Incompressible limit of mechanical model of tumor growth with viscosity

Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case.Comment: 17 page

A simple derivation of BV bounds for inhomogeneous relaxation systems

We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence. We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general

Variational Ansatz for an Abelian to non-Abelian Topological Phase Transition in ν=1/2+1/2\nu = 1/2 + 1/2 Bilayers

We propose a one-parameter variational ansatz to describe the tunneling-driven Abelian to non-Abelian transition in bosonic $\nu=1/2+1/2$ fractional quantum Hall bilayers. This ansatz, based on exact matrix product states, captures the low-energy physics all along the transition and allows to probe its characteristic features. The transition is continuous, characterized by the decoupling of antisymmetric degrees of freedom. We futhermore determine the tunneling strength above which non-Abelian statistics should be observed experimentally. Finally, we propose to engineer the inter-layer tunneling to create an interface trapping a neutral chiral Majorana. We microscopically characterize such an interface using a slightly modified model wavefunction.Comment: 5 pages, 4 Figures and Supplementary Materials. Comments are welcome

Transversal instability for the thermodiffusive reaction-diffusion system

The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two by two reaction-diffusion systems. Our interest is the extension of the Fisher/KPP equation to a two species reaction which represents reactant concentration and temperature when used for flame propagation, bacterial population and nutrient concentration when used in biology. We study in which circumstances instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, that means a one dimensional wave propagating in two space dimensions. Then, explicit analytical formulas give explicitely the range of paramaters for instability.Comment: 13 page

Radiation stability of EUV multilayer mirrors

This thesis describes the development and characterization of high thermal and radiation stable Si-based multilayer mirrors for their application in extreme ultraviolet lithographic steppers. Although EUV Lithography (EUVL) is an optical lithography, the reduced wavelength introduces new challenges because all materials are absorbing at 13.5 nm: The application of multilayer mirrors in EUVL requires not only the highest possible normal-incidence reflectivity but also a long-term thermal and radiation stability at operating temperatures. This requirement is most important in the case of the collector mirror of the illumination system close to the EUV source where a short-time decrease in reflectivity is most likely

Radiation stability of EUV multilayer mirrors

This thesis describes the development and characterization of high thermal and radiation stable Si-based multilayer mirrors for their application in extreme ultraviolet lithographic steppers. Although EUV Lithography (EUVL) is an optical lithography, the reduced wavelength introduces new challenges because all materials are absorbing at 13.5 nm: The application of multilayer mirrors in EUVL requires not only the highest possible normal-incidence reflectivity but also a long-term thermal and radiation stability at operating temperatures. This requirement is most important in the case of the collector mirror of the illumination system close to the EUV source where a short-time decrease in reflectivity is most likely