Exact Solutions of the Boltzmann Equation

  • H. Cornille
Part of the NATO ASI Series book series (ASIC, volume 310)


We review the exact solutions of the discrete and continuous Boltzmann Equations.

For the discrete B.E. the velocity \( {\vec V} \) can only take discrete values \( {{\vec V}_i} \). The discrete models equations are nonlinear but they include the linear conservation laws. The simplest solutions are the exponential similarity shock waves and the solutions found in 1+1, 2+1, 2D, 3D dimensions are sums of similarity waves. For a well-defined class of (1+1)-dimensional rational functions it is proved that they are the only possible solutions if two linear conservation laws are present. Different solutions are presented: with nonuniform Maxwellians, shock wave with or without specular reflection, periodic, semiperiodic.

The conservation laws being not directly included into the nonlinear continuous B.E. the exact solutions are different. For the Kac model we construct the even velocity BKW (Bobylev-Krook-Wu) solution, its odd velocity partner and an inhomogeneous solution, while for the homogeneous B.E. we derive the BKW solution. For the inhomogeneous B.E. a time-dependent harmonic potential, which vanishes at infinite time, is introduced, but the equilibrium state is an absolute Maxwellian. This is the only known exact inhomogeneous example relaxing toward the true equilibrium state and not toward the trivial vacuum state


Arbitrary Parameter Dimensional Solution Similarity Wave Inhomogeneous Solution Hypercubic Model 
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Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • H. Cornille
    • 1
  1. 1.Service de Physique Théorique de SaclayLaboratoire de l’Institut de Recherche Fondamentaledu Commissariat à l’Energie AtomiqueGif-sur-Yvette CedexFrance

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