Forthcoming events in this series
11:30 Refreshments (tea, coffee and homemade biscuits)
12:00 Talks (main room)
13:15 Buffet Style Lunch (incl. tea, coffee and homemade cakes)
15:00 End
We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.
We will discuss recent results regarding generation and propagation of summability of moments to solution of the Boltzmann equation for variable hard potentials.
These estimates are in direct connection to the understanding of high energy tails and decay rates to equilibrium.
We discuss shock reflection problem for compressible gas dynamics, various patterns of reflected shocks, and von Neumann conjectures on transition between regular and Mach reflections. Then
we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock
reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.
Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids.
In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the
relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same.
In such problems, the geometric nonlinearities of the fluid-structure interface
and the significant exchange in the energy between a moving fluid and a structure
require sophisticated ideas for the study of their solutions.
In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with
different mechanical characteristics.
No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers.
In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions
for a class of problems
describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid,
giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.}
A stable, modular, loosely coupled scheme will be presented, and an existence proof
showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed.
Our results reveal a new physical regularizing mechanism in
FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution.
All theoretical results will be illustrated with numerical examples.
This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).
I will discuss the well-posedness of a new nonlinear model for nematic
elastomers. The main novelty is that the Frank energy penalizes
spatial variations of the nematic director in the deformed, rather
than in the reference configuration, as it is natural in the case of
large deformations.
We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.
In this talk, I will introduce and analyse an elastic
surface energy recently introduced by G. Napoli and
L. Vergori to model thin films of nematic liquid crystals.
As it will be clear, the topology and the geometry of
the surface will play a fundamental role in understanding
the behavior of thin films of liquid crystals.
In particular, our results regards the existence of
minimizers, the existence of the gradient flow
of the energy and, in the case of an axisymmetric
toroidal particle, a detailed characterization of global and local minimizers.
This last item is supplemented with numerical experiments.
This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).
TBA
In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.
We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.
In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.
Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting,
it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function
for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.
This talk will focus on extremizers for
a family of Fourier restriction inequalities on planar curves. It turns
out that, depending on whether or not a certain geometric condition
related to the curvature is satisfied, extremizing sequences of
nonnegative functions may or may not have a subsequence which converges
to an extremizer. We hope to describe the method of proof, which is of
concentration compactness flavor, in some detail. Tools include bilinear
estimates, a variational calculation, a modification of the usual
method of stationary phase and several explicit computations.
We will present three different recent applications of cell motion in biology: (i) Movement of epithelial sheets and rosette formation, (ii) neural crest cell migrations, (iii) acid-mediated cancer cell invasion. While the talk will focus primarily on the biological application, it will be shown that all of these processes can be represented by reaction-diffusion equations with nonlinear diffusion term.
I will introduce the physical phenomena of transonic shocks, and review the progresses on related boundary value problems of the steady compressible Euler equations. Some Ideas/methods involved in the studies will be presented through specific examples. The talk is based upon joint works with my collaborators.
TBA
In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. 2-D and M-D potential stability will be presented.
This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.
We study the behavior of atomistic models under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to- continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. We present examples of mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.
Calculus of Variations for $L^{\infty}$ functionals has a successful history of 50 years, but until recently was restricted to the scalar case. Motivated by these developments, we have recently initiated the vector-valued case. In order to handle the complicated non-divergence PDE systems which arise as the analogue of the Euler-Lagrange equations, we have introduced a theory of "weak solutions" for general fully nonlinear PDE systems. This theory extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. A central ingredient is the discovery of a vectorial notion of extremum for maps which is a vectorial substitute of the "Maximum Principle Calculus" and allows to "pass derivatives to test maps" in a duality-free fashion. In this talk we will discuss some rudimentary aspects of these recent developments.
We prove existence of solution for evolutionary variational and quasivariational inequalities defined by a first order quasilinear operator and a variable convex set, characterized by a constraint on the absolute value of the gradient (which, in the quasi-variational case, depends on the solution itself). The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates.
Uniqueness of solution is proved for the variational inequality. We also obtain existence of stationary solutions, by studying the asymptotic behaviour in time. We shall illustrate a simple “sand pile” example in the variational case for the transport operator were the problem is equivalent to a two-obstacles problem and the solution stabilizes in finite time. Further remarks about these properties of the solution will be presented.This is a joint work with Lisa Santos.
If times allows, using similar techniques, I shall also present the existence, uniqueness and continuous dependence of solutions of a new class of evolution variational inequalities for incompressible thick fluids. These non-Newtonian fluids with a maximum admissible shear rate may be considered as a limit class of shear-thickening or dilatant fluids, in particular, as the power limit of Ostwald-deWaele fluids.
Dislocations are line defects in crystals, and were first posited as the carriers of plastic flow in crystals in the 1934 papers of Orowan, Polanyi and Taylor. Their hypothesis has since been experimentally verified, but many details of their behaviour remain unknown. In this talk, I present joint work with Christoph Ortner on an infinite lattice model in which screw dislocations are free to be created and annihilated. We show that configurations containing single geometrically necessary dislocations exist as global minimisers of a variational problem, and hence are globally stable equilibria amongst all finite energy perturbations.
We study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with time dependent inhomogeneous metric g. Based on a new approach for proving the decay of solutions of linear wave equations, we give several applications to nonlinear problems. In particular, we show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any particular stationary metric.
Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists that there is a positive cosmological constant. I shall review both sets of ideas within the framework of general relativity, and emphasise how the second set solves a problem posed by the first.
The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data.
\\ \\
This is a joint work with Andrew Stuart and Kody Law (Warwick)
Inspired by some recents developments in the theory of small-strain elastoplasticity, we
both revisit and generalize the formulation of the quasistatic evolutionary problem in
perfect plasticity for heterogeneous materials recently given by Francfort and Giacomini.
We show that their definition of the plastic dissipation measure is equivalent to an
abstract one, where it is defined as the supremum of the dualities between the deviatoric
parts of admissible stress fields and the plastic strains. By means of this abstract
definition, a viscoplastic approximation and variational techniques from the theory of
rate-independent processes give the existence of an evolution statisfying an energy-
dissipation balance and consequently Hill's maximum plastic work principle for an
abstract and very large class of yield conditions.
Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.
In this talk, I will present two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics. Using this result on the spreading of rarefaction waves, the rate of convergence for vanishing viscosity approximations to hyperbolic balance laws will also be established. The proof relies on error estimates that measure the interaction of waves using suitable Lyapunov functionals. If time allows, a further application of the recent developments in the theory of balance laws to differential geometry will be addressed.
We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.
The proof of several properties of solutions of hyperbolic systems of conservation laws in one space dimension (existence, stability, regularity) depends on the existence of a decreasing functional, controlling the nonlinear interactions of waves. In a special case (genuinely nonlinear systems) the interaction functional is quadratic, while in the general case it is cubic. Several attempts to prove the existence of a a quadratic functional also in the most general case have been done. I will present the approach we follow in order to prove this result, an some of its implication we hope to exploit.
\\ \\
Work in collaboration with Stefano Modena.
The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all
models of turbulence. In particular, K41 and corrections to it provide
estimates of small-scale quantities such as increments and energy spectrum
for a 3D turbulent flow. However, because of the well-known difficulties
involved in studying 3D turbulent flow, there are no rigorous results
confirming or infirming those predictions. Here, we consider a well-known
simplified model for 3D turbulence: Burgulence, or turbulence for the 1D
Burgers equation. In the space-periodic case with a stochastic white in
time and smooth in space forcing term, we give sharp estimates for
small-scale quantities such as increments and energy spectrum.
We will present a regularity result for degenerate elliptic equations in nondivergence form.
In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.
I will discuss the motion of screw dislocations in an elastic body under antiplane shear. In this setting, dislocations are viewed as points in a two-dimensional domain where the strain field fails to be a gradient. The motion is determined by the Peach-Koehler force and the slip-planes in the material. This leads to a system of discontinuous ODE, where the vector field depends on the solution to an elliptic PDE with Neumann data. We show short-time existence of solutions; we also have uniqueness for a restricted class of domains. In general, global solutions do not exist because of collisions.
{\bf This seminar is at ground floor!}
\\
An important question in geometry and analysis is to know when two $k-$forms
$f$ and $g$ are equivalent. The problem is therefore to find a map $\varphi$
such that%
\[
\varphi^{\ast}\left( g\right) =f.
\]
We will mostly discuss the symplectic case $k=2$ and the case of volume forms
$k=n.$ We will give some results when $3\leq k\leq n-2,$ the case $k=n-1$ will
also be considered.
\\
The results have been obtained in collaboration with S. Bandyopadhyay, G.
Csato and O. Kneuss and can be found, in part, in the book below.\bigskip
\\
\newline
Csato G., Dacorogna B. et Kneuss O., \emph{The pullback equation for
differential forms}, Birkha\"{u}ser, PNLDE Series, New York, \textbf{83} (2012).
We first provide a brief overview of some of the key properties of the space $\textrm{BV}(\Omega;\mathbb{R}^{N})$ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral
\[
F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,}
\]
where $\Omega\subset\mathbb{R}^{n}$ is open and bounded, and $f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is a continuous function satisfying the growth condition $0\leq f(\xi)\leq L(1+|\xi|^{r})$ for some exponent $r$. When $u\in\textrm{BV}(\Omega;\mathbb{R}^{N})$, we extend the definition of $F(u;\Omega)$ by introducing the functional
\[
\mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left|
\!\!\begin{array}{r}
(u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\
u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N})
\end{array} \right. \bigg\} \,\textrm{.}
\]
\noindent For $r\in [1,\frac{n}{n-1})$, we prove that $\mathscr{F}$ satisfies the lower bound
\[
\mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,}
\]
provided $f$ is quasiconvex, and the recession function $f_{\infty}$ ($:= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $r=1$; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.
In this talk, we will look at two non-linear wave equations in 2+1 dimensions, whose elliptic parts exhibit conformal invariance.
These equations have their origins in prescribing the Gaussian and mean curvatures respectively, and the goal is to understand well-posedness, blow-up and bubbling for these equations.
This is a joint work with Sagun Chanillo.
H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.\newline
In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.\newline
We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).
We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).
We establish the exact boundary controllability of nodal profile for general first order quasi linear hyperbolic systems in 1-D. And we apply the result in a tree-like network with general nonlinear boundary conditions and interface conditions. The basic principles of choosing the controls and getting the controllability are given.
