A Notion of Rectifiability Modeled on Carnot Groups

Avner Friedman, Chaocheng Huang, Averaged Motion of Charged Particles under Their Self-Induced Electric Field, Indiana Univ. Math. J. 43 (1994), 1167-1225

Abstract

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 ${\Delta} \varphi (x,t) = - P(x,t),\, d^2 \psi/dt^2 = - {\nabla} \varphi (\psi ,t)$ subject to $\psi (x,0) = x$ , $\psi _t (x,0) = \psi _1 (x)$ where $\varphi$ 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 $P(x,t) = P_0 ( {\psi ^{ - 1} (x,t)} )J( {\psi ^{ - 1} } )(x,t)$ where $P_0$ is the initial concentration, ${\psi ^{ - 1} }$ is the inverse of the mapping $x \to \psi (x,t)$ and $J(\psi ^{ - 1} )$ is its Jacobian. It is proved that if the initial data are in $C^{1 + \alpha }$ then there exists a unique local solution with $\psi$ in $C^{1 + \alpha }$ . The solution can be extended globally in time as long as $\psi$ and ${\psi ^{ - 1} }$ remain uniformly bounded in $C^1$ . 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 $|P_0 - \chi _{B_1 } |$ and $|\psi _1 (x) - b_0 x|$ are small enough, where $\chi _{B_1 }$ is the characteristic function of the unit ball and $b_0$ is a positive real number, then there exists a unique global solution.

References

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