Cauchy Problem for the Vlasov–Poisson–Boltzmann System

The dynamics of dilute electrons can be modelled by the fundamental Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.

[1]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[2]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[3]  S. Ukai,et al.  On the existence of global solutions of mixed problem for non-linear Boltzmann equation , 1974 .

[4]  Tai-Ping Liu,et al.  Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles , 2004 .

[5]  Yan Guo,et al.  The Vlasov-Maxwell-Boltzmann system near Maxwellians , 2003 .

[6]  Yan Guo.  The Vlasov–Poisson–Boltzmann System Near Vacuum , 2001 .

[7]  R. Glassey,et al.  The Cauchy Problem in Kinetic Theory , 1987 .

[8]  Stéphane Mischler,et al.  On the Initial Boundary Value Problem for the Vlasov–Poisson–Boltzmann System , 2000 .

[9]  François Golse,et al.  On a boundary layer problem for the nonlinear Boltzmann equation , 1988 .

[10]  Laurent Desvillettes,et al.  On long time asymptotics of the vlasov—poisson—boltzmann equation , 1991 .

[11]  Harold Grad,et al.  Asymptotic Theory of the Boltzmann Equation , 1963 .

[12]  Huijiang Zhao,et al.  Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation , 2006 .

[13]  Tai-Ping Liu,et al.  Energy method for Boltzmann equation , 2004 .

[14]  W. Strauss,et al.  Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system , 1999 .

[15]  P. Lions,et al.  Global weak solutions of Vlasov‐Maxwell systems , 1989 .

[16]  Yan Guo,et al.  The Vlasov‐Poisson‐Boltzmann system near Maxwellians , 2002 .

[17]  David Hoff,et al.  Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids , 1997 .

[18]  W. Strauss,et al.  Decay of the linearized Boltzmann-Vlasov system , 1999 .

[19]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[20]  Shuichi Kawashima,et al.  Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion , 1985 .

[21]  Ludwig Boltzmann,et al.  Lectures on Gas Theory , 1964 .