The analysis seminar covers a
wide range of topics in analysis with particular emphasis on
partial differential equations. Many of the speakers are
Courant Institute visitors and postdocs. A seminar talk may
cover original research or report on an interesting paper. The
seminar meets on ** Thursdays** at **11:00 am** in **room
1302** of Warren Weaver Hall at 251 Mercer Street, New
York. Talks generally last an hour. A few special analysis
seminars may be held at other times and locations.

The most reliable and inclusive
list of weekly seminars and events is to be found in the
weekly bulletin that is posted on a day-by-day basis on the
CIMS home page.

December 2, 2015

November 17, 2015

November 12, 2015

Ioannis Angelopoulos, Princeton UniversityNovember 5, 2015

Edriss TitiOctober 22, 2015

Mohameden Ahmedou, Giessen UniversityA critical point at Infinity approach to singular mean field equation of Liouville type

Hoai-Minh Nguyen, EPFL

This talk is devoted to various properties and applications of the Helmholtz equations with sign changing coefficients. These equations are used to model negative index materials which are artificial structures whose refractive index are negative over some frequency range. These materials were first

investigated theoretically by Veselago in 1964 and their existence was confirmed experimentally by Shelby et al. in 2001. The study of these equations faces two difficulties. First the ellipticity and the compactness are lost in general due to the changing sign coefficients. Second, the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss (the viscosity) goes to 0, might occur. In this talk, I will present several remarkable properties of these equations and their applications such as cloaking and superlensing using complementary media and cloaking a source and an object via anomalous localized resonance. Various conditions under which the equations are stable are also mentioned.

Loewner and Nirenberg discussed complete metrics conformal to the Euclidean metric and with a constant scalar curvature in bounded domains in the Euclidean space. The conformal factors blow up on boundary. In this talk, we discuss the optimal boundary expansions for the conformal factors in the context of the finite regularity. We also discuss the convergence of such expansions.

September

A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "quaternion Kahler": this leads to the weaker notion of "almost-stationarity", without however affecting our compactness results. This is a joint work with G. Tian (Princeton).

May 21, 2015

We discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions below this threshold. The primary stability mechanisms are an anisotropic enhanced dissipation effect and an inviscid damping effect of the velocity component normal to the shear, both a result of the mixing caused by the large mean shear. After detailing these linear effects, we will discuss some of the important steps in the proof, such as the analysis of the weakly nonlinear (potential) instabilities connected to the non-normal nature of the linearization. Joint work with Pierre Germain and Nader Masmoudi.

Anne-Sophie De Suzzoni, Paris 13

I will present an equation on random variables which is an extension of the cubic defocusing Schrödinger equation on density operators. I will then compute solutions of this equation whose laws are invariant both on the sphere and the Euclidean space. I will then give results of well-posedness and explain the consequences at the level of density operators.

Jameson Graber, University of Texas at Dallas

Mean field games have attracted lots of attention recently due to their applications in the social sciences and systems theory. In this study we develop a theory of existence and uniqueness a solutions to a coupled system of nonlinear PDEs describing mean field Nash equilibrium. It turns out that the system can be characterized as an optimality condition for the optimal control of the Fokker-Planck equation, where the adjoint state satisfies a Hamilton-Jacobi equation. Our variational methods are of particular interest when the PDEs are of degenerate parabolic type, for which traditional fixed-point methods break down due to lack of regularity.

Hatem Zaag (CNRS and University Paris 13)

In a series of papers with Mohamed Ali Hamza (University of Tunis el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

Hatem Zaag (CNRS and University Paris 13)

In a series of papers with Mohamed Ali Hamza (University of Tunis el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

David Lannes, University of Bordeaux

Motivated by the study of nonlinear wave-current interactions (such as rip-currents) we study the influence of vorticity on surface water waves. We first derive a generalization of the classical Hamiltonian Zakharov-Craig-Sulem formulation of irrotational water waves that takes into account the effects of the vorticity. The canonical variables for this formally Hamiltonian generalization are the surface elevation, the "gradient component" of the horizontal component of the tangential vector field, and the vorticity. It allows therefore to keep track of the influence of vorticity on the flow. We show the local well-posedness of these equations, and establish their stability with respect to shallow water limits. Based on the bounds thus obtained, we turn to derive shallow water asymptotic models. The big difference with irrotational flows is that the dynamics of the vorticity is fully three dimensional, while shallow water models are typically two-dimensional (through vertical averaging). We show however that the vorticity contribution can be reduced to two-dimensional equations; the idea, based on an analogy with turbulence theory, is that the vorticity contributes to the averaged momentum equation through a Reynolds-like tensor. A cascade of equations is then derived for this tensor, but contrary to standard turbulence theory, closure of the equations is obtained after a finite number of steps.

