In this contribution we give an overview of various lifting strategies for Boltzmann models. A lifting operator finds for given macroscopic variables the corresponding distribution functions, microscopic variables of the Boltzmann models. There are several applications where macroscopic variables need to be mapped to these distribution functions. For example, starting a Boltzmann model from given macroscopic initial conditions includes some arbitrariness. This initialization procedure then requires a lifting operator. Another application of a lifting operator is found in coupled Boltzmann and macroscopic partial differential equation models.

Hybrid simulations divide the spatial domain into subdomains. These calculations replace, locally in the domain, the kinetic models with a partial differential equation in the regions where it is justified.

The mathematical question is how to couple these heterogeneous models in a correct way. Because a kinetic model has typically more variables than a model based on a partial differential equation, at each interface we are faced with a missing data problem. The lifting operator provides the correct boundary conditions for the Boltzmann domain at the interfaces.

We report on different lifting operators and focus on some numerical results of various strategies like the Chapman-Enskog expansion, the Constrained Runs algorithm and the numerical Chapman-Enskog expansion.