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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 ${e}^{-\lambda {t}^{p}}$ for some $\lambda >0$ and $0. 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:
 76P05 Rarefied gas flows, Boltzmann equation 82B40 Kinetic theory of gases (equilibrium statistical mechanics)
##### References:
 [1] Caflisch Russel E. (1980) The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Comm. Math. Phys. 74, 71–95 · Zbl 0434.76065 · doi:10.1007/BF01197579 [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 [3] Carlen Eric A., Lu Xuguang (2003) Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Statist. Phys. 112, 59–134 · Zbl 1079.82012 · doi:10.1023/A:1023623503092 [4] Degond P., Lemou M. (1997) Dispersion relations for the linearized Fokker-Planck equation. Arch. Ration Mech. Anal. 138:137–167 · Zbl 0888.35084 · doi:10.1007/s002050050038 [5] Desvillettes L., Villani Cédric (2005) On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159:245–316 · Zbl 1162.82316 · doi:10.1007/s00222-004-0389-9 [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