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Long-time asymptotics for the coupled Maxwell-Lorentz equations. (English) Zbl 0970.35149

The authors consider the coupled Maxwell-Lorentz equations \[ \begin{aligned} & \text{div } E(x,t)=\rho (x-q(t)),\quad \text{rot } E(x,t)=-\dot B (x,t),\quad \text{div }B(x,t)=0,\\ & \text{rot } B(x,t)=\dot E(x,t)+\rho (x-q(t))\dot q(t),\quad \dot q(t)=(1+p^2(t))^{-1/2}p(t),\\ & \dot p(t)=E_{\text{ex}}(q(t))+\dot q(t)\wedge B_{ex}(q(t))+ \int d^3x\rho (x-q(t))[E(x,t)+\dot q(t)\wedge B(x,t)].\end{aligned} \] Here \(q(t)\in \mathbb{R}^3\) denotes the position of charge at time \(t\), and all derivatives are understood in the sense of distributions. The external potentials satisfy \(\Phi_{\text{ex}},A_{\text{ex}}\in C_{\text{loc}}^{\infty }(\mathbb{R}^3)\), \(\inf_{q\in\mathbb{R}^3}\Phi_{\text{ex}}(q)>-\infty \), \(\rho\in C_0^{\infty }(\mathbb{R}^3)\) (\(\rho \) is the charge distribution which is smooth and radially symmetric). The Wiener condition also holds, \[ \hat\rho (k)=\int d^3xe^{ikx}\rho (x)\neq 0\quad (k\in \mathbb{R}^3). \] The main result is the long-time asymptotics of the finite energy solutions to the considered system presented in the Fréchet topology.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35B40 Asymptotic behavior of solutions to PDEs
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