Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 12/10/2009
17:00 		Benoît Perthame (Universite Pierre & Marie Curie) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Adaptive evolution and concentrations in parabolic PDE
Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variation of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait? We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that desribe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed. This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon, S. Mirahimmi and G. Barles.
Mon, 19/10/2009
17:00 		Grégoire Allaire (Ecole Polytechnique) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Diffractive behavior of the wave equation in periodic media
We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider inital data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation. This is a joint work with M. Palombaro and J. Rauch.
Mon, 26/10/2009
17:00 		Juan Velasquez (Universidad Complutense Madrid) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the construction of singular solutions for kinetic equations
In this talk I will present the rigorous construction of singular solutions for two kinetic models, namely, the Uehling-Uhlenbeck equation (also known as the quantum Boltzmann equation), and a class of homogeneous coagulation equations. The solutions obtained behave as power laws in some regions of the space of variables characterizing the particles. These solutions can be interpreted as describing particle fluxes towards or some regions from this space of variables. The construction of the solutions is made by means of a perturbative argument with respect to the linear problem. A key point in this construction is the analysis of the fundamental solution of a linearized problem that can be made by means of Wiener-Hopf transformation methods.
Mon, 02/11/2009
17:00 		Thomas Schmidt (Universität Erlangen-Nürnberg) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
A uniqueness result for graphs of least gradient
We investigate the minimization problem for the variational integral
$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$
in Dirichlet classes of vector-valued functions $ w $. It is well known that the existence of minimizers can be established if the problem is formulated in a generalized way in the space of functions of bounded variation. In this talk we will discuss a uniqueness theorem for these generalized minimizers. Actually, the theorem holds for a larger class of variational integrals with linear growth and was obtained in collaboration with Lisa Beck (SNS Pisa).
Mon, 09/11/2009
17:00 		Reza Pakzad (University of Pittsburgh) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Elastic models for growing tissues: scaling laws and derivation by Gamma convergence
Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria. We further discuss the scaling laws and $ \Gamma $-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $ W^{2,2} $ isometric immersion of a given $ 2 $d metric on a bounded domain into $ \mathbb R3 $.
Mon, 16/11/2009
17:00 		Mark Peletier (Technical University Eindhoven) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Probing the origin of Wasserstein gradient flows
The talk starts with the observation that many well-known systems of diffusive type can be written as Wasserstein gradient flows. The aim of the talk is to understand _why_ this is the case. We give an answer that uses a connection between diffusive PDE systems and systems of Brownian particles, and we show how the Wasserstein metric arises in this context. This is joint work with Johannes Zimmer, Nicolas Dirr, and Stefan Adams.
Mon, 23/11/2009
17:00 		Charles A. Stuart (Ecole Polytechnique Federale de Lausanne) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Planar modes in a stratified dielectric, existence and stability
We consider monochromatic planar electro-magnetic waves propagating through a nonlinear dielectric medium in the optical regime. Travelling waves are particularly simple solutions of this kind. Results on the existence of guided travelling waves will be reviewed. In the case of TE-modes, their stability will be discussed within the context of the paraxial approximation.
Mon, 30/11/2009
10:30 		Denis Serre (École Normale Supérieure de Lyon) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
L1-contraction in viscous scalar conser vation laws: Unconditional stability
Several dissipative scalar conservation laws share the properties of $ L1 $-contraction and maximum principle. Stability issues are naturally posed in terms of the $ L1 $-distance. It turns out that constants and travelling waves are asymptotically stable under zero-mass initial disturbances. For this to happen, we do not need any assumption (smallness of the TW, regularity/smallness of the disturbance, tail asymptotics, non characteristicity, ...) The counterpart is the lack of a decay rate.

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