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Exponential decay for soft potentials near Maxwellian. (English) Zbl 1130.76069
Summary: We consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches zero at the rate of e -λt p for some λ>0 and 0<p<1. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical result [R. E. Caflisch, Commun. Math. Phys. 74, 97–109 (1980; Zbl 0434.76066)] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by L. Desvilletes and C. Villani [Invent. Math. 159, No. 2, 245–316 (2005; Zbl 1162.82316)] and R. M. Strain and Y. Guo [Commun. Partial Differ. Equations 31, 417–429 (2006)].
MSC:
76P05Rarefied gas flows, Boltzmann equation
82B40Kinetic theory of gases (equilibrium statistical mechanics)
References:
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[2]Caflisch Russel E. (1980) The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Comm. Math. Phys. 74, 97–109 · Zbl 0434.76066 · doi:10.1007/BF01197752
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[6]Glassey, Robert T.: The Cauchy problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996
[7]Grad, Harold.: Asymptotic theory of the Boltzmann equation. II. Rarefied Gas Dynamics. Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962, Vol. I 26–59, (1963)
[8]Guo Yan. (2002) The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55:1104–1135 · Zbl 1027.82035 · doi:10.1002/cpa.10040
[9]Guo Yan. (2002) The Landau equation in a periodic box. Comm. Math. Phys. 231: 391–434 · Zbl 1042.76053 · doi:10.1007/s00220-002-0729-9
[10]Guo Yan (2003) Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169:305–353 · Zbl 1044.76056 · doi:10.1007/s00205-003-0262-9
[11]Guo Yan (2003) The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153:593–630 · Zbl 1029.82034 · doi:10.1007/s00222-003-0301-z
[12]Guo Yan (2004) The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53:1081–1094 · Zbl 1065.35090 · doi:10.1512/iumj.2004.53.2574
[13]Guo Yan (2006) Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. 59:626–687 · Zbl 1089.76052 · doi:10.1002/cpa.20121
[14]Strain Robert M., Guo Yan. (2006) Almost exponential decay near Maxwellian. Comm. Partial Differential Equations 31:417–429 · Zbl 1096.82010 · doi:10.1080/03605300500361545
[15]Strain Robert M., Guo Yan. (2004) Stability of the relativistic Maxwellian in a Collisional Plasma. Comm. Math. Phys. 251:263–320 · Zbl 1113.82070 · doi:10.1007/s00220-004-1151-2
[16]Strain Robert M. (2006) The Vlasov-Maxwell-Boltzmann System in the Whole Space. Comm. Math. Phys. 268:543–567 · Zbl 1129.35022 · doi:10.1007/s00220-006-0109-y
[17]Ukai Seiji. (1974) On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. Ser. A Math. Sci. 50:179–184
[18]Ukai Seiji., Asano Kiyoshi. (1982) On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18:477–519 · Zbl 0538.45011 · doi:10.2977/prims/1195183569