Global Solution of the Electromagnetic Field-Particle System of Equations

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.

[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 [1996] 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 [2000] and Bauer and Duerr [2001]; the latter result has been developed further in [BauDecDur 2013] to consider weighted \(L^2\) spaces. Imaykin, Komech and Mauser [2004] 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 [2004].

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.