Concentration compactness principle says that when compactness of an imbedding between two normed vector spaces is lost due to invariance of the norms with respect to action of a non-compact group, it can be partially restored by accounting to "blowups" by the group action. The formal expression of it is the profile decomposition: every bounded sequence has a subsequence that converges (in a stronger than weak topology) up to a sum of the blowup terms. We give a general profile decomposition result for uniformly convex Banach spaces.

Surprisingly, the underlying convergence is not the weak convergence, but Delta-convergence of T.-C.Lim, known to experts in the fixed point theory, and related to the notion of asymptotic centers. In Hilbert spaces Delta convergence and weak convergence coincide. We discuss properties of Delta-convergence (including Delta-compactness) and its relation to weak convergence. We also give an analog of Brezis-Lieb Lemma where the condition of a.e. convergence is replaced by Delta- and weak convergence to the same limit. This is a joint work with Sergio Solimini.

Stefan Steinerberger, Yale University

If u is a Laplacian eigenfunction on a manifold, then there are easy heuristic arguments predicting the size of {x:u(x) = 0} (in two dimensions, this set can be created using a metal plate, sand and a violin bow and has already amazed Napoleon). These heuristic arguments, though almost certainly accurate, are difficult to make rigorous. We survey existing results and give a new proof for the currently best known lower bound (proved independently by Colding & Minicozzi as well as Sogge & Zelditch). Our argument exploits the fact that the heat equation with the eigenfunction as initial value can (a) be explicitely solved for trivial reasons and (b) studied using the stochastic interpretation based on Brownian motion. We also describe some other applications of our argument.

Vuk Milisic, Universite Paris 13

In this talk we present the starting mechanical model of the lamellipodial actin-cytoskeleton meshwork. The model is derived starting from the microscopic description of mechanical properties of ﬁlaments and cross-links and also of the life-cycle of cross-linker molecules. We introduce a simplified system of equations that accounts for adhesions created by a single point on which we apply a force. We present the adimensionalisation that led to a singular limit that motivated our mathematical study. Then we explain the mathematical setting and results already published. In the last part we present the latest developments: we give results for the fully coupled system with unbounded non-linear oﬀ-rates. This leads to two possible regimes : under certain hypotheses on the data there is global existence, out of this range we are able to prove blow-up in ﬁnite time.

We present recent results on existence of solitary waves solutions for semilinear elliptic systems in the whole plane. We assume nonlinearities may exhibit supercritical growth with respect to the critical growth prescribed by the Pohozaev-Trudinger-Moser inequality. A suitable variational framework provides linking type solitons which have finite energy in a Sobolev-Lorentz function space setting.

April 9, 2015

We consider families of commuting quantum Hamiltonian which are pertubations of constant Hamiltonians on a torus, which may have a dense pure point spectrum. We show that some of these families are unitary equivalent to a quantum Birkhoff normal form. As a consequence, we show that the Rayleigh-Schrödinger series are convergent in a neighborhood of each eigenvalue of the unperturbed Hamiltonians and that the spectrum of the perturbation is pure point. The results are uniform in the Planck constant. This a joint work with Thierry Paul.

We report on a series of recent mathematical results in the description of semiclassical quantum dynamics, jointly obtained (in different parts) with: P. Markowich, C. Klein, A. Figalli, and T. Paul. We consider a class of phase-space measures, which naturally arise from the Bohmian interpretation of quantum mechanics. These, so-called, Bohmian measures describe the time-evolution of the quantum mechanical position and velocity densities and are given by the push-forward under the Bohmian flow on phase space. The latter can be seen as a perturbation of the classical Hamiltonian flow. We study the classical limit of this flow and of the associated Bohmian measure, whenever the dimensionless Planck's constant tends to zero. Connections to the, by now, classical theory of Wigner measures are also discussed.

