Past Partial Differential Equations Seminar

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

The study of wave equations on black hole backgrounds provides important insights for the non-linear stability problem for black holes. I will illustrate this in the context of asymptotically anti de Sitter black holes and present both stability and instability results. In particular, I will outline the main ideas of recent work with J. Smulevici (Paris) establishing a logarithmic decay in time for solutions of the massive wave equation on Kerr-AdS black holes and proving that this slow decay rate is in fact sharp.

We discuss several singular limits for a scaled system of equations

(barotropic Navier-Stokes system), where the characteristic numbers become

small or ``infinite''. In particular, we focus on the situations relevant

in certain geophysical models with low Mach, large Rossby and large

Reynolds numbers. The limit system is rigorously identified in the

framework of weak solutions. The relative entropy inequality and careful

analysis of certain oscillatory integrals play crucial role.

The talk is based on the joint papers [{\it Bourgain J., Korobkov

M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik

(Crelles

Journal).

DOI: 10.1515/crelle-2013-0002] \ and \

[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:

arXiv:1302.0731, 4 Feb 2013]

We establish Luzin $N$ and Morse--Sard

properties for functions from the Sobolev space

$W^{n,1}(\mathbb R^n)$. Using these results we prove

that almost all level sets are finite disjoint unions of

$C^1$-smooth compact manifolds of dimension

$n-1$. These results remain valid also within

the larger space of functions of bounded variation

$BV_n(\mathbb R^n)$.

As an application, we study the nonhomogeneous boundary value problem

for the Navier--Stokes equations of steady motion of a viscous

incompressible fluid in arbitrary bounded multiply connected

plane or axially-symmetric spatial domains. We prove that this

problem has a solution under the sole necessary condition of zero total

flux through the boundary.

The problem was formulated by Jean Leray 80 years ago.

The proof of the main result uses Bernoulli's law

for a weak solution to the Euler equations based on the above-mentioned

Morse-Sard property for Sobolev functions.

Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:

diffusion, transport, and reaction of substances in the fluid and the solid phase,

mechanical interactions of the fluid and solid phase,

change of the mechanical properties of the solid phase by chemical reactions,

volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.

The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.

A time-invariant level surface is a (codimension one)

spatial surface on which, for every fixed time, the solution of an

evolution equation equals a constant (depending on the time). A

relevant and motivating case is that of the heat equation. The

occurrence of one or more time-invariant surfaces forces the solution

to have a certain degree of symmetry. In my talk, I shall present a

set of results on this theme and sketch the main ideas involved, that

intertwine a wide variety of old and new analytical and geometrical

techniques.

Self-gravitating elastic bodies provide models for extended

objects in general relativity. I will discuss constructions of static

and rotating self-gravitating bodies, as well as recent results on the

initial value problem for self-gravitating elastic bodies.

In this talk we focus on the incompressible Navier–Stokes equations with variable

density. The aim is to prove existence and uniqueness results in the case of a discontinuous

initial density (typically we are interested in discontinuity along an interface).

In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical

functional framework, under some smallness condition over the initial data,

In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded

and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though.

In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.

Defect measures have successfully been used, in a variety of

contexts, as a tool to quantify the lack of compactness of bounded

sequences of square-integrable functions due to concentration and

oscillation effects. In this talk we shall present some results on the

structure of the set of possible defect measures arising from sequences

of solutions to the linear Schrödinger equation on a compact manifold.

This is motivated by questions related to understanding the effect of

geometry on dynamical aspects of the Schrödinger flow, such as

dispersive effects and unique continuation.

It turns out that the answer to these questions depends strongly on

global properties of the geodesic flow on the manifold under

consideration: this will be illustrated by discussing with a certain

detail the examples of the the sphere and the (flat) torus.

Let ${T_1,...,T_l}$ be a collection of differential operators

with constant coefficients on the torus $\mathbb{T}^n$. Consider the

Banach space $X$ of functions $f$ on the torus for which all functions

$T_j f$, $j=1,...,l$, are continuous. The embeddability of $X$ into some

space $C(K)$ as a complemented subspace will be discussed. The main result

is as follows. Fix some pattern of mixed homogeneity and extract the

senior homogeneous parts (relative to the pattern chosen)

${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there

are two nonproportional operators among the $\tau_j$ (for at least one

homogeneity pattern), then $X$ is not isomorphic to a complemented

subspace of $C(K)$ for any compact space $K$.

The main ingredient of the proof is a new Sobolev-type embedding

theorem. It generalises the classical embedding of

${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that

now the integrability condition is imposed on certain linear combinations

of derivatives of different order of several functions rather than on the

first order derivatives of one function.

