Group analysis of the generalized Burnett equations

In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the classical Burnett equations these equations are well-posed and therefore can be used in applications. We consider the one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian and Lagrangian coordinates and perform the complete group analysis of these equations. In particular, this includes finding and analyzing admitted Lie groups. Our classifications of the Lie symmetries of the Navier-Stokes equations of compressible gas and generalized Burnett equations provide a basis for finding invariant solutions of these equations. We also consider representations of all invariant solutions. Some particular classes of invariant solutions are studied in more detail by both analytical and numerical methods.