Global Newtonian Limit for the Relativistic Boltzmann Equation near Vacuum

We study the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data. Unique global-in-time mild solutions are obtained uniformly in the speed of light parameter $c\geq1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon\in(0,2)$. This may be the first proof of unique global-in-time validity of the Newtonian limit for a kinetic equation.

[1]  The Newtonian limit of the relativistic Boltzmann equation , 2004, math-ph/0404043.

[2]  J. Polewczak.  Classical solution of the nonlinear Boltzmann equation in allR3: Asymptotic behavior of solutions , 1988 .

[3]  Yan Guo.  Classical Solutions to the Boltzmann Equation for Molecules with an Angular Cutoff , 2003 .

[4]  Yan Guo,et al.  The Vlasov-Maxwell-Boltzmann system near Maxwellians , 2003 .

[5]  K. Bichteler.  On the Cauchy problem of the relativistic Boltzmann equation , 1967 .

[6]  J. J. Dijkstra,et al.  Mathematical aspects of relativistic kinetic theory , 1978 .

[7]  Werner Israel.  The relativistic Boltzmann equation. , 1972 .

[8]  G. Naber.  The geometry of Minkowski spacetime : an introduction to the mathematics of the special theory of relativity , 1992 .

[9]  B. Wennberg.  The Geometry of Binary Collisions and Generalized Radon Transforms , 1997 .

[10]  W. T. Grandy,et al.  Kinetic theory : classical, quantum, and relativistic descriptions , 2003 .

[11]  S. Ukai,et al.  On the Vlasov-Poisson Limit of the Vlasov-Maxwell Equation , 1985 .

[12]  L. Hsiao,et al.  Asymptotic stability of the relativistic Maxwellian , 2006 .

[13]  Yan Guo,et al.  The Vlasov‐Poisson‐Boltzmann system near Maxwellians , 2002 .

[14]  G. Toscani,et al.  On the Cauchy problem for the nonlinear Boltzmann equation global existence uniqueness and asymptotic stability , 1985 .

[15]  Robert M. Strain.  Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials , 2010, 1003.4893.

[16]  Simone Calogero,et al.  The non-relativistic limit of the Nordstr\"om-Vlasov system , 2003, math-ph/0309030.

[17]  The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system , 1993, gr-qc/9303027.

[18]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[19]  Nicola Bellomo,et al.  Mathematical Topics In Nonlinear Kinetic Theory , 1988 .

[20]  P. Lions,et al.  On the Cauchy problem for Boltzmann equations: global existence and weak stability , 1989 .

[21]  R. Wagoner,et al.  Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity , 1973 .

[22]  R. J. Crewther,et al.  Introduction to quantum field theory , 1995, hep-th/9505152.

[23]  Harold Grad,et al.  Asymptotic Theory of the Boltzmann Equation , 1963 .

[24]  ON THE RELATIVISTIC BOLTZMANN EQUATION , 1998 .

[25]  M. Ekiel-Jeżewska,et al.  The relativistic Boltzmann equation - mathematical and physical aspects , 2007 .

[26]  W. Strauss,et al.  Asymptotic stability of the relativistic Maxwellian via fourteen moments , 1995 .

[27]  G. Naber.  The geometry of Minkowski spacetime , 1992 .

[28]  R. Illner,et al.  The Boltzmann equation: Global existence for a rare gas in an infinite vacuum , 1984 .

[29]  W. Strauss,et al.  On the derivatives of the collision map of relativistic particles , 1991 .

[30]  S. Kaniel,et al.  The Boltzmann equation , 1978 .

[31]  J. Stewart.  Non-Equilibrium Relativistic Kinetic Theory , 1971 .

[32]  R. Glassey.  Global Solutions to the Cauchy Problem for the Relativistic Boltzmann Equation with Near–Vacuum Data , 2006 .

[33]  M. Ekiel-Jeżewska,et al.  On the linearized relativistic Boltzmann equation , 1988 .

[34]  W. Israel.  Relativistic Kinetic Theory of a Simple Gas , 1963 .

[35]  M. Bostan.  Asymptotic behavior of weak solutions for the relativistic Vlasov–Maxwell equations with large light speed , 2006 .

[36]  S. R. de Groot,et al.  Relativistic kinetic theory , 1974 .

[37]  B. Boisseau,et al.  Relativistic Boltzmann theory in D + 1 spacetime dimensions , 1989 .

[38]  Yan Guo,et al.  The Boltzmann equation in the whole space , 2004 .

[39]  L. Tartar.  Some Existence Theorems for Semilinear Hyperbolic Systems in One Space Variable. , 1981 .

[40]  Jack Schaeffer.  The classical limit of the relativistic Vlasov-Maxwell system , 1986 .

[41]  P. Lions,et al.  Compactness in Boltzmann’s equation via Fourier integral operators and applications. III , 1994 .

[42]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[43]  Zhenglu Jiang.  Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions , 2008, 0901.0372.

[44]  Dudynski,et al.  Causality of the linearized relativistic Boltzmann equationp. , 1985, Physical review letters.

[45]  M. Peskin,et al.  An Introduction To Quantum Field Theory , 1995 .

[46]  R. Glassey,et al.  The Cauchy Problem in Kinetic Theory , 1987 .

[47]  M. Dudynski.  On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics , 1989 .

[48]  H. Andréasson.  Regularity of the gain term and strong L 1 convergence to equilibrium for the relativistic Boltzmann equation , 1996 .

[49]  Francis Filbet,et al.  Global existence for the Vlasov–Darwin system in ℝ3 for small initial data , 2003 .

[50]  H. Neunzert,et al.  Local existence of solutions of the vlasov‐maxwell equations and convergence to the vlasov‐poisson equations for infinite light velocity , 1986 .

[51]  Stéphane Mischler,et al.  Homogeneous Boltzmann equation in quantum relativistic kinetic theory , 2003, Electronic Journal of Differential Equations.

[52]  The Darwin Approximation of the Relativistic Vlasov-Maxwell System , 2004, math-ph/0401012.

[53]  Philip M. Morse,et al.  The Relativistic Gas , 1958 .

[54]  On blowup for gain‐term‐only classical and relativistic Boltzmann equations , 2004, math-ph/0402024.