Strong solutions in the dynamical theory of compressible fluid mixtures

In this paper we investigate the compressible Navier-Stokes-Cahn-Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinowsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn-Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic-parabolic system. We establish a local existence and uniqueness result for strong solutions

Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology

We study the existence of classical solutions of a taxis-diffusion-reaction model for tumour-induced blood vessel growth. The model in its basic form has been proposed by Chaplain and Stuart (IMA J. Appl. Med. Biol. (10), 1993) and consists of one equation for the endothelial cell-density and another one for the concentration of tumour angiogenesis factor (TAF). Here we consider the special and interesting case that endothelial cells are immobile in the absence of TAF, i.e. vanishing cell motility. In this case the mathematical structure of the model changes significantly (from parabolic type to a mixed hyperbolic-parabolic type) and existence of solutions is by no means clear. We present conditions on the initial and boundary data which guarantee local existence, uniqueness and positivity of classical solutions of the problem. Our approach is based on the method of characteristics and relies on known maximal Lp and Hölder regularity results for the diffusion equation