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Global solutions to the relativistic Landau-Maxwell system in the whole space. (English. French summary) Zbl 1251.35161
The purpose of this paper is to prove the global existence of classical solutions to the relativistic Landau-Maxwell system. Some additional results are obtained on the convergence to the steady-state for the simpler Landau-Poisson system. The equations being studied are \[ \partial_t F_+ + \frac{p}{p^0} \cdot \nabla_x F_ + + \Big (E + \frac{p}{p^0} \times B \Big ) \cdot \nabla_p F_+ = {\mathcal C}(F_+,F_+) + {\mathcal C}(F_+,F_-), \]
\[ \partial_t F_- + \frac{p}{p^0} \cdot \nabla_x F_ - - \Big (E + \frac{p}{p^0} \times B \Big ) \cdot \nabla_p F_- = {\mathcal C}(F_-,F_-) + {\mathcal C}(F_-,F_+), \] with \(F_{\pm}(0,x,p) = F_{0,\pm}(x,p)\). In (1) \(F_{\pm}(t,x,p)\) are number densities for ions (\(+\)) and electrons (\(-\)) with position \(x=(x_1,x_2,x_3) \in {\mathbb R}^3\), momentum \(p=(p_1,p_2,p_3) \in {\mathbb R}^3\), and \(p_0 = \sqrt{1+|p|^2}\). Equations (1) are coupled with Maxwell’s equations for the internally consistent electric field \(E=E(t,x)\) and magnetic field \(B=B(t,x)\). The relativistic collision operator given in normalized form is \[ {\mathcal C}(g,h)(p) = \nabla_p \cdot \Big \{ \int_{{\mathbb R}^3} \Phi(P,Q)\{\nabla_pg(p)h(q) - g(p) \nabla_qh(q) \} dq \Big \}, \] where \(P = (p_0,p_1,p_2,p_3),\; Q = (q_0,q_1,q_2,q_3)\), and the collision kernel \(\Phi(P,Q)\) is a non negative \(3 \times 3\) matrix.
The authors consider the system for small perturbations of the equilibrium state. The normalized global relativistic Maxwellian is \(J(p) = e^{-p_0}\). The small perturbation \(f_{\pm}(t,x,p)\) around \(J(p)\) is defined by \(F_{\pm} = J(p) + \sqrt{J(p)} f_{\pm}\). The relativistic Landau-Maxwell system in then reformulated in terms of \(f(t,x,p) = [f_+(t,x,p),f_-(t,x,p)]\). In terms of derivatives of \(f,E,B\), and an orthogonal projection operator \({\mathbf P}\) the instant energy functional \({\mathcal E}(t)\) and dissipation rate \({\mathcal D}(t)\) are defined. The main result of the paper is a theorem which proves that if \({\mathcal E}(0)\) is sufficiently small then there exists a unique global classical solution to the reformulated Landau-Maxwell system. In addition \({\mathcal E},{\mathcal D}\) satisfy \[ {\mathcal E}(t) + \int_0^t {\mathcal D}(s) ds \leq {\mathcal E}(0). \] The type of analysis applied to the Landau-Maxwell system is also applied to the simpler Landau-Poisson system. In this case in addition to the existence and uniqueness of solutions the authors also derive an optimal convergence rate for the solution sufficiently close to the equilibrium.

MSC:
35Q61 Maxwell equations
35Q75 PDEs in connection with relativity and gravitational theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
82D10 Statistical mechanics of plasmas
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