Jianjun HUANG
,
Zhenglu JIANG
. AVERAGE REGULARITY OF THE SOLUTION TO AN EQUATION WITH THE RELATIVISTIC-FREE TRANSPORT OPERATOR[J]. Acta mathematica scientia, Series B, 2017
, 37(5)
: 1281
-1294
.
DOI: 10.1016/S0252-9602(17)30073-5
[1] Andréasson H. Regularity of the gain term and strong L1 convergence to the equilibrium for the relativistic Boltzmann equation. SIAM J Math Anal, 1996, 27:1386-1405
[2] Andréasson H, Calogero S, Illner R. On blowup for gain-term-only classical and relativistic Boltzmann equations. Math Methods Appl Sci, 2004, 27:2231-2240
[3] Bachouri H, Chemin J, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Berlin Heidelberg:Springer-Verlag, 2011
[4] Bergh J, Lofstrom J. Interpolation Spaces, New York, Berlin:Springer-Verlag, 1976
[5] Bichteler K. On the Cauchy problem of the relativistic Boltzmann equations. Commun Math Phys, 1967, 4:352-364
[6] Bouchut F, Golse F, Pulvirenti M. Kinetic Equations and Asymptotic Theory. Ser Appl Math (Paris). Paris:Gauthiers-Villars, 2000
[7] Bouchut F, Desvillettes L. Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc Roy Soc Edinburgh Sect A, 1999, 129:19-36
[8] de Groot S R, van Leeuwen W A, van Weert Ch G. Relativistic Kinetic Theory. Amsterdam:North-Holland, 1980
[9] Debbasch F, Leeuwen W A V. General relativistic Boltzmann equation, 1:covariant treatment. Phys A:Stat Mech Appl, 2009, 388:1079-1104
[10] Debbasch F, Leeuwen W A V. General relativistic Boltzmann equation, 2:manifestly covariant treatment. Phys A:Stat Mech Appl, 2009, 388:1818-1834
[11] Denicol G S, Ulrich H, Mauricio M, Jorge N, Michael S. A new exact solution of the relativistic Boltzmann equation and its hydrodynamic limit. Phys Review Lett, 2014, 113(20):202301-202301
[12] DiPerna R J, Lions P L. On the Cauchy problem for Boltzmann equations:global existence and weak stability. Ann Math, 1989, 130:321-366
[13] Dolan P, Zenios A C. A construction of the general relativistic Boltzmann equation. Int J Math Math Sci, 1984, 7:591-597
[14] Dudyński M, Ekiel-Je·zewska M L. On the linearized relativistic Boltzmann equation 1:existence of solutions. Commun Math Phys, 1988, 115:607-629
[15] Dudyński M, Ekiel-Je·zewska M L. Global existence proof for the relativistic Boltzmann equation. J Stat Phys, 1992, 66:991-1001
[16] Esobedo M, Michler S, Valle M A. Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electronic J Differ Equ, 2003, 4:1-85
[17] Glassey R. Global solutions to the Cauchy problem for the relativistic Boltzmann equation with nearvacuum data. Commun Math Phys, 2006, 264:705-724
[18] Glassey R. Strauss W A. Asymptotic stability of the relativistic Maxwellian. Publ Res Inst Math Sci, 1993, 29:301-347
[19] Glassey R, Strauss W A. Asymptotic stability of the relativistic Maxwellian via fourteen moments. Trans Theory Stat Phys, 1995, 24:657-678
[20] Golse F, Lions P L, Perthame B, Sentis R. Regularity of the moments of the solution of a transport equation. J Funct Anal, 1988, 76:110-125
[21] Ha S, Kim Y, Lee H, Noh S. Asympotic completeness for relativistic kinetic equations with short-range interaction forces. Meth Appl Anal, 2007, 14:251-262
[22] Hsiao L, Yu H. Asympotic stability of relativistic Maxwellian. Math App Sci, 2006, 29:1481-1499
[23] Israel W. Relativistic kinetic theory of simple gas. J Math Phys, 1963, 4:1163-1181
[24] Lichnerowicz A, Marrot R. Propriétés statistiques des ensembles de particules en relativité restreinte. Compt Rend Acad Sci (Paris), 1940, 210:759-761
[25] Liu S Q, Xiao Q H. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinet Relat Mod, 2016, 9:515-550
[26] Jiang Z. Compactness related to the transport operator of the relaivistic Boltzmann equation. Acta Math Sci (in Chinese), 1997, 17A(3):330-335
[27] Jiang Z. On the relatvistic Boltzmann equation. Acta Math Sci, 1998, 13:348-360
[28] Jiang Z. Existence of global solution to the Cauchy problem for the relativistic Boltzmann equation in a periodic box. Acta Math Sci, 1998, 18:375-384
[29] Jiang Z. On the Cauchy problem for the relatvistic Boltzmann equationin a periodic box:global existence. Trans Theory Stat Phys, 1999, 28:617-628
[30] Jiang Z. Global existence proof for relativistic Boltzmann equation with hard interactions. J Stat Phys, 2008, 130:535-544
[31] Jüttner F. Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativitheorie. Ann Phys, 1911, 34:856-882
[32] Rein G. Global weak solutions to the relativistic Vlasov-Maxwell system revisited. Comm Math Sci, 2004, 2:145-158
[33] Strain R M. Global Newtonian limit for the relativistic Boltzmann equation near vacuum. SIAM J Math Anal, 2010, 42(4):1568-1601
[34] Strain R M. Asymptotic stability of the relativistic Boltzmann equantion for the soft potentials. Commun Math Phys, 2010, 300(2):529-597
[35] Strain R M. Coordinates in the relativistic Boltzmann theory. Kinet Relat Mod, 2011, 4(1):345-359
[36] Stewart J. Non-equilibrium Relativistic Kinetic Theory, Vol 10, Lectures Notes in Physics. Berlin:SpringerVerlag, 1971
[37] Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam:North-Holland, 1978
[38] Tsumura K, Kunihiro T. Derivation of relativistic hydrodynamic equation consitient with relativistic Boltzmann equation by renormalizatio-group method. Eur Phys J A, 2012, 48:162-173
=f,where f=f (t,x,p) belongs to Lp ((0,T)×R3×R3) for 1 < p < ∞ and 
is the relativisticfree transport operator from the relativistic Boltzmann equation.We show the regularity of ∫R3 u(t,x,p) dp using the same method as given by Golse,Lions,Perthame and Sentis.This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.