Communications in Mathematical Sciences

Volume 5 (2007)

Number 4

Weak shock waves for the general discrete velocity model of the Boltzmann equation

Pages: 815 – 832



Niclas Bernhoff

Alexander Bobylev


We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks.

We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity.

Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive, more straightforward, proof that suits the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corre- sponding stable, unstable and center manifolds) for singular points of general dynamical systems of the same type as in the shock wave problem for DVMs.

The same approach can also be applied for DVMs for mixtures.


Boltzmann equation; discrete velocity models; shock waves

2010 Mathematics Subject Classification

76P05, 82C40

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