March 26, 2015

A particular kind of weak solutions for a 2D active scalar are the so called “sharp fronts”, i.e., solutions for which the scalar is a step function. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this setting we will present several analytical results for the surface quasi-geostrophic equation (SQG): the existence of convex $C^{\infinity}$ global rotating solutions, elliptical shapes are not rotating solutions (as opposed to 2D Euler equations) and the existence of convex solutions that lose their convexity in finite time.

March 12, 2015

Guido de Phillipis, University of Lyon

In this talk we present a sharp quantitative improvement of the celebrated Faber-Krahn inequality. The latter asserts that balls uniquely minimize the ﬁrst eigenvalue of the Dirichlet-Laplacian, among sets with given volume. We prove that indeed more can be said: the diﬀerence between the ﬁrst eigenvalue λ(Ω) of a set Ω and that of a ball of the same volume controls the deviation from spherical symmetry of Ω. Moreover, such a control is the sharpest possible. This settles a conjecture by Bhattacharya, Nadirashvili and Weitsman.

Takahisa Inui, Kyoto University

We consider the focusing cubic nonlinear Klein-Gordon equation in 3-d space. It is well known that this initial value problem is locally well posed. So we are concerned with what initial values cause scattering or blow-up. Ibrahim, Masmoudi and Nakanishi gave a sufficient and necessary condition for the real valued solution with energy below the ground state to scatter or blow up. We will extend their result to complex valued solutions below the standing waves by using the charge. If time permits, we will consider an extension of the classification by the momentum.

Yu Deng, Princeton University

We study the Euler-Maxwell electron system in 2D and prove global stability of the constant equilibrium. The proof combines modified energy estimates, spacetime resonance analysis, vector field method, and a crucial linear $L

Juergen Jost, Max Planck Institute (Leipzig)

February 19, 2015

A brief history of elliptic boundary value problems arising in Yang-Mills theory will be given, as well as their application to on-going work in collaboration with Moncrief (Yale) and Maitra (Wenthworth Institute of Technology) on a new method for Euclidean-signature semi-classical Yang-Mills fields, directed at a possible new approach to the mass gap problem.

The review
will include the Dirichlet, Neumann, and *generalized*
Neumann boundary conditions for Yang-Mills in
dimension 4, as well as the proof of existence of a
very rich structure for the space of solutions,
including non-minimal solutions, for the Dirichlet
problem with small boundary data (joint work with
Isobe, Tokyo Institute of Technology).

The
difficulties inherent in these *natural*
boundary value problems and their main points of
interest will be discussed, as well as the extension
to Yang-Mills fields of the modified semi-classical
method developed for the analysis of finite
dimensional, nonlinear quantum oscillations systems
(with Moncrief and Maitra).

Gaoyong Zhang, NYU Polytechnic School of Engineering

The
logarithmic Minkowski problem, like the classical Minkowski
problem, is a fundamental

question in
the affine geometry of convex bodies. For symmetric convex
bodies, it asks what are the necessary and sufficient
conditions for a measure in (n-1)-dimensional projective
space to be the cone-volume measure of the unit ball in an
n-dimensional normed space. Solving this problem is
equivalent to establishing existenceof a solution to a
Monge-Ampere equation. This talk outlines a complete
solution to the symmetric logarithmic Minkowski problem
and will present related open problems.

I will
describe some recent results with Noah Linden and Huw Wells
concerning the density of states in quantum spin chains, the
entanglement of most of the eigenstates in these systems,
and an associated random-matrix model.

**February 5, 2015
*special seminar at 1:30p***

We analyze the long time behavior of solutions of the Boltzmann equation in the vicinity of global Maxwellian initial data, and show the existence of a scattering regime that leads to the construction of eternal solutions that do not coincide with a global Maxwellian. This long time behavior is a consequence of decay conditions in space, showing that dispersion takes over the collisional dissipative effect by increasing the rarefaction effect.

This is work in collaboration with C. Bardos, F. Golse and C.D. Levermore.

The incompressible Magneto-Hydro-Dynamic (MHD) system is a classical and fundamental model in plasma physics. Although well known, its derivation from Navier-Stokes type equations has been so far formal. In this talk and after reviewing the results about the well-posedness, I show how an asymptotic analysis of such equations can rigorously lead to a such a derivation. The key points is a precise study of the weak stability in the Lorentz.

