# Global Solution of the Electromagnetic Field-Particle System of Equations

### J. Math. Phys. 55 10152

The full published version is available (under subscription) at AIPScitation.

### arXiv 1311.1675

The preprint in pdf is available at arXiv.org.

## Abstract

In this paper we discuss global existence of the solution of the Maxwell and Newton system of equations, describing the interaction of a rigid charge distribution with the electromagnetic field it generates. A unique solution is proved to exist (for regular charge distributions) on suitable homogeneous and non-homogeneous Sobolev spaces, for the electromagnetic field, and on coordinate and velocity space for the charge; provided initial data belong to the subspace that satisfies the divergence part of Maxwell's equations.

## Introduction

[The introduction is copied here for rapid consultation]

We are interested in the following system of equations: let $$e\in\mathbb{R}$$, $$\varphi$$ a sufficiently regular function; then the Maxwell-Newton equations are written in three spatial dimensions as

\begin{equation} \label{eq:1} %\left\{ \begin{aligned} &\left\{\begin{aligned} \partial_t &B + \nabla\times E=0\\ \partial_t &E - \nabla\times B=-j \end{aligned}\right. \mspace{20mu} \left\{\begin{aligned} \nabla\cdot &E=\rho\\ \nabla\cdot &B=0 \end{aligned}\right.\\ &\left\{\begin{aligned} \dot{\xi}&= v\\ \dot{v}&= e[(\varphi*E)(\xi)+v\times(\varphi*B)(\xi)] \end{aligned}\right. \end{aligned} %\right. \qquad ; \end{equation}

with

\begin{equation} \label{eq:13} j=ev \varphi (\xi-x)\; ,\quad \rho=e \varphi (\xi-x)\; . \end{equation}

This system can be used to describe motion of a non-relativistic rigid particle, with an extended charge distribution $$e\varphi$$, interacting with its own electromagnetic field (in this case we could need some additional physical conditions, such as $$\int d x \varphi=1$$, however these conditions are not necessary for the existence of a solution). So $$\xi,v\in\mathbb{R}^3$$ will be the position and velocity of the charge's center of mass, $$E,B$$ the electric and magnetic field vectors. We remark that in \eqref{eq:1} charge is conserved, i.e.

\begin{equation} \label{eq:2} \partial_t\rho+\nabla\cdot j=0\; . \end{equation}

It is useful to construct the electromagnetic tensor $$F^{\mu\nu}$$:

\begin{equation*} F^{\mu\nu}=\left( \begin{array}{cccc} 0&E_1&E_2&E_3\\ -E_1&0&B_3&-B_2\\ -E_2&-B_3&0&B_1\\ -E_3&B_2&-B_1&0 \end{array}\right)\; . \end{equation*}

Therefore we make the following identifications:

\begin{gather*} E_j=\sum_{j=1}^3\delta_{ij}F^{0j}\; ,\\ B_j=\frac{1}{2}\sum_{k,l=1}^3\epsilon_{jkl}F^{kl}\; ; \end{gather*}

where $$\delta_{ij}$$ is the Kronecker's delta and $$\epsilon_{ijk}$$ is the three-dimensional Levi-Civita symbol. From now on, we adopt the following notation: whenever an index is repeated twice, a summation over all possible values of such index is intended. Define $$R^j_{kl}=-\delta_l^j\partial_k+\delta_k^j\partial_l$$ and $$(R^*)^{kl}_j=\delta_j^l\partial^k-\delta^k_j\partial^l$$, and let $$\Omega=(RR^*)^{1/2}$$, then

\begin{equation} \label{eq:4} U(t)\equiv\left( \begin{array}{cc} \;\cos\Omega t\;&\;\frac{\sin\Omega t}{\Omega}R\;\\ \; -R^*\frac{\sin\Omega t}{\Omega}\; &\; \cos(R^*R)^{1/2}t\; \end{array} \right)\; ; \end{equation}

also, define

\begin{equation} \label{eq:5} W(t)\equiv 1+t\left(\begin{array}{cc} \;0\;&\;1\;\\\;0\;&\;0\; \end{array}\right )\; . \end{equation}

