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Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system. (English) Zbl 1245.35130
The authors address the problem of the existence and uniqueness of classical solutions to the fully Lorentz invariant, relativistic Vlasov-Maxwell-Boltzmann system. The two-species relativistic equations studied are \[ \begin{cases} \partial_t F_+ + c\frac{p}{p_+^0} \cdot \nabla_x F_ + + e_+\Big (E + \frac{p}{p_+^0} \times B \Big ) \cdot \nabla_p F_+ &= {\mathcal Q}^+(F) + {\mathcal Q}^{\pm}(F), \\ \partial_t F_- + c\frac{p}{p_-^0} \cdot \nabla_x F_ - - e_-\Big (E + \frac{p}{p_-^0} \times B \Big ) \cdot \nabla_p F_- &= {\mathcal Q}^-(F) + {\mathcal Q}^{\mp}(F).\end{cases}\tag{1} \] Here \({\mathcal Q}\) represents the collision operator in which \({\mathcal Q}^+(F) = {\mathcal Q}(F_+,F_+),\; {\mathcal Q}^{\pm}(F) = {\mathcal Q}(F_+,F_-)\), and similarly with \({\mathcal Q}^-(F),\; {\mathcal Q}^{\mp}(F)\). The “hard ball” condition is assumed throughout for the collision kernel of \({\mathcal Q}\). Equations (1) are coupled with Maxwell’s equations for the internally consistent electric field, \(E(t,x)\), and magnetic field, \(B(t,x)\). The functions \(F_{\pm}(t,x,p)\) are spatially periodic number density functions for ions (+) and electrons (\(-\)) in which position is \(x=(x_1,x_2,x_3) \in {\mathcal T}^3 = [-\pi,\pi]^3\), and momentum is \(p=(p_1,p_2,p_3) \in \mathbb R^3\). The energy of a particle is given by \(p_{\pm}^0 = \sqrt{(m_{\pm}c)^2 + |p|^2}\) where \(m_{\pm}\) are the particle masses and \(c\) the speed of light. The charge is \(e_{\pm}\). A steady state solution to (1) is the global relativistic Maxwellian (a.k.a the Juttner solution) represented by \(J_{\pm}(p)\) and with \(E(t,x) = 0\) and \(B(t,x) = \overline B\), constant.
The goal of this paper is to prove the existence of solutions to (1) in the context of perturbations around the steady state. The standard perturbation \(f_{\pm}(t,x,p)\) to \(J_{\pm}\) is \(F_{\pm} = J_{\pm} + \sqrt{J_{\pm}} f_{\pm}\). The relativistic Vlasov-Maxwell-Boltzmann system is then written in terms of \(f_{\pm}\). The main result of the paper is a theorem which guarantees that there are unique, global, classical solutions \([f_{\pm}(t,x,p),E(t,x),B(t,x)]\) to the perturbed relativistic Vlasov-Maxwell-Boltzmann system. Furthermore, for initial data sufficiently close to the steady state then \([f_{\pm}(t,x,p),E(t,x),B(t,x)] \rightarrow [J_{\pm},0,\overline B]\) as \(t \rightarrow \infty\), and the rate of convergence is \(O((1+t)^{-k})\) for some \(k > 0\). A main element of the proof of this result is to establish the momentum regularity of the solution. This is done by splitting the domain of integration in the collision operator into two regions. For one region the collision operator is represented in the Glassey-Strauss frame and for the other region the representation is in the center of mass frame.

MSC:
35Q83 Vlasov equations
35Q20 Boltzmann equations
35Q61 Maxwell equations
82D10 Statistical mechanics of plasmas
83A05 Special relativity
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