×

Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. (English) Zbl 1147.76054

Summary: This paper is about the propagation of singularities in the solutions to Cauchy problem for spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of L. Boudin and L. Desvillettes [Monatsh. Math. 131, No. 2, 91–108 (2000; Zbl 0984.76076)] on the propagation of singularities in solutions near vacuum. It is shown that for the solution near a global Maxwellian, singularities in the initial data propagate like free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of initial data, and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, we give the dependence of the regularity on the cross-section.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
45K05 Integro-partial differential equations

Citations:

Zbl 0984.76076
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams R., Sobolev Spaces (1985)
[2] DOI: 10.1007/s002050000083 · Zbl 0968.76076
[3] DOI: 10.1142/S0218202505000613 · Zbl 1161.35331
[4] Bouchut F., Rev. Mat. Iber. 14 pp 47–
[5] DOI: 10.1007/s006050070015 · Zbl 0984.76076
[6] Desvillettes L., Riv. Mat. Univ. Parma 7 pp 1–
[7] DOI: 10.1081/PDE-120028847 · Zbl 1103.82020
[8] DiPerna R. J., Ann. Math. 130 pp 312–
[9] DOI: 10.1142/S0218202506001613 · Zbl 1096.76050
[10] DOI: 10.1016/0022-1236(88)90051-1 · Zbl 0652.47031
[11] Grad H., Flügge’s Handbuch Phys. 12 pp 205–
[12] DOI: 10.1007/BF03167035 · Zbl 0653.76053
[13] Illner R., Comm. Math. Phys. 95 pp 117–
[14] Imai K., Publ. RIMS Kyoto Univ. 12 pp 229–
[15] DOI: 10.1007/BF01624788 · Zbl 0371.76061
[16] Lions P.-L., J. Math. Kyoto Univ. 34 pp 391–
[17] DOI: 10.1006/jmaa.1998.6141 · Zbl 0913.76081
[18] Mouhot C., Arch. Rational. Mech. Anal. 173 pp 169–
[19] DOI: 10.1142/S0218202597000517 · Zbl 0907.76079
[20] DOI: 10.3792/pja/1195519027 · Zbl 0312.35061
[21] DOI: 10.1016/S0168-2024(08)70128-0
[22] DOI: 10.1142/S0219530506000784 · Zbl 1096.35012
[23] DOI: 10.1016/S1874-5792(02)80004-0
[24] DOI: 10.1080/03605309408821082 · Zbl 0818.35128
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.