On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces.

*(English)*Zbl 1083.82530Summary: This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded body \(D\) with general boundary conditions. First, mild \(L^1\)-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with an \(H\)-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weak \(L^ 1\)-solutions are proved by compactness, using the \(H\)-theorem from the cutoff case. Finally, an \(H\)-theorem is given also for the infinite-range case.

##### MSC:

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

##### Keywords:

linear Boltzmann equation; transport equation; initial boundary value problem; boundary conditions; entropy function; \(H\)-functional; detailed balance relation; mild \(L^1\)-solution; higher moments; infinite-range forces
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\textit{R. Pettersson}, J. Stat. Phys. 59, No. 1--2, 403--440 (1990; Zbl 1083.82530)

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