# zbMATH — the first resource for mathematics

The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. (English) Zbl 1360.35132
Summary: We are concerned with the Cauchy problem of the relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov-Maxwell-Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. Our proof is based on a new time-velocity weighted energy method and some optimal temporal decay estimates on the solution itself. The results also extend the case of “hard ball” model considered by Y. Guo and R. M. Strain [Commun. Math. Phys. 310, No. 3, 649–673 (2012; Zbl 1245.35130)] to the short range interactions.

##### MSC:
 35Q20 Boltzmann equations 83A05 Special relativity 35B40 Asymptotic behavior of solutions to PDEs 35Q83 Vlasov equations 35Q61 Maxwell equations 35B35 Stability in context of PDEs
Full Text:
##### References:
 [1] S. Calogero, The Newtonian limit of the relativistic Boltzmann equation,, J. Math. Phys., 45, 4042, (2004) · Zbl 1064.82030 [2] C. Cercignani, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics,, 22, (2002) · Zbl 1011.76001 [3] S. R. de Groot, Relativistic Kinetic Theory. Principles and Applications,, North-Holland Publishing Co., (1980) [4] R. J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions,, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 31, 751, (2014) · Zbl 1305.82057 [5] R. J. Duan, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory,, Electronic Journal of Differential Equations, (2003) · Zbl 1103.82022 [14] R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996) · Zbl 0858.76001 [15] R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data,, Comm. Math. Phys., 264, 705, (2006) · Zbl 1107.82047 [16] R. T. Glassey, On the derivatives of the collision map of relativistic particles,, Transport Theory Statist. Phys., 20, 55, (1991) · Zbl 0793.45008 [17] R. T. Glassey, Asymptotic stability of the relativistic Maxwellian,, Publ. Res. Inst. Math. Sci., 29, 301, (1993) · Zbl 0776.45008 [18] R. T. Glassey, Asymptotic stability of the relativistic Maxwellian via fourteen moments,, Transport Theory Statist. Phys., 24, 657, (1995) · Zbl 0882.35123 [19] Y. Guo, The Vlasov-Poisson-Landau system in a periodic box,, J. Amer. Math. Soc., 25, 759, (2012) · Zbl 1251.35167 [20] Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153, 593, (2003) · Zbl 1029.82034 [21] Y. Guo, Momentum regularity and stability of the relativistic Vlasov- Maxwell-Boltzmann system,, Comm. Math. Phys., 310, 649, (2012) · Zbl 1245.35130 [22] S. Y. Ha, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces,, Methods Appl. Anal., 14, 251, (2007) · Zbl 1163.35433 [23] L. Hsiao, Asymptotic stability of the relativistic Maxwellian,, Math. Methods Appl. Sci., 29, 1481, (2006) · Zbl 1109.35115 [24] L. Hsiao, Global classical solutions to the initial value problem for the relativistic Landau equation,, J. Differential Equations, 228, 641, (2006) · Zbl 1122.35146 [25] T. Hosono, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system,, Math. Models Methods Appl. Sci., 16, 1839, (2006) · Zbl 1108.35014 [26] N. A. Krall, Principles of Plasma Physics,, McGraw-Hill, (1973) [27] Y. J. Lei, The Vlasov-Maxwell-Boltzmann system with a uniform ionic background near Maxwellians,, J. Differential Equations, 260, 2830, (2016) · Zbl 1382.35303 [28] S. Q. Liu, Optimal large-time decay of the relativistic Landau-Maxwell system,, J. Differential Equations, 256, 832, (2014) · Zbl 1320.35344 [29] A. Majda, Vorticity and Incompressible Flow,, Cambridge University Press, (2002) · Zbl 0983.76001 [30] R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268, 543, (2006) · Zbl 1129.35022 [31] R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory,, Phd Thesis, (2005) [32] R. M. Strain, Stability of the Relativistic Maxwellian in a Collisional Plasma,, Comm. Math. Phys., 251, 263, (2004) · Zbl 1113.82070 [33] R. M. Strain, Almost exponential decay near Maxwellian,, Comm. Partial Differential Equations, 31, 417, (2006) · Zbl 1096.82010 [34] R. M. Strain, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187, 287, (2008) · Zbl 1130.76069 [35] R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum,, SIAM J. Math. Anal., 42, 1568, (2010) · Zbl 1429.76095 [36] R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials,, Comm. Math. Phys., 300, 529, (2010) · Zbl 1214.35072 [37] R. M. Strain, Coordinates in the relativistic Boltzmann theory,, Kinetic and Related Models, 4, 345, (2011) · Zbl 1429.76094 [38] R. M. Strain, Large-time decay of the soft potential relativistic Boltzmann equation in $$\mathbb R^3$$,, Kinetic and Related Models, 5, 383, (2012) · Zbl 1247.76071 [39] C. Villani, A review of mathematical topics in collisional kinetic theory,, North-Holland, I, 71, (2002) · Zbl 1170.82369 [40] L. S. Wang, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions,, Acta Mathematica Scientia, (2016) · Zbl 1363.35266 [41] Q. H. Xiao, Large-time behavior of the two-species relativistic Landau-Maxwell system in $$\mathbb R^3_x$$,, J. Differential Equations, 259, 3520, (2015) · Zbl 1330.35431 [42] Q. H. Xiao, The Vlasov-Poisson-Boltzmann system with angular cutoff for soft potentials,, J. Differential Equations, 255, 1196, (2013) · Zbl 1284.35442 [43] Q. H. Xiao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,, Sci. China Math., 57, 515, (2014) · Zbl 1307.35089 [44] Q. H. Xiao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials,,
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.