The Cauchy problem for the linear Maxwell-Boltzmann equation
It is supposed that two kinds of particles, labelled M and N are present in physical space. Type M particles, not influenced by external forces, are in equilibrium and condition the physical behaviour of the type N particles without being conditioned themselves. It is also supposed that an external force \(\underset \tilde{} F\), dependent on the position vector \(\underset \tilde{} r\) and on the velocity \(\underset \tilde{} v\) but not on time t, acts on each N particle and that the collision of N particles amongst themselves can be neglected compared with those of the M particles. The one-particle distribution function f(\(\underset \tilde{} r,\underset \tilde{} v,t)\) for N particles satisfies the linear Maxwell-Boltzmann equation. The author discusses the critical value problem for this equation in the Banach space L' using the theory of linear semigroups, under the following assumptions: (1) the external force \(\underset \tilde{} F\) is defined on the whole phase space and the mapping F: \(R^ 6\to \tilde R^ 3\) is Lipschitz continuous, (2) the collision frequency \(\nu\) also depends on the state variables and the function \(\nu\) : \(R^ 6\to R\) is measurable and non-negative, (3) \(\nu\) is either bounded or Lipschitz continuous, (4) the collision kernel k is measurable and a non-negative real valued function in \(R^ 9\). For each (\(\underset \tilde{} r,\underset \tilde{} v)\in R^ 6\) it satisfies the condition \[ \int_{R^ 3}k(\underset \tilde{} r,\underset \tilde{} v',v)dv'=\nu (\underset \tilde{} r,\underset \tilde{} v). \]
scientific article; zbMATH DE number 4069457
scientific article