This is a joint work with D. Maksimov and D. Stolyarov.

The talk will discuss generalizations of the classical Hadamard jump  condition to general locally Lipschitz maps, and applications to
polycrystals. This is joint work with Carsten Carstensen.

I will report my work on rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, in which we obtain the local well-posedness result in $H^s$, $s$>$2$. The novelty of this approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\bf{g}$, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $\Box_{\bf{g} } \phi=0$ directly. If time allows, I will talk about my work in progress on the sharp results for the more general quasilinear wave equations by vector fields approach.

In this talk we show how degree N maps of the form $u_{N}(z) = \frac{z^{N}}{|z|^{N-1}}$ arise naturally as stationary points of functionals like the Dirichlet energy. We go on to show that the $u_{N}$ are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N. Meyers in the 60s, and another where the constraint $\det \nabla u = 1$ a.e. plays a prominent role.

Crystalline solids are descibed by a material manifold endowed

with a certain structure which we call crystalline. This is characterized by

a canonical 1-form, the integral of which on a closed curve in the material manifold

represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines

enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold

becomes a Lie group upon the choice of an identity element. In this talk crystalline solids

with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,

shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group

and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a

mapping from the material domain into Euclidean space. The equilibrium configurations correspond

to mappings which minimize a certain energy integral. The static problem is solved in the case of

a small density of dislocations.

This presentation concerns five topics in computational partial differential equations with the overall goals of reliable error control and efficient simulation.

The presentation is also an advertisement for nonstandard discretisations in linear and nonlinear Computational PDEs with surprising advantages over conforming

finite element schemes and the combination

of the two. The equivalence of various first-order methods is explained for the linear Poisson model problem with conforming

(CFEM), nonconforming (NC-FEM), and mixed finite element methods (MFEM) and others discontinuous Galerkin finite element (dGFEM). The Stokes

equations illustrate the NCFEM and the pseudo-stress MFEM and optimal convergence of adaptive mesh-refining as well as for guaranteed error bounds.

An optimal adaptive CFEM computation of elliptic eigenvalue

problems and the computation of guaranteed upper and lower eigenvalue bounds based on NCFEM. The obstacle problem and its guaranteed error

control follows another look due to D. Braess with guaranteed error bounds and their effectivity indices between 1 and 3. Some remarks on computational

microstructures with degenerate convex minimisation

problems conclude the presentation.

We review potential theoretic aspects of degenerate parabolic PDEs of p-Laplacian type.

Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat

equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic

functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined

as lower semicontinuous functions obeying the comparison principle. The superparabolic

functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss

their existence, structural, convergence and Sobolev space properties. We also consider the

definition and properties of the nonlinear parabolic capacity and show that the infinity set of a superparabolic

function is of zero capacity.

An intriguing, and largely open, question in mathematical fluid dynamics is whether solutions of the Navier-Stokes equations converge in some sense to a solution of the Euler equations in the zero viscosity limit. In fact this convergence could conceivably fail due to oscillations and concentrations occuring in the sequence.

In the late 1980s, R. DiPerna and A. Majda extended the classical concept of Young measure to obtain a notion of measure-valued solution of the Euler equations, which records precisely these oscillation and concentration effects. In this talk I will present a result recently obtained in joint work with L. Székelyhidi, which states that any such measure-valued solution is generated by a sequence of distributional solutions of the Euler equations.

The result is interesting from two different viewpoints: On the one hand, it emphasizes the huge flexibility of the concept of weak solution for Euler; on the other hand, it provides an example of a characterization theorem for Young measures in the tradition of D. Kinderlehrer and P. Pedregal where the differential constraint on the generating sequence does not satisfy the constant rank condition.

Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena.

We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces.

Recent results (starting with Scheffer and Shnirelman and continuing with De Lellis and Szekelhyhidi ) underline the importance of considering solutions of the incompressible Euler equations as limits of solutions of more physical examples like Navier-Stokes or Boltzmann.
I intend to discuss several examples illustrating this issue.

based on joint work with Charles Fefferman (Princeton) and Kevin Luli (Yale).

In 1932 Signorini formulated the first variational inequality as a model of an elastic body laying on a rigid surface. In this talk we will revisit this problem from the point of view of regularity theory.

We will sketch a proof of optimal regularity and regularity of the contact set. Similar result are known for scalar equations. The proofs for scalar equations are however based on maximum principles and thus not applicable to Signorini's problem which is modelled by a system of equations.

We consider the stationary flow of Prandtl-Eyring fluids in two

dimensions. This model is a good approximation of perfect plasticity.