This is a joint work with D. Arsenio (Paris 7) & N. Masmoudi (Courant)

André Lisibach, ETH Zurich

The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but possesses the additional property of having singular initial data due to the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

Manuel Victor Gnann, University of Michigan

We investigate a free boundary problem for a thin-film equation with quadratic mobility and a zero contact angle condition at the triple point where air, liquid, and solid meet. This problem can be derived by a lubrication approximation from the Navier-Stokes system with a Navier-slip condition at the substrate. By treating the model problem of source-type solutions, we motivate why general solutions to this problem are generically singular. The method for proving well-posedness therefore requires to subtract the leading-order singular expansion at the free boundary in the maximal regularity estimates for the linearized evolution. We also discuss the regularizing effect of the degenerate-parabolic operator to arbitrary orders of the singular expansion. Many of the presented results are joint with Lorenzo Giacomelli, Hans Knüpfer, and Felix Otto.

------------------------------------------------------------------------

Mohamed Majdoub, Faculte de Sciences de Tunis

Title: Existence, uniqueness and regularity of solutions for the thin-film equation

Abstract: We show the uniqueness of strong solution for the thin-film equation: $u_t + (u u_{xxx})_x =0, t>0$ with initial data $u(0)=m\delta, m>0$

where $\delta$ is the Dirac mass at
the origin. We also investigate the boundary
regularity of source-type self-similar solutions to the
thin-film equation with gravity:

$h_t=-(h^nh_{xxx})_x+(h^{n+3})_{xx},$
$ t>0, h(0)= m \delta$ where $n\in (3/2,3)$. The existence
of these solutions has been established by E. Beretta. It is
also shown that the leading order expansion near the edge of
the support coincides with that of a travelling-wave solution
for the standard TFE: $h_t=-(h^n h_{xxx})_x$. We sharpen this
result by proving that the higher order corrections are
analytic with respect to three variables: the first one is
just the spacial variable, whereas the second and third
(except for $n = 2$) are irrational powers of it. We also
prove the uniqueness of solutions. This is a joint work with
Nader Masmoudi and Slim Tayachi.

This talk is concerned with energy minimizers in an orbital-free density functional theory that models the response of massless fermions in a graphene monolayer to an out-of-plane external charge. The considered energy functional generalizes the Thomas-Fermi energy for the charge carriers in graphene layers by incorporating a von-Weizsaecker-like term that penalizes gradients of the charge density. Contrary to the conventional theory, however, the presence of the Dirac cone in the energy spectrum implies that this term should involve a fractional Sobolev norm of the square root of the charge density. We formulate a variational setting in which the proposed energy functional admits minimizers in the presence of an out-of-plane point charge. The associated Euler- Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero to non-zero response at a finite threshold value of the external charge is proved. This is joint work with J. Lu and V. Moroz.

December 4, 2014

We will revisit, using our techniques, Contact Form Geometry and we will describe the results that we can derive using these techniques. Time permitting, we will also discuss some open problems in Conformal Geometry. which come naturally to our understanding.

November 13, 2014

Philippe LeFloch, Univ. Paris 6 and CNRS

Weakly regular Einstein spacetimes with symmetry

Regularity results for general nonlocal equations of quasilinear type

I will consider an equation which can be seen as a fractional p-laplacian, with measure data. I will introduce a method to prove the existence of suitable classes of solutions. More importantly, I will prove some estimates relating the solution to the associated Wolff potentials of the measure.

November 6, 2014 *special seminar taking place at 2pm in WWH 1302*

Benjamin Harrop-Griffiths, Univ. of California - Berkeley

We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets. This is joint work with Mihaela Ifrim and Daniel Tataru.

Mircea Petrache, ETH Zurich

The celebrated results of K. K. Uhlenbeck furnished the bases for a complete variational study of the Yang-Mills functional in dimension 4, leading to the definition of new differential invariants of 4-manifolds by S. K. Donaldson. In this setting the objects of study were Sobolev connections over smooth bundles. In the first part of the talk I will explain how this setting works, and show why it is not sufficient for studying the Yang-Mills functional in dimensions higher than 4.

