Séminaire EDP-Analyse ICJ

Journée “Jeunes analystes et modélisateurs lyonnais”.

by Billel Guelmame (UMPA, ENS Lyon.) , Laurent Lafleche (ICJ - Université Claude Bernard Lyon 1) , Maria Eugenia Martinez (CMM) , Michèle Romanos (Institut de Mathématiques de Toulouse (UT3))

Fokko (ICJ)




14h00. Michèle Romanos : "Multi-tissue viscous models for tissue growth: incompressible limit, qualitative behaviour at the limit, and applications to the elongation of the vertebrate embryo."

14h45. Maria Eugenia Martinez : " The soliton problem for the Zakharov water waves system with a slowly varying bottom."

15h30. Pause.

15h45. Laurent Laflèche : "On Semiclassical Sobolev inequalities."

16h30. Billel Guelmame : "On some regularized nonlinear hyperbolic equations."



Title: Multi-tissue viscous models for tissue growth: incompressible limit, qualitative behaviour at the limit, and applications to the elongation of the vertebrate embryo.

Abstract: during vertebrate embryo elongation, neural and muscular tissues grow in contact while remaining segregated, and the live imaging of these tissues reveals cellular turbulent behavior. To understand such behaviors, we introduce two 2D mechanical models modeling the evolution of two viscous tissues in contact. Their main property is to model the swirling cell motions while keeping the tissues segregated, as observed during embryonic development. Segregation is encoded differently in the two models: by passive or active segregation (based on a mechanical repulsion pressure). We formally compute the incompressible limits of the two models, and obtain strictly segregated solutions. The two models thus obtained are compared and a well-posedness and regularity analysis is conducted. Two striking features in the active segregation model are revealed: the persistence of the repulsion pressure at the limit (ghost effect) and a pressure jump at the tissues' boundaries. The results are supported by numerical simulations in 2D and confronted to the biological data. Inspired by these models, we exhibit a final model which incorporates additional biological terms such as the addition of new cells into the tissues. We calibrate this model using the biological data at hand, and simulate the elongation of the vertebrate embryo. Interesting biological hypotheses arise from the numerical exploration of the model parameters, which we then confirm experimentally on quail embryos.

Title: The soliton problem for the Zakharov water waves system with a slowly varying bottom.


Abtract: Zakharov water waves arises as a free surface model for an irrotational and incompressible fluid under the influence of gravity. Such fluid is considered in a domain with rigid bottom (described as ha(x)) and a free surface. When considering the pressure over the surface, Amick-Kirchgässner proved the existence of solitary waves Qc (solutions that maintain its shape as they travel in time) of speed c for the flat-bottom case (a=1). 

In this talk, we are interested in the analysis of the behavior of the solitary wave solution of the flat-bottom problem when the bottom actually presents a (slight) change at some point. We construct a solution to the Zakharov water waves system with non-flat bottom that is time assympotic (as time t tends to - infinity) to the Amick-Kirchgässner soliton Q_c.



Title: On Semiclassical Sobolev inequalities.

Abstract: in the context of combined mean-field and semiclassical limits, such as the limit from the N-body Schrödinger equation to the Hartree--Fock and Vlasov equations, it is useful to obtain inequalities uniform in the Planck constant and the number of particles. It is therefore important to obtain analogous tool and inequalities in the context of quantum mechanics, such as operator versions of Wasserstein, Lebesgue and Sobolev distances, and the corresponding classical inequalities.
In this talk, I will present the quantum version of the phase space Sobolev spaces, as well as the corresponding Sobolev inequalities, other related inequalities and some applications to quantum optimal transport, and mean-field and semiclassical limits.


Title: On some regularized nonlinear hyperbolic equations.

Abstract: it is known that the solutions of nonlinear hyperbolic partial differential equations develop discontinuous shocks in finite time even with smooth initial data. Those shock are problematic in the theoretical study and in the numerical computations. To avoid these shocks, many regularizations have been studied in the literature. For example, adding diffusion and/or dispersion to the equation. In this talk, we present and study some non-diffusive and non-dispersive regularizations of the Burgers equation and the barotropic Euler equations that have similar properties as the classical equations.