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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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References:
 [1] Belyaev, S.T.; Budker, G.I., The relativistic kinetic equation, Sov. phys.-dokl. proc. acad. sci. USSR, 1, 218-222, (1956) · Zbl 0072.22901 [2] Cercignani, C.; Illner, R.; Pulvirenti, M., The mathematical theory of dilute gases, Applied mathematical sciences, vol. 106, (1994), Springer-Verlag New York · Zbl 0813.76001 [3] Desvillettes, L.; Villani, C., On the spatially homogeneous Landau equation for hard potentials (I-II), Comm. P.D.E., 25, 1-2, 179-298, (2000) · Zbl 0951.35130 [4] Duan, R.-J.; Ukai, S.; Yang, T.; Zhao, H.-J., Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Comm. math. phys., 277, 189-236, (2008) · Zbl 1175.82047 [5] Glassey, R.T., The Cauchy problem in kinetic theory, (1996), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0372.35009 [6] Glassey, R.T.; Strauss, W.A., Decay of the linearized Boltzmann-Vlasov system, Transport theory statist. phys., 28, 2, 135-156, (1999) · Zbl 0983.82018 [7] Glassey, R.T.; Strauss, W.A., Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system, Discrete contin. dynam. systems, 5, 3, 457-472, (1999) · Zbl 0951.35102 [8] Glassey, R.T.; Strauss, W.A., Asymptotic stability of the relativistic Maxwellian, Publ. res. inst. math. sci., 29, 301-347, (1993) · Zbl 0776.45008 [9] Glassey, R.T.; Strauss, W.A., Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport theory statist. phys., 24, 657-678, (1995) · Zbl 0882.35123 [10] Guo, Y., The Landau equation in periodic box, Comm. math. phys., 231, 391-434, (2002) · Zbl 1042.76053 [11] Guo, Y., The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. math., 153, 3, 593-630, (2003) · Zbl 1029.82034 [12] Hinton, F.L., Collisional transport in plasma, (), 147 [13] Hsiao, L.; Yu, H.-J., Global classical solutions to the initial value problem for the relativistic Landau equation, J. differential equations, 228, 2, 641-660, (2006) · Zbl 1122.35146 [14] Hsiao, L.; Yu, H.-J., On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. appl. math., 65, 2, 281-315, (2007) · Zbl 1143.35085 [15] Kawashima, S., The Boltzmann equation and thirteen moments, Japan J. appl. math., 7, 301-320, (1990) · Zbl 0702.76090 [16] Lemou, M., Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. methods appl. sci., 23, 12, 1093-1119, (2000) · Zbl 1018.82012 [17] Lifschitz, E.M.; Petaevski, L.P., Equations cinétiques, vol. 10, (1975), Mir Moscow [18] Strain, R.M., The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. math. phys., 268, 2, 543-567, (2006) · Zbl 1129.35022 [19] Strain, R.M.; Guo, Y., Stability of the relativistic Maxwellian in a collisional plasma, Comm. math. phys., 251, 2, 263-320, (2004) · Zbl 1113.82070 [20] Strain, R.M.; Guo, Y., Almost exponential decay near Maxwellian, Comm. P.D.E., 31, 1-3, 417-429, (2006) · Zbl 1096.82010 [21] Villani, C., A survey of mathematical topics in kinetic theory, (), 71-305 · Zbl 1170.82369 [22] Yang, T.; Yu, H.-J., Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. differential equations, 248, 6, 1518-1560, (2010) · Zbl 1397.35166 [23] Yang, T.; Yu, H.-J., Optimal convergence rates of Landau equation with external forcing in the whole space, Acta math. sci. ser. B engl. ed., 9, 4, 1035-1062, (2009) · Zbl 1212.35490 [24] Yang, T.; Yu, H.-J., Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. math. anal., 42, 1, 459-488, (2010) · Zbl 1219.35302 [25] Yang, T.; Yu, H.-J., Optimal convergence rates of classical solutions for Vlasov-Poisson-Boltzmann system, Comm. math. phys., 301, 2, 319-355, (2011) · Zbl 1233.35040 [26] Yu, H.-J., Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. differential equations, 246, 10, 3776-3817, (2009) · Zbl 1170.35096 [27] Yu, H.-J., Global classical solution of the Vlasov-Maxwell-Landau system near maxwellians, J. math. phys., 45, 11, 4360-4376, (2004) · Zbl 1064.82035 [28] Zhan, M., Local existence of classical solutions to the Landau equations, Transport theory statist. phys., 23, 4, 479-499, (1994) · Zbl 0810.35095 [29] Zhan, M., Local existence of solutions to the Landau-Maxwell system, Math. methods appl. sci., 17, 8, 613-641, (1994) · Zbl 0803.35114
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