I will then define a space of weak connections introduced in collaboration with Tristan Riviere, and present our weak closure and approximability results in that setting. These results lead to the an existence-and-regularity theory for Yang-Mills minimizers in dimensions greater than 4. This bears a strong analogy to the work of Federer-Fleming on the Plateau problem. The existence of minimizers requires the definition of new weak connection spaces and we have a proof that they correspond to smooth connections over classical bundles, up to a codimension 5 singular set. This gives a first step for connecting the variational theory to the algebraic geometry framework which motivated some conjectures of Gang Tian. The optimality of the codimension 5 is shown by an explicit example of a minimizer, and the proof of its minimality uses a new combinatorial technique together with a decomposition theorem for vector fields by S. Smirnov.

**October 30, 2014****
**Mark Wilkinson, CIMS

**October 23, 2014
**Michael Jenkinson,
Columbia U.

We construct several families of
symmetric localized standing waves (solitons) to the one-,
two-, and three-dimensional discrete nonlinear Schrödinger
equation (DNLS) with cubic nonlinearity using bifurcation
methods about the continuum limit. Such waves and their energy
differences play a role in the propagation of localized states
of DNLS across the lattice. The energy differences, which we
prove to exponentially small in a natural parameter, are
related to the "Peierls-Nabarro Barrier" in discrete systems,
first investigated by M. Peyrard and M.D. Kruskal (1984).
These results may be generalized to different lattice
geometries and inter-site coupling parameters. This is joint
work with Michael I. Weinstein.

**October 16, 2014
**Tom Trogdon, CIMS

The classical Gibbs phenomenon is an artifact of non-uniform convergence. More precisely, it arises from the approximation of a discontinuous function with an analytic partial sum of the Fourier series. It is known from the work of DiFranco and McLaughlin (2005) that a similar phenomenon occurs when a box initial condition is taken for the free Schrödinger equation in the short-time limit. This talk is focused extending the linear theory of this work in two ways. First, we establish sufficient conditions for the classical smoothness of the solutions of linear dispersive equations for positive times. Second, we derive a highly-oscillatory and computable short-time asymptotic expansion of the solution of general linear dispersive PDEs with a large class of discontinuous initial data. Boundary-value problems can also be treated.

**October 9, 2014
**Shouhong Wang, U. of
Indiana

Second, we briefly introduce a field theory coupling the four fundamental interactions, based only on a few fundamental principles and symmetries. This theory leads to insights to the understanding of such challenging problems as the dark matter and dark energy and quark confinement. This is joint work with Tian Ma.

**October 2, 2014**

Yuri Latushkin, U. of Missouri

*The Morse and Maslov indices for multidimensional
Schroedinger operators with matrix-valued potentials
*

**
September 25, 2014
**Miles Wheeler, CIMS

** ****September 18, 2014
**Tristan Buckmaster, CIMS

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy.

The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

**PREVIOUS SEMINARS**

**January 16, 2014**

Slim Tayachi ,
University Tunis El Manar

*The nonlinear heat equation with high order mixed
derivatives of the Dirac delta as initial values and
Applications *

In this talk we prove
local existence of solutions for the nonlinear heat equation
$$u_t = \Delta u + `|u|`

^\alpha
u, \; t\in(0,T),\; x\in \R^N,\; \alpha>0,$$ with
initial values in high order negative Sobolev spaces. In
particular, we consider high order mixed derivatives of the
Dirac Delta as initial values.

As an application, we prove the existence of initial values
$u_0 = \lambda f$ for which the resulting solution blows up in
finite time if $\lambda>0$ is sufficiently small and
$\alpha<2/(N+m).$ Here, $f$ satisfies in particular $f\in
C_0(\R^N)\cap L^{^1}(\R^N)$
and is anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\;
x_m,\; 1\leq

m \leq N,$ where $x:=(x_1,x_2,\cdots,x_N)\in \R^N.$ Moreover
we require, $\int_{\R^N} x_1\cdots x_mf(x) dx\not=0$. This
extends known ``small lambda" blowup results.

