The instationary motion of a Navier-Stokes fluid through a vessel with an elastic cover

We study here the time-dependent movement of a fluid through a vessel having an elastic cover and inflow and outflow sections, the rest of the boundary being rigid and fixed. The two media interact with each other. The fluid domain is moving in time. For the elastic structure we use plate equations and in order to describe the behavior of the fluid we consider Navier-Stokes equations with prescribed pressures at the inflow and at the outflow sides of the vessel. These are nonstandard boundary conditions. We prove the existence of a solution for the coupled problem

Mathematical analysis of the time dependent motion of a fluid through a tube with flexible walls

We study the motion of a Stokes fluid through an elastic cylinder. The fluid is driven by a small time-dependent pressure drop between the outflow and the inflow ends of the tube. We consider small displacements of the elastic structure, thus the domains involved are not moving in time. We prove existence and uniqueness of a weak solution for this three dimensional fluid-elastic structure interaction problem

Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis

We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence our setting features two interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations in a two-dimensional setting.Comment: arXiv admin note: text overlap with arXiv:1512.0428

On a mathematical model for cancer invasion with repellent pH-taxis and nonlocal intraspecific interaction

Starting from a mesoscopic description of cell migration and intraspecific interactions we obtain by upscaling an effective reaction-difusion-taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions

A flux-limited model for glioma patterning with hypoxia-induced angiogenesis

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction-diffusion-taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced [34] description of glioma pseudopalisade formation, with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects