10
1 Introduction The relativistic Vlasov-Darwin (RVD) system describes the evolution of a collisionless plasma whose particles interact only through the electromagnetic field they induce. In contrast to the Vlasov-Maxwell system, the particle interaction is assumed to be a low-order relativistic correction (i.e., the Darwin approximation) of the full Maxwell case. In terms of the electromagnetic field, the RVD system can be introduced as follows. Consider an ensemble of identical charged particles with mass m and charge q, which we set to one for simplicity. Let f (t, x, ξ ) denote the density of the particles in the phase space R3x × R3ξ at a time t ∈ ]0, ∞[, where x denotes position and ξ momentum. If collisions are neglected, the time evolution of f is given by the Vlasov equation (1) ∂t f + v · ∇x f + E + c−1 v × B · ∇ξ f = 0,
R. Sospedra-Alfonso Institute of Applied Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver V6T 1Z2, BC, Canada e-mail: [email protected] M. Agueh () Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria V8W 3R4, BC, Canada e-mail: [email protected]
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R. Sospedra-Alfonso, M. Agueh
where v = √
ξ
is the relativistic velocity and c is the speed of light. The electric
1+c−2 |ξ |2
field E = E(t, x), and magnetic field B = B(t, x), are induced by the particles and satisfy Maxwell’s equations ∇ × B − c−1 ∂t E = 4πc−1 j, ∇ × E + c−1 ∂t B = 0, where
(2)
∇ · E = 4πρ,
ρ(t, x) =
∇ · B = 0,
(3)
and
f (t, x, ξ )dξ R3
j (t, x) =
v(ξ )f (t, x, ξ )dξ
(4)
R3
are respectively the charge and current densities. The coupled nonlinear system (1)–(4) is known as the relativistic Vlasov-Maxwell (RVM) system, which is essential in the study of dilute hot plasmas; see [7] and the references therein. The electric field E can be further decomposed into a longitudinal component EL , and a transverse component ET , as E = EL + ET ,
∇ × EL = 0,
∇ · ET = 0.
(5)
The Darwin approximation consists of neglecting the transverse part of the displacement current, ∂t ET , in Maxwell-Ampère’s law (2). Maxwell’s equations (2)–(3) then reduce to ∇ × B − c−1 ∂t EL = 4πc−1 j, ∇ × ET + c−1 ∂t B = 0,
∇ · B = 0,
(6)
∇ · EL = 4πρ.
(7)
The RVD system is defined as the Vlasov equation (1) coupled with (6)–(7) via the charge and cu
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