Strong solutions in the dynamical theory of compressible fluid mixtures
2009, Kotschote, Matthias, Zacher, Rico
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
2007, Gerisch, Alf, Kotschote, Matthias, Zacher, Rico
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 L p and Hölder regularity results for the diffusion equation.
Strong Well-Posedness for a Korteweg-Type Model for the Dynamics of a Compressible Non-Isothermal Fluid
2009, Kotschote, Matthias
The aim of this work is to prove an existence and uniqueness result for a non-isothermal model of capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985). The proof is essentially based on the maximal regularity result of the associated linear problem, where we can fall back upon useful results proved before. Using the maximal regularity the nonlinear problem can be approached by the contraction mapping principle.
Strong solutions for a compressible fluid model of Korteweg type
2008, Kotschote, Matthias
We prove existence and uniqueness of local strong solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985). This nonlinear problem is approached by proving maximal regularity for a related linear problem in order to formulate a fixed point equation, which is solved by the contraction mapping principle. Localising the linear problem leads to model problems in full and half space, which are treated by Dore–Venni Theory, real interpolation and H∞-calculus. For these steps, it is decisive to find conditions on the inhomogeneities that are necessary and sufficient.