If $f$ is also in $H^{^1}(\R^N)$,
then by standard energy arguments, the solution with initial
value $u_0 = \lambda f$ blows up in finite time if $\lambda
> 0$ is sufficiently large. We prove the existence of a
function $f_0$ for which, the solution with initial
value $u_0 = \lambda f_0,\; \alpha<2/(N+m)$ would
blow up in finite time for large and small $\lambda >
0$, but would be global for $\lambda = 1.$

We show also that the condition $\alpha<2/(N+m)$ is sharp.
If $\alpha>2/(N+m)$, then all initial data $u_0 = \lambda
f$ anti-symmetric in $x_1,x_2,\cdots, x_m,$

$\lambda>0$ is sufficiently small and $f$ satisfying
the same conditions as above produce solutions which are
global in time.

This is a joint work with Fred B. Weissler.

**January 30,
2014**

Sergiu
Klainerman, Princeton University

*Are Black
Holes Real ? *

**February 4, 2014 **

Igor Kukavica
(USC)

*On the well-posedness of an interface damped free boundary
fluid-structure model*

We address a
fluid-structure system which consists of the incompressible
Navier-Stokes equations and a damped linear wave equation
defined on two dynamic domains. The equations are coupled
through transmission boundary conditions and additional
boundary stabilization effects imposed on the free moving
interface separating the two domains. We will discuss
local existence and uniqueness of solutions and establish
global existence for small initial data. This is a joint work
with M. Ignatova, I. Lasiecka, and A. Tuffaha.

**February 6, 2014
**Eric
Carlen, Rutgers

We prove a
quantitative Brunn-Minkowski inequality for sets E and K, one
of which, K, is assumed convex, but without assumption
on the other set. We are

primarily interested in the case in which K is a ball. We use
this to prove an estimate on the remainder in the Reisz
rearrangement inequality under certain conditions on the three
functions involved that are relevant to a problem arising in
statistical mechanics: This is joint work with Franceso
Maggi.

February 13, 2014

Thierry Goudon, Inria, FranceModels for “mixtures”, multifluid flowsFebruary 20, 2014

Thomas H. Otway, Yeshiva UniversityA weak Dirichlet problem for the cold plasma model

The open Dirichlet problem for elliptic—hyperbolic equations, in which data are prescribed on a proper subset of the boundary, has been studied for decades and arises, for example, in nozzle flow. But the closed Dirichlet problem for such equations, in which data are prescribed on the entire boundary, is not well known although it also arises naturally, e.g., in transonic flow about a profile and in plasma physics. Recently, Lupo, Morawetz, and Payne showed the existence of weak solutions to a closed Dirichlet problem for a class of equations having strong regularity in comparison to other elliptic—hyperbolic equations. We consider the same problem for an elliptic—hyperbolic equation, introduced by Weitzner in 1984, for which regularity is more problematic.

Andrea Nahmod,University of Massachusetts, Amherst *** Please note unusual time and location for this event: 2:30 pm in room WWH 517Almost sure well-posedness for the periodic 3D quintic NLS below the energy space

In this talk we first review recent progress in the study of certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem) which stems from incorporating to the deterministic toolbox, powerful but still classical tools from probability as well. We will explain some of these ideas and describe in more detail recent work, joint with Gigliola Staffilani on the almost sure well-posedness for the periodic 3D quintic nonlinear Schrodinger equation in the supercritical regime; that is, below the critical space $H^1(\mathbb T^3)$.February 27, 2014Pierre Germain, CIMSKdV limit for geometric Schrodinger equations with potential

It is well known that the KdV equation arises as a long wave limit of the Gross-Pitaevskii equation. In this talk, I will present a generalization of this derivation to geometric Schrodinger equations (wave maps). This has applications to various physical situtations : spin models, coupled nonlinear Schrodinger equations.March 4, 2014 *****11 am in room 1302Erwan Faou, INRIA Rennes and ENS ParisLandau damping in Sobolev spaces for the Vlasov-HMF modelWe consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of dampingMarch 6, 2014Tej-eddine Ghoul, CIMSStability of infinite time aggregation for Keller-Seguel equation

In this talk we give a sharp description and we proof the stability of the collapse in infinite time for the parabolic elliptic Patlak-Keller-Seguel equation when the mass of the solution is critical equals to 8\pi.March 20, 2014