For the construction of $$U(t)$$ we have followed [GinVel 1981]. Then \eqref{eq:1} can be rewritten as an integral equation: set

\begin{equation} \label{eq:3} \vec{u}(t)= \begin{pmatrix} F_0(t)\\ F(t) \\ \xi(t)\\ v(t) \end{pmatrix}\; , \end{equation}

and let $$\vec{u}(t_0)=\vec{u}_{(0)}$$ (we define the electric part as $$F_0=F^{0j}$$, the magnetic part as $$F=(F^{jk})_{j< k}$$ ); also, let

\begin{gather*} j(t)=ev(t)\varphi(\xi(t)-x)\; ,\\ (f_{em}(t))_i=e\sum_{j=1}^3\delta_{ij}\bigl[\bigl(\varphi*F^{0j}(t)\bigr)(\xi(t))+\sum_{k=1}^3 v_k(t)\bigl(\varphi*F^{jk}(t)\bigr)(\xi(t))\bigr]\; . \end{gather*}

Then we can write the integral equation:

\begin{equation} \label{eq:6} \vec{u}(t)=\left( \begin{array}{cc} U(t-t_0)&0\\0&W(t-t_0) \end{array} \right)\vec{u}_{(0)}+\int_{t_0}^td\tau\left( \begin{array}{cc} U(t-\tau)&0\\0& W(t-\tau) \end{array} \right) \left( \begin{array}{c} -j(\tau)\\0\\0\\f_{em}(\tau) \end{array} \right)\; . \end{equation}

The second couple of Maxwell's equations (namely $$\nabla\cdot E=\rho$$ and $$\nabla\cdot B=0$$) have to be dealt with separately.

We are interested in solutions belonging to the following spaces: let $$\mathscr{X}_{s}$$, for $$-\infty < s < 3/2$$, to be

\begin{equation*} \mathscr{X}_{s}\equiv \bigl(\dot{H}^s(\mathbb{R}^3) \bigr)^3\oplus \bigl(\dot{H}^s(\mathbb{R}^3) \bigr)^3\oplus \mathbb{R}^3\oplus\mathbb{R}^3\; . \end{equation*}

If $$\vec{u}\in\mathscr{X}_{s}$$ has the form \eqref{eq:3}, then $$\mathscr{X}_{s}$$ is a Hilbert space if equipped with the norm

\begin{equation*} \lVert\vec{u}\rVert_{\mathscr{X}_{s}}^2=\lVert F_0\rVert^2_{(\dot{H}^s)^3}+\lVert F\rVert^2_{(\dot{H}^s)^3}+\lvert \xi \rvert^2+\lvert v \rvert^2\; ; \end{equation*}

where

\begin{equation*} \lVert f \rVert^2_{(\dot{H}^s)^3}=\sum_{j=1}^3\lVert \omega^s f_j\rVert^2_{L^2}=\sum_{j=1}^3\int_{\mathbb{R}^3}d{k}\,\lvert k \rvert^{2s}\lvert\hat{f_j}(k)\rvert^2\; , \end{equation*}

with

\begin{equation*} \omega=\lvert \nabla \rvert\; . \end{equation*}

The homogeneous Sobolev spaces $$\dot{H}^s(\mathbb{R}^d)$$ are Hilbert spaces for all $$s < d/2$$ [see BahCheDan 2011]. We will use as well the non-homogeneous Sobolev spaces $$H^r(\mathbb{R}^d)$$, $$r\geq 0$$, complete with the norm

\begin{equation*} \lVert f \rVert^2_{H^r}=\int_{\mathbb{R}^d}d{k}\,(1+\lvert k \rvert^2)^r\lvert\hat{f}(k)\rvert^2\; . \end{equation*}

Let $$I\subseteq \mathbb{R}$$; then we define

\begin{equation*} \mathscr{X}_{s}(I)=\mathscr{C}^0(I,\mathscr{X}_{s})\; ; \end{equation*}

complete with the norm

\begin{equation*} \lVert \vec{u}(\cdot) \rVert_{\mathscr{X}_{s}(I)}=\sup_{t\in I}\lVert \vec{u}(t)\rVert_{\mathscr{X}_{s}}\; . \end{equation*}

Our goal will be to prove that a unique global solution of \eqref{eq:1} exists on $$\mathscr{X}_{s}(\mathbb{R})$$ whenever the initial datum belongs to (a subspace of) $$\mathscr{X}_{s}$$ (theorem 1); for all $$s < 3/2$$ and suitably regular $$\varphi$$ (by the result for $$s=0$$ global existence on non-homogenous Sobolev spaces is also proved, theorem 2).

