Section: Research Program

Design of complex models

Project team positioning

The originality of our work is the quantitative description of propagation phenomena accounting for several time and spatial scales. Here, propagation has to be understood in a broad sense. This includes propagation of invasive species, chemotactic waves of bacteira, evoluation of age structures populations ... Our main objectives are the quantitative calculation of macroscopic quantities as the rate of propagation, and microscopic distributions at the edge and the back of the front. These are essential features of propagation which are intimately linked in the long time dynamics.

Recent results

• Mixed evolution - propagation models:

  Vincent Calvez is studying propagation phenomena at the mesoscale, including travelling waves, accelerating fronts, evolutionary adaptation of a population to a changing environment. The common feature between these projects is the strong heterogeneity inside the propagating front. It is a great mathematical challenge to be able to keep track of this heterogeneity throughout the mathematical analysis.

  This research is structured in several axes, which all belong to the ERC starting grant MESOPROBIO (2015-2020).

  • Modeling and data analysis of species' invasion:

    We are currently developing two directions of research on the case study of cane toads invasion in Northern Australia. The first direction is based on a dataset provided by Australian group of biologists. They recorded individual trajectories of individuals at a fixed location during ten consecutive years (2005-2015). We are using our expertise about waves of expansion in kinetic transport equations in order to calibrate some dedicated mesoscopic models on data. So far we experienced some difficulties because any approach severely underestimate the real speed of expansion. We are revisiting the problem based on this apparent counter-intuitive results. This was the purpose of Nils Caillerie's PhD thesis (defended July 2017).

    The second direction is more theoretical. Namely we are seeking the true rate of expansion of a minimal model for wave front acceleration inspired from the cane toad study. This mathematical question appeared to much more difficult than expected. We are attacking it using a mixture of theoretical and numerical computations (in collaboration with T. Dumont, C. Henderson, S. Mirrahimi and O. Turanova).

  • Analysis of subdiffusive processes:

    H. Berry, V. Calvez and T. Lepoutre co-supervised the PhD of Alvaro Mateos Gonzalez. He studied large scale asymptotics of age-structured sub-diffusive models.

  • Concentration waves of chemotactic bacteria at the mesoscale:

    Vincent Calvez achieved a long standing goal in 2015, by proving the existence of traveling wave in a coupled kinetic/parabolic system of equations modeling bacteria chemotaxis. This project was grounded on biological experiments. There was several parallel sub-projects, including the PhD thesis of Emeric Bouin, and Nils Caillerie, as well as Hélène Hivert's post-doctoral project. VC is also collaborating with Laurent Gosse and Monika Twarogowska on well balanced schemes for kinetic traveling waves.

    This project led to new types of non-local Hamilton-Jacobi equations, including a collaboration with E. Bouin, G. Nadin and E. Grenier.

  • Adaptation of a population to a changing environment:

    Vincent Calvez moved recently to evolutionary biology, by some collaboration with biologists at Montpellier. The goal is to analyse quantitative genetics model using modern tools of PDE analysis, in particular WKB expansions, as initiated by Perthame and co-authors in 2005. We made two breakthroughs: firstly, we obtained quantitative results for age-structured population models, which add a level of complexity (the population is described by a phenotypic trait and age of individuals). Secondly, we realized that WKB expansions, which are well designed for linear equations (here, asexual mode of reproduction), could be extended to some non linear equations, including some sexual mode of reproduction. This paves the way for new mathematical challenges, as the asymptotic analysis requires new tools, and new quantitative results in evolutionary biology.

    Vincent Calvez is currently at UBC, Vancouver, over the period 08/2017-03/2018, to consolidate the last project and initiate new collaborations there.

• Inviscid limit of Navier Stokes equations.

  The question of the behavior of solutions of Navier Stokes equations in a bounded domain as the viscosity goes to 0 is a classical and highly difficult open question in Fluid Mechanics. A small boundary layer, called Prandtl layer, appears near the boundary, which turns out to be unstable if the viscosity is small enough. The stability analysis of this boundary layer is highly technical and remained open since the first formal analysis in the 1940's by physicists like Orr, Sommerfeld, Tollmien, Schlichting or Lin. E. Grenier recently made a complete mathematical analysis of this spectral problem, in collaboration with T. Nguyen and Y. Guo. We rigorously proved that any shear layer is spectrally and linearly unstable if the viscosity is small enough, which is the first mathematical result in that field. We also get some preliminary nonlinear results. A book on this subject is in preparation, already accepted by Springer.

• Qualitative properties.

  Sometimes a beautiful mathematical question arises from modeling. For instance the question to know whether the apparent size of a tumor on a MRI can decrease long after a radiotherapy is linked to the following mathematical question: if $u$ is a solution of the classical KPP equation, does it satisfies ${\partial }_{t}u>0$ everywhere provided $t$ is large enough ? Despite its simplicity, this natural question appeared open and delicate. With the help of F. Hamel, E. Grenier managed to prove that this is true (to be published in JMPA) [13].

• Numerical analysis of complex fluids: the example of avalanches.

  This deals with the development of numerical schemes for viscoplastic materials (namely with Bingham or Herschell-Bulkley laws). Recently, with other colleagues, Paul Vigneaux finished the design of the first 2D well-balanced finite volume scheme for a shallow viscoplastic model. It is illustrated on the famous Taconnaz avalanche path in the Mont-Blanc, Chamonix, in the case of dense snow avalanches. The scheme deals with general Digital Elevation Model (DEM) topographies, wet/dry fronts and is designed to compute precisely the stopping state of avalanches, a crucial point of viscoplastic flows which are able to rigidify [21].

Collaborations

• Mixed evolution: N. Bournaveas (Edinburgh), B. Perthame (Paris 6), C. Schmeiser (Vienna), P.Silberzan (Institut Curie), S. Mirrahimi (Toulouse).

• Inviscid limit of Navier Stokes equations: Brown University (Y. Guo, B. Pausader), Penn State University (T. Nguyen), Orsay University (F. Rousset).

• Qualitative properties: F. Hamel (Marseille University).

• Numerical analysis of complex fluids: Enrique D. Fernandez - Nieto (Univ. de Sevilla, Spain), Jose Maria Gallardo (Univ. de Malaga, Spain).