ON THE ASYMPTOTIC DYNAMICS OF THE VLASOV-YUKAWA-BOLTZMANN SYSTEM NEAR VACUUM

doi:10.1016/S0252-9602(15)30027-8

Abstract

Abstract:

In this paper, we study uniform L¹-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann (V-Y-B) system. For a sufficiently small and smooth initial data, we show that classical solutions exist globally and satisfy dispersion estimates, uniform L¹-stability with respect to initial data and scattering type estimate. We show that the short range nature of interactions due to the Yukawa potential enables us to construct robust Lyapunov type functional to derive scattering states. In the zero mass limit of force carrier particles, we also show that the classical solutions to the V-Y-B system converge to that of the Vlasov-Poisson-Boltzmann (V-P-B) system in any finite time interval.

Key words: asymptotic completeness, collision operator, the Vlasov-Yukawa-Boltzmann system, uniform L¹-stability

CLC Number:

92D25

Sun-Ho CHOI, Seung-Yeal HA. ON THE ASYMPTOTIC DYNAMICS OF THE VLASOV-YUKAWA-BOLTZMANN SYSTEM NEAR VACUUM[J].Acta mathematica scientia,Series B, 2015, 35(4): 887-905.

Trendmd

References

[1] Alonso R J, Gamba I M. Distributional and classical solutions to the Cauchy-Boltzmann problem for soft potentials with integrable angular cross section. J Stat Phys, 2009, 137: 1147-1165
[2] Bardos C, Degond P. Global existence for the Vlasov-Poisson equation in three space variables with small initial data. Ann Inst Henri P′oincare C, 1985, 2: 101-118
[3] Chae M, Ha S -Y. New Lyapunov functionals of the Vlasov-Poisson system. SIAM J Math Anal, 2006, 37(6): 1709-1731
[4] Chae M, Ha S -Y, Hwang H J. Time-asymptotic behavior of the Vlasov-Poisson-Boltzmann system near vacuum. J Differential Equations, 2006, 230: 71-85
[5] Choi S -H, Ha S -Y. Asymptotic behavior of the nonlinear Vlasov equation with a self-consistent force. SIAM J Math Anal, 2011, 43: 2050-2077
[6] Choi S -H, Ha S -Y, Lee H. Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data. J Differential Equations, 2011, 250(1): 515-550
[7] Desvillettes L, Dolbeault J. On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Commun Partial Diffential Equations, 1991, 16: 451-489
[8] Duan R -J, Zhang M, Zhu C. L¹-stability for the Vlasov-Poisson-Boltzmann system around vacuum. Math Models Methods Appl Sci, 2006, 16: 1505-1526
[9] Glassey R, Strauss W A. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson- Boltzmann system. Discrete Contin Dynam Systems, 1999, 5: 457-472
[10] Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians. Commun Pure Appl Math, 2002, 55: 1104-1135
[11] Guo Y. The Vlasov-Poisson-Boltzmann system near vacuum. Commun Math Phys, 2001, 218: 293-313
[12] Ha S -Y. Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J Differential Equations, 2005, 215: 178-205
[13] Ha S -Y. ¹-stability of the Boltzmann equation for the hard-sphere model. Arch Rat Mech Anal, 2004, 173: 279-296
[14] Ha S -Y, Ha T, Hwang C -O, Lee H. Nonlinear instability of the one-dimensional Vlasov-Yukawa system. J Math Phys, 2011, 52: 033301
[15] Ha S -Y, Kim Y, Lee H, Noh S. Asymptotic completeness for relativistic kinetic equations with short-range interaction forces. Methods Appl Anal, 2007, 14: 251-262
[16] Ha S -Y, Lee H. Global well posedness of the relativistic Vlasov-Yukawa system with small data. J Math Phys, 2007, 48: 123508
[17] Lions P L, Perthame B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent Math, 1991, 105: 415-430
[18] Mischler S. On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun Math Phys, 2000, 210: 447-466
[19] Pfaffelmoser K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J Differential Equations, 1992, 95: 281-303
[20] Piazza F, Marinoni, C. Model for gravitational interaction between dark matter and baryons. Phys Rev Lett, 2003, 91: 141301
[21] Rein G. Collisionless kinetic equations from astrophysics-the Vlasov-Poisson system//Handbook of Differ- ential Equations: Evolutionary Equations, Vol III. North-Holland, Amsterdam: Elsevier, 2007: 383-476
[22] Saito K, Maruyama T, Soutome K. Collective modes in hot and dense matter. Phys Rev C, 1989, 40: 407-431
[23] Yang T, Yu H -J, Zhao H -J. Cauchy problem for the Vlasov-Poisson-Boltzmann system. Arch Rat Mech Anal, 2006, 182(3): 415-470
[24] Yukawa H. On the interaction of elementary particles. Proc Phys Math Soc Jpn, 1935, 17: 48-57