Past Partial Differential Equations Seminar

Electronic structure calculations are commonly used to understand and predict the electronic, magnetic and optic properties of molecular systems and materials. They are also at the basis of ab initio molecular dynamics, the most reliable technique to investigate the atomic scale behavior of materials undergoing chemical reactions (oxidation, crack propagation, ...).

In the first part of my talk, I will briefly review the foundations of the density functional theory for electronic structure calculations. In the second part, I will present some recent achievements in the field, as well as open problems. I will focus in particular on the mathematical modelling of defects in crystalline materials.

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface

into $R^n$, $n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $W_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension $1$, we know $W_{3,1} = 8\pi$ . Joint work with Enrst Kuwert.

The talk will survey some recent and not so recent work on the

Hausdorff and box dimension of self-affine sets and related

attractors and repellers that arise in certain dynamical systems.

A classical problem in differential geometry is to deform the metric of a given manifold so that some of its curvatures become prescribed functions. Classical examples are the Uniformization problem for compact surfaces and the Yamabe problem for compact manifolds of dimension greater than two.
We address a similar problem for the so-called Q-curvature, which plays an important role in conformal geometry and is a natural higher order analogue of the Gauss curvature. The problem is tackled using a variational and Morse theoretical approach.

he study of polycrystals of shape-memory alloys and rigid-perfectly plastic materials gives rise to problems of nonlinear homogenization involving degenerate energies. We present a characterisation of the strain and stress fields for some classes of problems in plane strain and also for some three-dimensional situations. Consequences for shape-memory alloys and rigid-perfectly plastic materials are discussed through model problems. In particular we explore connections to previous conjectures characterizing those shape-memory polycrystals with non-trivial recoverable strain.

Y. Brenier, R. Natalini and M. Puel have considered a ``relaxation" of the Euler equations in R². After an approriate scaling, they have obtained the following hyperbolic version of the Navier-Stokes equations, which is similar to the hyperbolic version of the heat equation introduced by Cattaneo, $$\varepsilon u_{tt}^\varepsilon + u_t^\varepsilon -\Delta u^\varepsilon +P (u^\varepsilon \nabla u^\varepsilon) \, = \, Pf~, \quad (u^\varepsilon(.,0), u_t^\varepsilon(.,0)) \, = \, (u_0(.),u_1(.))~, \quad (1) $$ where $P$ is the classical Leray projector and $\varepsilon$ is a small, positive number. Under adequate hypotheses on the forcing term $f$, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1}({\bf R}^2) \times L^2({\bf R}^2))$ of (1), for large initial data $(u_0,u_1)$ in $H^{1}({\bf R}2) \times L^2({\bf R}2)$, provided that $\varepsilon>0$ is small enough, thus improving the global existence results of Brenier, Natalini and Puel (actually, we can work in less regular Hilbert spaces). The proof uses appropriate Strichartz estimates, combined with energy estimates. We also show that $(u^\varepsilon,u_t^\varepsilon)$ converges to $(v,v_t)$ on finite intervals of time $[t_0,t_1]$, $0 <+ \infty$, when $\varepsilon$ goes to $0$, where $v$ is the solution of the corresponding Navier-Stokes equations $$ v_t -\Delta v +P (v\nabla v) \, = \, Pf~, \quad v(.,0) \, = \, u_0~. \quad (2) $$ We also consider Equation (1) in the three-dimensional case. Here we expect global existence results for small data. Under appropriate assumptions on the forcing term, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1+\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3))$ of (1), for initial data $(u_0,u_1)$ in $H^{1 +\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3)$ (where $\delta >0 $ is a small positive number), provided that $\varepsilon > 0$ is small enough and that $u_0$ and $f$ satisfy a smallness condition. (Joint work with Marius Paicu)

In space dimension 2, it is well-known that the Smoluchowski-Poisson

system (also called the simplified or parabolic-elliptic Keller-Segel

chemotaxis model) exhibits the following phenomenon: there is a critical

mass above which all solutions blow up in finite time while all solutions

are global below that critical mass. We will investigate the case of the

critical mass along with the stability of self-similar solutions with

lower masses. We next consider a generalization to several space

dimensions which involves a nonlinear diffusion and show that a similar

phenomenon takes place but with some different features.

I will talk about recent work concerning the Onsager equation on metric

spaces. I will describe a framework for the study of equilibria of

melts of corpora -- bodies with finitely many

degrees of freedom, such as stick-and-ball models of molecules.

In the lecture, I am going to explain a connection between

local regularity theory for the Navier-Stokes equations

and Liouville type theorems for bounded ancient solutions to

these equations.

The Willmore Functional for surfaces has been introduced for the first time almost one century ago in the framework of conformal geometry (though it's one dimensional version already appears in thework of Daniel Bernouilli in the XVIII-th century). Maybe because of its simplicity and the depth of its mathematical relevance, it has since then played a significant role in various fields of sciences and technology such as cell biology, non-linear elasticity, general relativity...optical design...etc.

Critical points to the Willmore Functional are called Willmore Surfaces. They satisfy the so called Willmore Equations introduced originally by Gerhard Thomsen in 1923 . This equation, despite the elegance of it's formulation, is very inappropriate for dealing with analysis questions such as regularity, compactness...etc. We will present a new formulation of the Willmore Euler-Lagrange equation and explain how this formulation, together with the Integrability by compensation theory, permit to solve fundamental analysis questions regarding this functional, which were untill now totally open.

I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin &amp; Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, &amp; Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).

We compute the high frequency limit of the Hemholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case.

The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.

we present some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, under the no-slip boundary condition. This is a classical turbulence model, considered by von Neumann and Richtmeyer in the 50's, and by Smagorinski in the beginning of the 60's (for p= 3). The model was extended to other physical situations, and deeply studied from a mathematical point of view, by Ladyzhenskaya in the second half of the 60's. We consider the shear thickening case p&gt;2. We are interested in regularity results in Sobolev spaces, up to the boundary, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. In spite of the very rich literature on the subject, sharp regularity results up to the boundary are quite new.

Multi-well relaxation problem emerges e.g. in characterising effective properties of composites and in phase transformations. This is a nonlinear problem and one approach uses its reformulation in Fourier space, known in the theory of composites as Hashin-Shtrikman approach, adapted to nonlinear composites by Talbot and Willis. Characterisation of admissible mixtures, subjected to appropriate differential constraints, leads to a quasiconvexification problem. The latter is equivalently reformulated in the Fourier space as minimisation with respect to (extremal points of) H-measures of characteristic functions (Kohn), which in a sense separates the microgeometry of mixing from the differential constraints. For three-phase mixtures in 3D we obtain a full characterisation of certain extremal H-measures. This employs Muller's Haar wavelet expansion estimates in terms of Riesz transform to establish via the tools of harmonic analysis weak lower semicontinuity of certain functionals with rank-2 convex integrands. As a by-product, this allows to fully solve the problem of characterisation of quasiconvex hulls for three arbitrary divergence-free wells. We discuss the applicability of the results to problems with other kinematic constraints, and other generalisations. Joint work with Mariapia Palombaro, Leipzig.