Past OxPDE Lunchtime Seminar

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.

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This is a joint work with Andrew Stuart and Kody Law (Warwick)

In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation, \[ u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy. \]

First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $t$ goes to infinity.

Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.

In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.

Finally, we present some numerical experiments that illustrate our results.

This is a joint work with J. D. Rossi.

In 1985 Moffatt suggested that stationary flows of the 3D Euler equations with non-trivial topology could be obtained as the time-asymptotic limits of certain solutions of the equations of magnetohydrodynamics. Heuristic arguments also suggest that the same is true of the system
\[ -\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad \]
\[ B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u \] when $\eta=0$.

In this talk I will discuss well posedness of this coupled elliptic-parabolic equation in the two-dimensional case when $B(0)\in L^2$ and $\eta$ is positive.

Crucial to the analysis is a strengthened version of the 2D Ladyzhenskaya inequality: $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|\nabla f\|_{L^2}^{1/2}$, where $L^{2,\infty}$ is the weak $L^2$ space. I will also discuss the problems that arise in the case $\eta=0$.

This is joint work with David McCormick and Jose Rodrigo.

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.

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.

Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces.

The proof follows from the well-posedness of the homogeneous linearized problem and a basic a priori energy estimate, the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.

This is a joint work with Y. Trakhinin (Novosibirsk).

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.

[1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254:125–140, 2013.

[2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundaries. Submitted

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.

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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.

The new schedule will follow shortly

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!}

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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.

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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.

Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.

We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.

Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann's principle $S = k\log W$, does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold's `cat' map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

We study minimizers of the Ginzburg-Landau (GL) functional \[E_\epsilon(u):=\frac{1}{2}\int_A |\nabla u|^2 + \frac{1}{4\epsilon^2} \int_A(1-|u|^2)^2\] for a complex-valued order parameter $u$ (with no magnetic field). This functional is of fundamental importance in the theory of superconductivity and superuidity; the development of these theories led to three Nobel prizes. For a $2D$ domain $A$ with holes we consider “semistiff” boundary conditions: a Dirichlet condition for the modulus $|u|$, and a homogeneous Neumann condition for the phase $\phi = \mathrm{arg}(u)$. The principal

result of this work (with V. Rybalko) is a proof of the existence of stable local minimizers with vortices (global minimizers do not exist). These vortices are novel in that they approach the boundary and have bounded energy as $\epsilon\to0$.

In contrast, in the well-studied Dirichlet (“stiff”) problem for the GL PDE, the vortices remain distant from the boundary and their energy blows up as

$\epsilon\to 0$. Also, there are no stable minimizers to the homogeneous Neumann (“soft”) problem with vortices.

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Next, we discuss more recent results (with V. Rybalko and O. Misiats) on global minimizers of the full GL functional (with magnetic field) subject to semi-stiff boundary conditions. Here, we show the existence of global minimizers with vortices for both simply and doubly connected domains and describe the location of their vortices.

We present a variational model for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. We

allow for multiple damaged states. Moreover, unlike current formulations, the materials are allowed to be anisotropic and the

deformations are not restricted to anti-plane shear. The model can be formulated either energetically or through a strain threshold. We

explore the relationship between these formulations. This is joint work with Christopher Larsen, Worcester Polytechnic Institute.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also

be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.

In this seminar I will present two results regarding the uniqueness (and further properties) for the two-dimensional continuity equation

and the ordinary differential equation in the case when the vector field is bounded, divergence free and satisfies additional conditions on its distributional curl. Such settings appear in a very natural way in various situations, for instance when considering two-dimensional incompressible fluids. I will in particular describe the following two cases:\\

(1) The vector field is time-independent and its curl is a (locally finite) measure (without any sign condition).\\

(2) The vector field is time-dependent and its curl belongs to L^1.\\

Based on joint works with: Giovanni Alberti (Universita' di Pisa), Stefano Bianchini (SISSA Trieste), Francois Bouchut (CNRS &amp;

Universite' Paris-Est-Marne-la-Vallee) and Camillo De Lellis (Universitaet Zuerich).

In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.

