We study the initial-boundary value problems and the corresponding stationary problems of the one-dimensional discrete Boltzmann equation in a bounded region. The boundary conditions considered are of mixed type and involve both the reflection and diffusion parts. It is shown that a unique solution to the initial-boundary value problem exists globally in time under the general situation that the reflection parts of both the boundary conditions do not increase the number of gas particles. Furthermore, it is proved that stationary solutions exist under the restriction that the reflection part of the boundary condition on one side really decreases the number of gas particles. This restriction plays an essential role in proving the existence result.
"Solutions to the discrete Boltzmann equation with general boundary conditions." J. Math. Soc. Japan 51 (3) 757 - 779, July, 1999. https://doi.org/10.2969/jmsj/05130757