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The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of R d is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

The simplest system in Levermore's moment hierarchy involving moments higher than second order is the five-moment closure. It is obtained by taking velocity moments of the one-dimensional Boltzmann equation under the assumption that the velocity distribution is a maximum-entropy function. The moment vectors for which a maximum-entropy function exist consequently make up the domain of definition of the system. The aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum-entropy problem. The space homogeneous case of the equation and numerical aspects are also addressed.

Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equation. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equation are obtained.

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