Gibson 1st Floor SR

We present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the

space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. This is joint work with Carlos Kenig.

We consider the three dimensional Navier-Stokes equations with a large initial data and

we prove the existence of a global smooth solution. The main feature of the initial data

is that it varies slowly in the vertical direction and has a norm which blows up as the

small parameter goes to zero. In the language of geometrical optics, this type of

initial data can be seen as the ``ill prepared" case. Using analytical-type estimates

and the special structure of the nonlinear term of the equation we obtain the existence

of a global smooth solution generated by this large initial data. This talk is based on a

work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z.

Zhang.

I will discuss recent results on dispersive estimates for linear PDE's with time dependent coefficients. Then I will discuss how such

estimates can be used to study stability of nonlinear solitary waves and resonance phenomena.

According to the Ernst-Geroch reduction, in an axially symmetric stationary electrovac spacetime, the Einstein-Maxwell equations reduce to a harmonic map problem with singular boundary data. I will discuss the “regularity” of the reduced harmonic maps near the boundary and its implication on the regularity of the corresponding spacetimes.

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.

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 Schr\"{o}dinger 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 Schr\"{o}dinger equation. This is a joint

work with M. Palombaro and J. Rauch.

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.

When two inclusions (in a composite) get closer and their conductivities degenerate

to zero or infinity, the gradient of the solution to the

conductivity equation blows up in general. We show

that the solution to the conductivity equation can be decomposed

into two parts in an explicit form: one of them has a bounded

gradient and the gradient of the other part blows up. Using the

decomposition, we derive the best possible estimates for the blow-up

of the gradient. The decomposition theorem and estimates have an

important implication in computation of electrical field in

the presence of closely located inclusions.

The accuracy of the Nos-Hoover thermostat to sample the Gibbs measure depends on the

dynamics being ergodic. It has been observed for a long time that this dynamics is

actually not ergodic for some simple systems, such as the harmonic oscillator.

In this talk, we rigourously prove the non-ergodicity of the Nos-Hoover thermostat, for

the one-dimensional harmonic oscillator.

We will also show that, for some multidimensional systems, the averaged dynamics for the limit

of infinite thermostat "mass" has many invariants, thus giving

theoretical support for either non-ergodicity or slow ergodization.

Our numerical experiments for a two-dimensional central force problem

and the one-dimensional pendulum problem give evidence for

non-ergodicity.

We also present numerical experiments for the Nose-Hoover chain with

two thermostats applied to the one-dimensional harmonic

oscillator. These experiments seem to support the non-ergodicity of the

dynamics if the masses of the reservoirs are large enough and are

consistent with ergodicity for more moderate masses.

Joint work with Frederic Legoll and Richard Moeckel

The Cosserat brothers’ ingenuous and powerful idea (presented in several papers in the Comptes Rendus at the turn of the 20th century) of solving elasticity problems by considering the homogeneous Navier equations as an eigenvalue problem is presented. The theory was taken up by Mikhlin in the 1970’s who rigorously studied it in the context of spectral analysis of pde’s, and who also presented a representation theorem for the solution of the boundary-value problems of displacement and traction in elasticity as a convergent series of the ( orthogonal and complete in the Sobolev space H1) Cosserat eigenfunctions. The feature of this representation is that the dependence of the solution on geometry, material constants and loading is provided explicitly. Recent work by the author and co-workers obtains the set of eigenfunctions for the spherical shell and compares them with the Cosserat expressions, and further explores applications and a new variational principle. In cases that the loading is orthogonal to some of the eigenfunctions, the form of the solution can be greatly simplified. Applications, in addition to elasticity theory, include thermoelasticity, poroelesticity, thermo-viscoelasticity, and incompressible Stokes flow; several examples are presented, with comparisons to known solutions, or new solutions.