Forthcoming events in this series
We consider the motion of a viscous incompressible fluid described by
the Navier-Stokes equations in a bounded cylinder with slip boundary
conditions. Assuming that $L_2$ norms of the derivative of the initial
velocity and the external force with respect to the variable along the
axis of the cylinder are sufficiently small we are able to prove long
time existence of regular solutions. By the regular solutions we mean
that velocity belongs to $W^{2,1}_2 (Dx(0,T))$ and gradient of pressure
to $L_2(Dx(0,T))$. To show global existence we prolong the local solution
with sufficiently large T step by step in time up to infinity. For this purpose
we need that $L_2(D)$ norms of the external force and derivative
of the external force in the direction along the axis of the cylinder
vanish with time exponentially.
Next we consider the inflow-outflow problem. We assume that the normal
component of velocity is nonvanishing on the parts of the boundary which
are perpendicular to the axis of the cylinder. We obtain the energy type
estimate by using the Hopf function. Next the existence of weak solutions is
proved.
Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1
{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.
We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.
Due to the scaling properties of the Navier-Stokes equations,
homogeneous initial data may lead to forward self-similar solutions.
When the initial data is small enough, it is well known that the
formalism of mild solutions (through the Picard-Duhamel formula) give
such self-similar solutions. We shall discuss the issue of large initial
data, where we can only prove the existence of weak solutions; those
solutions may lack self-similarity, due to the fact that we have no
results about uniqueness for such weak solutions. We study some tools
which may be useful to get a better understanding of those weak solutions.
In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion.
\textbf{Theorem 1}. Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain (open connected set) $\Omega\subset\mathbb{R}^2$. Suppose
$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$
Then $\operatorname{meas}\nabla v(\Omega)=0$.
Here $\operatorname{Int}E$ is the interior of ${E}$, $\operatorname{meas} E$ is the Lebesgue measure of ${E}$. Theorem 1 is a straight consequence of the following two results.
\textbf{ Theorem 2 [2]}. Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb{R}^2$. Suppose (1) is fulfilled. Then the graph of $v$ is a normal developing surface.
Recall that a $C^1$-smooth manifold $S\subset\mathbb{R}^3$ is called a normal developing surface [3] if for any $x_0\in S$ there exists a straight segment $I\subset S$ (the point $x_0$ is an interior point of $I$) such that the tangent plane to $S$ is stationary along $I$.
\textbf{Theorem 3}. The spherical image of any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ has the area (the Lebesgue measure) zero.
Recall that the spherical image of a surface $S$ is the set $\{\mathbf{n}(x)\mid x\in S\}$, where $\mathbf{n}(x)$ is the unit normal vector to $S$ at the point~$x$. From Theorems 1--3 and the classical results of A.V. Pogorelov (see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a $C^1$-smooth surface $S\subset\mathbb{R}^3$ have no interior points. Then this surface is a surface of zero extrinsic curvature in the sense of Pogorelov.
\textbf{ Corollary 5}. Any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ is a surface of zero extrinsic curvature in the sense of Pogorelov.
\textbf{Theorem 6}. Let $K\subset\mathbb{R}^{2\times 2}$ be a compact set and the topological dimension of $K$ equals 1. Suppose there exists $\lambda> 0$ such that $\forall A,B\in K, \, \, |A-B|^2\le\lambda\det(A-B).$
Then for any Lipschitz mapping $f:\Omega\to\mathbb R^2$ on a domain $\Omega\subset\mathbb R^2$ such that $\nabla f(x)\in K$ a.e. the identity $\nabla f\equiv\operatorname{const}$ holds.
Many partial cases of Theorem 6 (for instance, when $K=SO(2)$ or $K$ is a segment) are well-known (see, for example, [5]).
The author was supported by the Russian Foundation for Basic Research (project no. 08-01-00531-a).
[1] {Korobkov M.\,V.,} {``Properties of the $C^1$-smooth functions whose gradient range has topological dimension~1,'' Dokl. Math., to appear.}
[2] {Korobkov M.\,V.} {``Properties of the $C^1$-smooth functions with nowhere dense gradient range,'' Siberian Math. J., \textbf{48,} No.~6, 1019--1028 (2007).}
[3] { Shefel${}'$ S.\,Z.,} {``$C^1$-Smooth isometric imbeddings,'' Siberian Math. J., \textbf{15,} No.~6, 972--987 (1974).}
[4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs. Vol. 35. Providence, R.I.: American Mathematical Society (AMS). VI (1973).}
[5] {M\"uller ~S.,} {Variational Models for Microstructure and Phase Transitions. Max-Planck-Institute for Mathematics in the Sciences. Leipzig (1998) (Lecture Notes, No.~2. http://www.mis.mpg.de/jump/publications.html).}
In this talk I will present recent developments of the obstacle type problems, with various applications ranging
from: Industry to Finance, local to nonlocal operators, and one to multi-phases.
The theory has evolved from a single equation
$$
\Delta u = \chi_{u > 0}, \qquad u \geq 0
$$
to embrace a more general (two-phase) form
$$
\Delta u = \lambda_+ \chi_{u>0} - \lambda_- \chi_{u0$.
The above problem changes drastically if one allows $\lambda_\pm$ to have the incorrect sign (that appears in composite membrane problem)!
In part of my talk I will focus on the simple {\it unstable} case
$$
\Delta u = - \chi_{u>0}
$$
and present very recent results (Andersson, Sh., Weiss) that classifies the set of singular points ($\{u=\nabla u =0\}$) for the above problem.
The techniques developed recently by our team also shows an unorthodox approach to such problems, as the classical technique fails.
At the end of my talk I will explain the technique in a heuristic way.
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.
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.
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.
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$.
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).
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.
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.
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.
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.
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 & 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, & 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>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.