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Two Problems on Existence and Approximation Related to the Boltzmann Equation

Doctoral thesis, 1999

In this thesis, two different types of problems related to the Boltzmann equation of kinetic theory are studied. The first part is devoted to establishing consistency and convergence for discrete-velocity models of the Boltzmann equation. For a new such model, introduced by the author jointly with A. Heintz, the formal consistency with the Boltzmann equation is established. The result is obtained by a more direct method than those previously used for such type of problems. Based on the consistency result, the convergence to solutions of the space inhomogeneous Cauchy problem is proved. The results of a numerical test for the space homogeneous equations are presented.
In the second part, the problem of existence for the stationary Povzner equation with large data is studied. The Povzner equation is a regularized model of the Boltzmann equation, in which the collision process is "smeared'' in space, so that the collision operator has better regularity properties. The existence of solutions for problems in bounded domains with smooth boundaries is established for a general class of the collision kernels. The boundary conditions with fixed in-data and with linear operators of diffuse reflection type are considered. The method is based on constructing approximate solutions by a strong fixed point argument and establishing their weak compactness using estimates for the entropy production.

diffuse reflection

existence for large data

discrete-velocity models

weak compactness

Boltzmann equation

Povzner type models

convergence

consistency