Global well-posedness of a conservative relaxed cross diffusion system

We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_i)$ where the $u_i, i=1,...,I$ represent $I$ density-functions, $\tilde{u}$ is a spatially regularized form of $(u_1,...,u_I)$ and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous

Entropy, Duality and Cross Diffusion

This paper is devoted to the use of the entropy and duality methods for the existence theory of reaction-cross diffusion systems consisting of two equations, in any dimension of space. Those systems appear in population dynamics when the diffusion rates of individuals of two species depend on the concentration of individuals of the same species (self-diffusion), or of the other species (cross diffusion)

All optical cooling of 39^{39}K to Bose Einstein condensation

We report the all-optical production of Bose Einstein condensates (BEC) of $^{39}$K atoms. We directly load $3 \times 10^{7}$ atoms in a large volume optical dipole trap from gray molasses on the D1 transition. We then apply a small magnetic quadrupole field to polarize the sample before transferring the atoms in a tightly confining optical trap. Evaporative cooling is finally performed close to a Feshbach resonance to enhance the scattering length. Our setup allows to cross the BEC threshold with $3 \times 10^5$ atoms every 7s. As an illustration of the interest of the tunability of the interactions we study the expansion of Bose-Einstein condensates in the 1D to 3D crossover

Synchronisation and control of proliferation in cycling cell population models with age structure

International audienceWe present and analyse in this article a mathematical question with a biological origin, the theoretical treatment of which may have far-reaching implications in the practical treatment of cancers. Starting from biological and clinical observations on cancer cells, tumourbearing laboratory rodents, and patients with cancer, we ask from a theoretical biology viewpoint questions that may be transcribed, using physiologically based modelling of cell proliferation dynamics, into mathematical questions. We then show how recent fluorescence-based image modelling techniques performed at the single cell level in proliferating cell populations allow to identify model parameters and how this may be applied to investigate healthy and cancer cell populations. Finally, we show how this modelling approach allows us to design original optimisation methods for anticancer therapeutics, in particular chronotherapeutics, by controlling eigenvalues of the differential operators underlying the cell proliferation dynamics, in tumour and in healthy cell populations. We propose a numerical algorithm to implement these principles

Scaling limit of a discrete prion dynamics model

International audienceThis paper investigates the connection between discrete and continuous models describing prion proliferation. The scaling parameters are interpreted on biological grounds and we establish rigorous convergence statements. We also discuss, based on the asymptotic analysis, relevant boundary conditions that can be used to complete the continuous model

The albedo-color diversity of transneptunian objects

We analyze albedo data obtained using the Herschel Space Observatory that reveal the existence of two distinct types of surface among midsized transneptunian objects. A color-albedo diagram shows two large clusters of objects, one redder and higher albedo and another darker and more neutrally colored. Crucially, all objects in our sample located in dynamically stable orbits within the classical Kuiper belt region and beyond are confined to the bright-red group, implying a compositional link. Those objects are believed to have formed further from the Sun than the dark-neutral bodies. This color-albedo separation is evidence for a compositional discontinuity in the young solar system.Comment: 16 pages, 4 figures, 1 table, published in ApJL (12 August 2014), The Astrophysical Journal (2014), vol. 793, L

Conservative cross diffusions and pattern formation through relaxation

Analyse mathématique et numérique de sytèmes "cross-diffusion"This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space-time Lebesgue spaces for a large class of 'diffusion' matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability can occur. Numerical simulations show that patterns arise indeed inthis range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory

Conservative cross diffusions and pattern formation through relaxation

Analyse mathématique et numérique de sytèmes "cross-diffusion"This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space-time Lebesgue spaces for a large class of 'diffusion' matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability can occur. Numerical simulations show that patterns arise indeed inthis range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory

Circadian rhythm and cell population growth

Molecular circadian clocks, that are found in all nucleated cells of mammals, are known to dictate rhythms of approximately 24 hours (circa diem) to many physiological processes. This includes metabolism (e.g., temperature, hormonal blood levels) and cell proliferation. It has been observed in tumor-bearing laboratory rodents that a severe disruption of these physiological rhythms results in accelerated tumor growth. The question of accurately representing the control exerted by circadian clocks on healthy and tumour tissue proliferation to explain this phenomenon has given rise to mathematical developments, which we review. The main goal of these previous works was to examine the influence of a periodic control on the cell division cycle in physiologically structured cell populations, comparing the effects of periodic control with no control, and of different periodic controls between them. We state here a general convexity result that may give a theoretical justification to the concept of cancer chronotherapeutics. Our result also leads us to hypothesize that the above mentioned effect of disruption of circadian rhythms on tumor growth enhancement is indirect, that, is this enhancement is likely to result from the weakening of healthy tissue that are at work fighting tumor growth