In this talk I will survey the results on the existence of solutions of the semigeostrophic system, a fully nonlinear reduction of the Navier-Stokes equation that constitute a valid model when the effect of rotation dominate the atmospheric flow. I will give an account of the theory developed since the pioneering work of Brenier in the early 90's, to more recent results obtained in a joint work with Mike Cullen and David Gilbert.

In this talk I will discuss the refraction of shocks on the interface for 2-d steady compressible flow. Particularly, the class of E-H type regular refraction is defined and its global stability of the wave structure is verified. The 2-d steady potential flow equations is employed to describe the motion of the fluid. The stability problem of the E-H type regular refraction can be reduced to a free boundary problem of nonlinear mixed type equations in an unbounded domain. The corresponding linearized problem has similarities to a generalized Tricomi problem of the linear Lavrentiev-Bitsadze mixed type equation, and it can be reduced to a nonlocal boundary value problem of an elliptic system. The later is finally solved by establishing the bijection of the corresponding nonlocal operator in a weighted H\"older space via careful harmonic analysis.

This is a joint work with CHEN Shuxing and HU Dian.

We study the mean-field models describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem describing the micro phase separation of diblock copolymer melts, where micro phase separation consists of an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.

Next we study the dynamics for a free boundary problem in two dimension, obtained as a gradient flow of Ohta- Kawasaki free energy, in the case that one component is a distorted disk with a small volume fraction. We show the existence of solutions that a small, almost circular interface moves along a curve determined via a Green’s function of the domain. This talk is partly based on a joint work with Xiaofeng Ren.

In this lecture I will report on joint work with J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol which was published in Numerische Mathematik, vol. 105, (2007), pp. 337-510.

In this talk, we first study the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. We review a famous monotonicity formula of Huisken and its application to classifying so-called Type I singularities. Then, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain how Huisken's classification of Type I singularities can be adopted to this intrinsic setting, using monotone quantities found by Perelman.

 Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon.

In the talk I will mention two regularity results: the SBV regularity for strictly hyperbolic, genuinely nonlinear 1D systems of conservation laws and the characterization of intrinsic Lipschitz codimension 1 graphs in the Heisenberg groups. In both the contexts suitable scalar, 1D balance laws arise with very low regularity. I will in particular highlight the role of characteristics.

This seminar will be based on joint works with G. Alberti, S. Bianchini, F. Bigolin and F. Serra Cassano, and the main previous literature.

Pseudo-differential operators (PDO's) are primarily defined in the familiar setting of the Euclidean space. For four decades, they have been standard tools in the study of PDE's and it is natural to attempt defining PDO's in other settings. In this talk, after discussing the concept of PDO's on the Euclidean space and on the torus, I will present some recent results and outline future work regarding PDO's on Lie groups as well as some of the applications to PDE's

In this seminar I will expose some results obtained jointly with P. Marcati, concerning the global existence of weak solutions for the Quantum Hydrodynamics System in the space of energy. We don not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.

I will then discuss some possible future extensions of the theory.

The aim of this talk is to explain how to construct solutions to a

relativistic transport equation via a time discrete scheme based on an

optimal transportation problem.

First of all, I will present a joint work with J. Bertrand, where we prove the existence of an optimal map

for the Monge-Kantorovich problem associated to relativistic cost functions.

Then, I will explain a joint work with Robert McCann, where

we study the limiting process between the discrete and the continuous

equation.

I will talk about $W^{2,1}$ regularity for strictly convex Aleksandrov solutions to the Monge Amp\`ere equation

\[

\det D^2 u =f

\]

where $f$ satisfies $\log f\in L^{\infty} $. Under the previous assumptions in the 90's Caffarelli was able to prove that $u \in C^{1,\alpha}$ and that $u\in W^{2,p}$ if $|f-1|\leq \varepsilon(p)$. His results however left open the question of Sobolev regularity of $u$ in the general case in which $f$ is just bounded away from $0$ and infinity. In a joint work with Alessio Figalli we finally show that actually $|D^2u| \log^k |D^2 u| \in L^1$ for every positive $k$.

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If time will permit I will also discuss some question related to the $W^{2,1}$ stability of solutions of Monge-Amp\`ere equation and optimal transport maps and some applications of the regularity to the study of the semi-geostrophic system, a simple model of large scale atmosphere/ocean flows (joint works with Luigi Ambrosio, Maria Colombo and Alessio Figalli).