Particles interacting with its electromagnetic field has been widely studied in physics. The study of a radiating point particle revealed the presence of divergencies. Therefore classically a radiating particle is always assumed to have extended charge distribution. The most used equations to describe such particle's (and corresponding fields) motion are \eqref{eq:1} above, or its semi-relativistic counterpart, called Abraham model [Spohn 2004] (also Maxwell-Lorentz equations in [BauDur 2001]), see Equation (8) below. For a detailed discussion of their physical properties, historical background and applications the reader can consult any classical textbook on electromagnetism. On a mathematical standpoint, almost all results deal with the semi-relativistic system, and are quite recent. We mention an early work of Bambusi and Noja  on the linearised problem; papers of Appel and Kiessling [1999-2001] on conservation laws and motion of a rotating extended charge. Concerning global existence of solutions, refer to Komech and Spohn  and Bauer and Duerr ; the latter result has been developed further in [BauDecDur 2013] to consider weighted $$L^2$$ spaces. Imaykin, Komech and Mauser  have investigated soliton-type solutions and asymptotics. For a comprehensive review on the classical and quantum dynamics of particles and their radiation fields the reader may refer to the book by Spohn .

The existence results [BauDecDur 2001-2013] are formulated for the semi-relativistic Maxwell-Lorentz system, but they should apply also to \eqref{eq:1}: existence of a differentiable solution holds on suitable subspaces of $$(L^2_w(\mathbb{R}^3))^3\oplus (L^2_w(\mathbb{R}^3))^3\oplus \mathbb{R}^3\oplus \mathbb{R}^3$$, with $$w$$ denoting an eventual weight on $$L^2$$ spaces; in this paper a continuous global solution is proved to exist on a wider class of spaces: $$\bigl(\dot{H}^s(\mathbb{R}^3) \bigr)^3\oplus \bigl(\dot{H}^s(\mathbb{R}^3) \bigr)^3\oplus \mathbb{R}^3\oplus\mathbb{R}^3$$, $$s<3/2$$ and $$\bigl(H^r(\mathbb{R}^3) \bigr)^3\oplus \bigl(H^r(\mathbb{R}^3) \bigr)^3\oplus \mathbb{R}^3\oplus\mathbb{R}^3$$, $$r\geq 0$$.

From a physical standpoint, it is a natural choice to consider the energy space $$\mathscr{X}_0(\mathbb{R})$$ to solve \eqref{eq:1}. Also, it is not necessary, in principle, to introduce new objects like the electromagnetic potentials $$\phi$$ and $$A$$. Nevertheless, there are plenty of situations where such potentials are either convenient or necessary: a simple example is in defining a Lagrangian or Hamiltonian function of the charge-electromagnetic field system. Once a gauge is fixed, investigating the regularity of the potentials $$\phi$$ and $$A$$ is equivalent to provide a solution to \eqref{eq:1} on a suitable space, often different form $$\mathscr{X}_0(\mathbb{R})$$. In this context, homogeneous Sobolev spaces emerge. For example, consider the vector potential $$A$$ in the Coulomb gauge. Then, given $$F^{ij}$$, we have $$A_{j}=\omega ^{-2}\sum_{i=1}^{3}\partial _i F^{ij}$$. So the requirement $$A\in \bigl(L^2 (\mathbb{R}^3 )\bigr)^3$$ is equivalent to $$B\in \bigl(\dot{H}^{-1}(\mathbb{R}^3)\bigr)^{3}$$. Another natural choice, if one wants to investigate the connection between the quantum and the classical theory, is to have $$A\in \bigl(\dot{H}^{1/2}(\mathbb{R}^3)\bigr)^{3}$$ and $$\partial _t A\in \bigl(\dot{H}^{-1/2}(\mathbb{R}^3)\bigr)^{3}$$ (because, roughly speaking, the classical correspondents of quantum creation/annihilation operators behave as $$\omega ^{1/2}A$$, $$\omega ^{-1/2}\partial _t A$$ and are required to be square integrable). This is equivalent to solve \eqref{eq:1} with $$E\in \bigl(\dot{H}^{-1/2}(\mathbb{R}^3)\bigr)^{3}$$ and $$B\in \bigl(\dot{H}^{-1/2}(\mathbb{R}^3)\bigr)^{3}$$. This led us to consider the existence of global solutions of \eqref{eq:1} on homogeneous Sobolev spaces, especially the ones with negative index $$s<0$$.

• Remark. In formulating the equations, we have restricted to one charge for the sake of simplicity; results analogous to those stated in Theorems 1 and 2 should hold also in the case of $$n$$ charges, even if they are subjected to mutual and external interactions, provided these interactions are regular enough.

The rest of the paper is organised as follows: in section II we summarise and discuss the results proved in this paper; in section III a local solution is constructed by means of Banach fixed point theorem; in section IV we prove uniqueness and construct the maximal solution; in section V we show that the maximal solution is defined for all $$t\in\mathbb{R}$$; finally in section VI we discuss the divergence part of Maxwell's equations.