The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Resistance to chemotherapies, particularly to anticancer treatments, is an increasingmedical concern. Among the many mechanisms at work in cancers, one of the most importantis the selection of tumor cells expressing resistance genes or phenotypes. Motivated bythe theory of mutation-selection in adaptive evolution, we propose a model based on acontinuous variable that represents the expression level of a resistance gene (or genes,yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects ofchemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work bydemonstrating how qualitatively different actions of chemotherapeutic and cytostatictreatments may induce different levels of resistance. The mathematical interest of ourstudy is in the formalism of constrained Hamilton–Jacobi equations in the framework ofviscosity solutions. We derive the long-term temporal dynamics of the fittest traits inthe regime of small mutations. In the context of adaptive cancer management, we alsoanalyse whether an optimal drug level is better than the maximal tolerated dose.

The standard multilayer Saint-Venant system consists in introducing fluidlayers that are advected by the interfacial velocities. As a consequence there is no massexchanges between these layers and each layer is described by its height and its averagevelocity.Here we introduce another multilayer system with mass exchanges between the neighboringlayers where the unknowns are a total height of water and an average velocity per layer.We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy andhyperbolicity properties of the model. We also give a kinetic interpretation leading toeffective numerical schemes with positivity and energy properties. Numerical tests showthe versatility of the approach and its ability to compute recirculation cases with windforcing.

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turinginstability and the interpretation refers to adaptive evolution. By analogy with other formalismsused in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sumof Dirac masses) will happen in the limit of small mutations. In the present work we study thisasymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation.We prove the convergence analytically and illustrate it numerically. We also illustrate numericallyhow the constraint is related to the concentration points. We investigate numerically some featuresof these concentration points such as their weights and their numbers. We show analytically howthe constrained Hamilton-Jacobi gives the so-called canonical equation relating their motion withthe selection gradient. We illustrate this point numerically.

We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

We consider a simple model for the immune systemin which virus are able to undergo mutations and are in competitionwith leukocytes. These mutations are related to several other concepts which havebeen proposed in the literature like those of shape or ofvirulence – a continuous notion. For a given species, the system admits aglobally attractive critical point. We prove that mutations do not affect thispicture for small perturbations and under strong structural assumptions.Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the K-method of real interpolation.