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Initial-boundary value problems for the Boltzmann equation: Global existence of weak solutions. (English) Zbl 0777.76084
The paper deals with two types of initial-boundary value problems in kinetic theory: the evolution of the distribution of a rarefied gas occupying a bounded spatial domain \(\Omega\) and that of a gas flowing past an obstacle \(\Omega_ 0\) in \(R^ 3\) \((\Omega\) and \(\Omega_ 0\) are regular bounded open sets of \(R^ 3)\). The intermolecular forces are derived from hard potentials. The boundary conditions considered are fairly general and comprise Maxwell types with specular or reverse reflections.
Among the results we quote an improved version of the averaged regularity lemma of F. Golse, P.-L. Lions, B. Perthame, and R. Sentis [J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)]. This is an useful tool to prove averaged stability of the collision term of the Boltzmann equation. Two weak formulations of the problems are introduced, the first one based on duality, the other on integration along the characteristic lines. Existence of at least one solution is claimed, but for sake of conciseness the lengthy proof is given only for the integral formulation. Finally uniqueness results for regularized solutions employed in the work are given by using the semigroup theory of maximal accretive operators.
Reviewer: G.Busoni (Firenze)

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
45K05 Integro-partial differential equations
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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