Averaged Motion of Charged Particles under Their Self-Induced Electric Field

• Published in: Indiana University mathematics journal Vol. 43; no. 4; pp. 1167 - 1225
• Main Authors: Friedman, Avner, Huang, Chaocheng
• Format: Journal Article
• Language: English
• Published: Department of Mathematics INDIANA UNIVERSITY 01-12-1994
• Subjects: A priori knowledge; Boundary conditions; Charged particles; Electric fields; Jacobians; Mathematical constants; Mathematical functions; Mathematical inequalities; Tangents; Uniqueness
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.1994.43.43052

In this paper we consider the averaged equations for a large number of small balls of uniform mass and charge moving under the force of their self-electric field. These equations are Δφ(x,t) = −P(x,t), d2ψ/dt2 = −∇φ(ψ,t) subject to ψ(x,0) = x, ψt(x,0) = ψ1(x) where φ is the electric potential and P is the limit concentration of the small balls as their number increases to infinity (and their radius goes to zero). The evolution of P is given by P(x,t) = P0(ψ−1(x,t))J(ψ−1)(x,t) where P0 is the initial concentration, ψ−1 is the inverse of the mapping x → ψ(x,t) and J(ψ−1) is its Jacobian. It is proved that if the initial data are in C1+α then there exists a unique local solution with ψ in C1+α. The solution can be extended globally in time as long as ψ and ψ−1 remain uniformly bounded in C1. There are however smooth initial data for which a global solution does not exist. One of the main results of the paper is that if |P0 − χB1| and |ψ1(x) − b0x| are small enough, where χB1 is the characteristic function of the unit ball and b0 is a positive real number, then there exists a unique global solution.