2017-07-14, Freistühler, Heinrich, Kotschote, Matthias

Aiming at an understanding of the nonlinear stability of moving fluidic phase boundaries, this project has provided (a) solution theories for three basic systems of nonlinear partial differential equations that model two-phase fluid flow as well as (b) results on the existence of corresponding traveling waves and the spectrum of the operators that result from linearizing the PDEs about these waves.

The three models are the Navier-Stokes-Korteweg (NSK), the Navier-Stokes-Allen-Cahn (NSAC), and the Navier-Stokes-Cahn-Hilliard (NSCH) equations. For compressible NSAC and NSCH, the theories of strong solutions obtained seem to be the first ones in the literature. For NSK, a new theory of strong solutions has been developed, which in particular provides an alternative to the ‘quasi-incompressible’ approach that Abels et al. pursue for NSCH in the case of two separately incompressible phases of different density.

While for NSK the existence of traveling waves representing phase boundaries was known before, corresponding results have been newly obtained for NSAC and NSCH. For all three contexts, the project has established the spectral stability of these traveling waves. Viscous shock waves providing useful heuristic inspiration, fluidic interfaces corresponding to phase boundaries turn out to have their own flavour.