Past OxPDE Lunchtime Seminar

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 &

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).

In the first part of the talk I will introduce a notion of convexity ($\mathcal{X}$-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the H\"ormander condition.

Then I will give a PDE-characterization for $\mathcal{X}$-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.\\

In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $\mathcal{X}$-subdifferential) and show that a non empty $\mathcal{X}$-subdifferential at any point characterizes the class of $\mathcal{X}$-convex functions.

As application I will prove a Jensen-type inequality for $\mathcal{X}$-convex functions in the case of Carnot-type vector fields. {\em (Joint work with Martino Bardi)}.

An overview is given of some key issues and definitions in the Calculus of Variations, with a focus on lower semicontinuity and quasiconvexity. Some well known results and instructive counterexamples are also discussed. We then move to consider variational problems in the BV setting, and present a new lower semicontinuity result for quasiconvex integrals of subquadratic growth. The proof of this requires some interesting techniques, such as obtaining boundedness properties for an extension operator, and exploiting fine properties of Sobolev maps.

We will give a survey on some recent results on travel tomography which consists in determining the index of refraction of a medium by measuring the travel times of sound waves going through the medium. In differential geometry this is known as the boundary rigidity problem. We will also consider the related problem of tensor tomography which consists in determining a function, a vector field or tensors of higher rank from their integrals along geodesics.

We consider an active scalar equation with singular drift velocity that is motivated by a model for the geodynamo. We show that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed. This work is joint with Vlad Vicol.

Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No previous knowledge of singular integral operators or the circle method will be assumed.

Solutions which are time-bounded in L^3 up to time T can be continued

past this time, by a landmark result of Escauriaza-Seregin-Sverak,

extending Serrin's criterion. On the other hand, the local Cauchy

theory holds up to solutions in BMO^-1; we aim at describing how one

can obtain intermediate regularity results, assuming a priori bounds

in negative regularity Besov spaces.

This is joint work with J.-Y. Chemin, Isabelle Gallagher and Gabriel

Koch.