Multiphase saturation equations, change of type and inaccessible regions

We identify a class of flux functions which give rise to conservation laws which are hyperbolic except along a codimension one subspace of state space. We show that a number of systems modeling porous medium flow can be regarded as perturbations of such systems, and describe the phenomenon of change of type for these perturbations. We also discuss a property of solutions of such systems, the existence of inaccessible regions - subsets of state space which appear to be avoided by solutions

Deterministic mathematical modelling for cancer chronotherapeutics: cell population dynamics and treatment optimisation

Chronotherapeutics has been designed and used for more than twenty years as an effective treatment against cancer by a few teams around the world, among whom one of the first is Francis Lévi's at Paul-Brousse hospital (Villejuif, France), in application of circadian clock physiology to determine best infusion times within the 24-hour span for anticancer drug delivery. Mathematical models have been called in the last ten years to give a rational basis to such optimised treatments, for use in the laboratory and ultimately in the clinic. While actual clinical applications of the theoretical optimisation principles found have remained elusive so far to improve chronotherapeutic treatments in use, mathematical models provide proofs of concepts and tracks to be explored experimentally, to progress from theory to bedside. Starting from a simple ordinary differential equation model that allowed setting and numerically solving a drug delivery optimisation problem with toxicity constraints, this modelling enterprise has been extended to represent the division cycle in proliferating cell populations with different molecular targets, to allow for the representation of anticancer drug combinations that are used in clinical oncology. The main point to be made precise in such a therapeutic optimisation problem is to establish, here in the frame of circadian chronobiology, physiologically based differences between healthy and cancer cell populations in their responses to drugs. To this aim, clear biological evidence at the molecular level is still lacking, so that, starting from indirect observations at the experimental and clinical levels and from theoretical considerations on the model, speculations have been made, that will be exposed in this review of cancer chronotherapeutics models with the corresponding optimisation problems and their numerical solutions, to represent these differences between the two cell populations, with regard to circadian clock control

Development of singularities in Riemann Invariants

Shocks form in finite time in systems of quasilinear hyperbolic equations in one space variable which are genuinely nonlinear. The authors write down a simple geometric construction for systems of two equations, and use it to obtain a priori estimates for the growth of the derivatives. They also find realistic bounds on the maximum and minimum time of existence of smooth solutions for large amplitude waves in a model system of an unusual type

Shock stability in systems that change type. Final grant report to the Department of Energy

The aim of the original project was to investigate systems of conservation laws that change type. Progress was made on this problem. During the last period of the grant, the author began an investigation of a multidimensional system related to Mach reflection which goes beyond the original work proposed. This has been fruitful direction in which to apply expertise on change of type. Some basic theoretical results have been found

A Geometric Study Of Shocks In Equations That Change Type

In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, holds, where now the real part of a complex characteristic replaces the characteristic speed itself. We test this criterion by a nonlinear inviscid perturbation. We prove that the perturbed Cauchy problem in the elliptic region has a solution for a uniform time if the data lie in a suitable class of analytic functions and show that under small perturbations of the data a perturbed shock and a perturbed solution in the hyperbolic region exist, also for a uniform time. © 1994 Plenum Publishing Corporation.6335139

The sonic line as a free boundary

We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be solved with the same technique. We identify a class of such problems

Existence and uniqueness results for the continuity equation and applications to the chromatography system

We discuss some new well-posedness results for the continuity equation in arbitrary space dimension and we then illustrate applications to a system of conservation laws in one space dimension known as the chromatography system. In the last section, we discuss some related open problems