The corresponding potential is only slightly super linear. Thus, many

severe problems arise in the existence theory of weak solutions. These

problems are overcome by use of a divergence free Lipschitz

truncation. As a second application of this technique, we generalize

the concept of almost harmonic functions to the Stokes system.

This talk will consist of a pure PDE part, and an applied part. The unifying topic is mean curvature flow (MCF), and particularly mean curvature flow starting at cones. This latter subject originates from the abstract consideration of uniqueness questions for flows in the presence of singularities. Recently, this theory has found applications in several quite different areas, and I will explain the connections with Harnack estimates (which I will explain from scratch) and also with the study of the dynamics of charged fluid droplets.

There are essentially no prerequisites. It would help to be familiar with basic submanifold geometry (e.g. second fundamental form) and intuition concerning the heat equation, but I will try to explain everything and give the talk at colloquium level.

Joint work with Sebastian Helmensdorfer.

In the first part, a variational model for composition of finitely many strongly elliptic

homogenous elastic materials in linear elasticity is considered. The notion of`universal coercivity' for the variational integrals is introduced which is independent of particular compositions of materials involved. Examples and counterexamples for universal coercivity are presented.

In the second part, some results of recent work with colleagues on image processing and feature extraction will be displayed.

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time ``splash'' singularity, wherein the evolving 2-D hypersurface intersects itself at a point. Our approach is based on the Lagrangian description of the free-boundary problem, combined with novel approximation scheme. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems. This is joint work with Daniel Coutand.

I will describe a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic

two-material composites with high contrast in a finite-strain setting. I will start by introducing the nonlinear

description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions. I shall then focus

on the loading regimes when the applied load is small or of order one in terms of the period of the composite structure.

I will show that this corresponds to the situation when the displacements on the stiff component are situated in the vicinity

of a rigid-body motion. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component

by its linearised version. As a main result, I derive (rigorously in the spirit of $\Gamma$-convergence) a limit functional

that allows to establish a precise two-scale expansion for minimising sequences. This is joint work with M. Cherdantsev and

S. Neukamm.

We describe several bifurcation properties corresponding to various classes of nonlinear elliptic equations The purpose of this talk is two-fold. First, it points out different competition effects between the terms involved in the equations. Second, it provides several non standard phenomena that occur according to the structure of the differential operator.

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.

We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. The equation was derived by Saxton as a model for liquid crystals. This equation shares many of the peculiarities of the Hunter–Saxton and the Camassa–Holm equations. In particular, the equation possesses two distinct classes of solutions denoted conservative and dissipative. In order to solve the Cauchy problem uniquely it is necessary to augment the equation properly. In this talk we describe how this is done for conservative solutions. The talk is based on joint work with X. Raynaud.

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).

The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.

The description of the problem is indeed extremely complex in its generality.

In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.

Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.

The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast.  The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. 

The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential.    

Please note that this seminar has been cancelled due to unforeseen circumstances.

In this talk, we will present some recent mathematical features around two-fluid models. Such systems may be encountoured for instance to model internal waves, violent aerated flows, oil-and-gas mixtures. Depending on the context, the models used for simulation may greatly differ. However averaged models share the same structure. Here, we address the question whether available mathematical results in the case of a single fluid governed by the compressible barotropic equations for single flow may be extended to two phase model and discuss derivations of well-known multi-fluid models from single fluid systems by homogeneization (assuming for instance highly oscillating density). We focus on existence of local existence of strong solutions, loss of hyperbolicity, global existence of weak solutions, invariant regions, Young measure characterization.

Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion\emph{ Poiseuille Free Boundary} PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \par\noindentBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, however\emph{surface waves of finite wavelengths are observed}, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.
Aim of this talk is the investigation of nonlinear stability ofPFB providing \emph{ a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane}, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz \emph{nonlinear stability}. We use\emph{a modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability}. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\par\noindentNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. \\ {\bf References}\\ [1]  M.K. Smith, \textit{The mechanism for the long-waveinstability in thin liquid films} J. Fluid Mech., \textbf{217},1990, pp.469-485.
\\ [2]  Benjamin T.B., \textit{Wave formation in laminar flow down aninclined plane}, J. Fluid Mech. \textbf{2}, 1957, 554-574.
\\ [3]  Yih Chia-Shun, \textit{Stability of liquid flow down aninclined plane}, Phys. Fluids, \textbf{6}, 1963, pp.321-334.
\\ [4] Padula M., {\it On nonlinear stability of MHD equilibriumfigures}, Advances in Math. Fluid Mech., 2009, 301-331.
\\ [5] Padula M., \textit{On nonlinear stability of linear pinch},Appl. Anal.  90 (1), 2011, pp. 159-192.