Past Partial Differential Equations Seminar

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.

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.

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.

Some results of R.Harvey and B.Lawson on the Dirichlet problem for a class of fully nonlinear elliptic equations will be presented.

No background is required; the talk will be expository.

For incompressible Navier-Stokes equations in a bounded domain, I will

first present a formula for the pressure that involves the commutator

of the Laplacian and Leray-Helmholtz projection operators. This

commutator and hence the pressure is strictly dominated by the viscous

term at leading order. This leads to a well-posed and computationally

congenial unconstrained formulation for the Navier-Stokes equations.

Based on this pressure formulation, we will present a new

understanding and design principle for third-order stable projection

methods. Finally, we will discuss the delicate inf-sup stability issue

for these classes of methods. This is joint work with Bob Pego and Jie Liu.

This work in collaboration with J. Casado-Díaz deals with the asymptotic behaviour of two-dimensional linear conduction problems for which the sequence of conductivity matrices is bounded from below but not necessarily from above. On the one hand, we prove an extension in dimension two of the classical div-curl lemma, which allows us to derive a H-convergence type result for any L¹-bounded sequence of conductivity matrices. On the other hand, we obtain a uniform convergence result satisfied by the minimisers of a sequence of two-dimensional diffusion energies. This implies the closure for the L²-strong topology of $\Gamma$-convergence of the sets of equicoercive diffusion energies without assuming any bound from above. A few counter-examples in dimension three, connected with the appearance of non-local effects, show the specificity of dimension two in the two previous compactness results.

We consider problems of elastic plastic deformation with isotropic and  kinematic hardening.

A dual formulation with stresses as principal variables is used. 

We obtain several results on Sobolev space regularity of the stresses  and strains.

In particular, we obtain the existence of a full derivative of the  stress tensor up to the boundary of the basic domain.

Finally, we present an outlook for obtaining further regularity  results in connection with general nonlinear evolution problems.

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.

One of the intrinsic methods in elasticity is to consider the Cauchy-Green tensor as the primary unknown, instead of the deformation realizing this tensor, as in the classical approach.

Then one can ask whether it is possible to recover the deformation from its Cauchy-Green tensor. From a differential geometry viewpoint, this amounts to finding an isometric immersion of a Riemannian manifold into the Euclidian space of the same dimension, say d. It is well known that this is possible, at least locally, if and only if the Riemann curvature tensor vanishes. However, the classical results assume at least a C2 regularity for the Cauchy-Green tensor (a.k.a. the metric tensor). From an elasticity theory viewpoint, weaker regularity assumptions on the data would be suitable.

We generalize this classical result under the hypothesis that the Cauchy-Green tensor is only of class W^{1,p} for some p&gt;d.

The proof is based on a general result of PDE concerning the solvability and stability of a system of first order partial differential equations with L^p coefficients.

We discuss a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition a thermodynamically admissible flow rule for F_p is described involving as plastic gradient Curl F_p. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependenceon the plastic metric C_p=F_p^T F_p.
The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied.
Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: d_tp.τ=0 on the microscopically hard boundary Γ_D⊂∂Ω and [Curlp].τ=0 on the microscopically free boundary ∂Ω\Γ_D, where τ are the tangential vectors at the boundary ∂Ω. Moreover, I show that a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasivariational inequality with symmetric and coercive bilinear form, following the abstract framework proposed by Reddy. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity.

The talk will discuss the modeling of multi-phase fluid membranes surrounded by a viscous fluid with a particular emphasis on the inner flow--the motion of the lipids within the membrane surface.

For this purpose, we obtain the equations of motion of a two-dimensional viscous fluid flowing on a curved surface that evolves in time. These equations are derived from the balance laws of continuum mechanics, and a geometric form of these equations is obtained. We apply these equations to the formation of a protruding bud in a fluid membrane, as a model problem for physiological processes on the cell wall. We discuss the time and length scales that set different regimes in which the outer or inner flow are the predominant dissipative mechanism, and curvature elasticity or line tension dominate as driving forces. We compare the resulting evolution equations for the shape of the vesicle when curvature energy and internal viscous drag are operative with other flows of the curvature energy considered in the literature, e.g. the $L_2$ flow of the Willmore energy. We show through a simple example (an area constrained spherical cap vesicle) that the time evolutions predicted by these two models are radically different.

Joint work with Antonio DeSimone, SISSA, Italy.

We study Onsager’s model of isotropic–nematic phase transition with orientation parameter on a circle and sphere. We show the axial symmetry and derive explicit formulae for all critical points. Using the information about critical points we investigate a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the free energy is a functional of spatially-dependent orientation distribution, however, the spatial variation effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. As a particular example we consider a simplified model with orientation parameter on a circle and illustrate its relation to complex Ginzburg-Landau theory.