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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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