Valeria Banica, Universite d'EvryVarious dynamics for several vortex filaments

We consider the Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega and concern the case of same circulations for two filaments and for a class of configurations of three filaments. We prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N filaments for any N larger or equal to 2. We also show the existence of travelling wave type dynamics, and we describe configurations leading to collision for N larger or equal to 3. Finally we consider the problem of collisions for perturbations of antiparallel translating pairs of filaments. This are joint works with Erwan Faou and Evelyne Miot.March 27, 2014

Scott ArmstrongHigher regularity in stochastic homogenization for uniformly elliptic equations

We study uniformly elliptic equations with random coefficients satisfying a finite range of dependence. We show that, with overwhelming probability, solutions possess much higher regularity than can be expected in general for equations with rapidly coefficients. The statements take the form of a priori estimates, except that in place of universal constants are random variables which, while not almost surely bounded, but have quite good integrability in the probability space. These results can be thought of as stochastic, quantitative analogues of the results Avellaneda-Lin developed in the late 80s with compactness arguments. In the stochastic setting, we don't have compactness, but we do have concentration inequalities.April 1, 2014

Bob Strichartz, CornellSpectral asymptotics for "April Fools" wave equations on compact spacetimes that pretend to be elliptic.The Weyl asymptotic law tells you about the spectrum of an elliptic pde on a compact manifold. You would not ordinarily think it would tell you anything about wave equations. Nevertheless, for certain compact spacetimes, such as the product of a circle and a 2-sphere, there is an asymptotic law for the spectrum (separately for the positive and negative part). However, "April Fools!", it gets the dimension wrong. The gist of the argument is elementary number theory (distictions between even and odd numbers). This is joint work with Jonathan Fox, and I will bring along some nice graphics that he has made.April 10, 2014

Erik Wahlen, University of LundSolitary water waves in three dimensions

I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localized in the sense that they converge to the undisturbed state of the water in every horizontal direction. The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili equation (for strong surface tension) or the Davey-Stewartson equation (for weak surface tension). These model equations play an important role in the theory.April 11, 2014****Please note this talk is at 10am

Mahir Hadzic, Kings College, LondonStability problem in the dust-Einstein system with a positive cosmological constantThe dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstroem's work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.April 17, 2014

Zhouping Xin, Chinese University of Hong KongOn Multi-Dimensional Compressible Navier-Stokes System with Possible Large Oscillations and Vacuum

In this talk, I will discuss some of recent results on the large time well-posedness of classical solutions to the multi-dimensional compressible Navier-Stokes system with possible large oscillations and vacuum.

The focus will be on finite-time blow-up of classical solutions for the 3-D full compressible Navier-Stokes system, and the global existence of classicla solutions to the isentropic compressible Navier-Stokes system in both 2-D and 3-D in the presence of vacuum and possible large oscillations. Some new estimates on the fast decay of the pressure in the presence of vacuum will be presented, which are crucial for the well-posedness theory in 2-dimension.April 24, 2014

Zhen Lei, FudanGlobal well-posedness of incompressible elastodynamics in 2D

In this talk I will report our recent result on the global well posedness of classical solutions to system of incompressible elastodynamics in 2D. The system is revealed to be inherently strong linearly degenerate and automatically satisfies astrongnull condition, due to the isotropic nature and the incompressible constraint.

May 1, 2014

Antoine GloriaQuantitative central limit theorem for the effective diffusion in iid environments

In this talk I shall present a quantitative central limit theorem for the approximation $A_L$ of the effective diffusion $A_{hom}$ by periodization (on the $L$-torus) in a discrete iid environment in dimension $d\geq 2$. On the one hand, using Stein's method, I shall prove a quantitative estimate on the decay of the Wasserstein distance between $A_L$ minus its expectation rescaled by the square-root of its variance and a normal random variable. On the other hand, I shall prove that the rescaled variance $\sigma_L^{^2}:= L^d \mathrm{var}(A_L)$ has a limit $\sigma^{^2}$ as $L$ goes to infinity, and shall quantify in terms of $L$ the convergence of the expectation of $A_L$ to $A_{hom}$ and the convergence of $\sigma_L^{^2}$ to $\sigma^{^2}$. The proofs of these results are based on a seriesof works in collaboration with Marahrens, Mourrat, Neukamm, and Otto. Combining both types of results then allows us to bound the Wasserstein distance between $\sigma^{-1} L^{d/2} (A_L-A_{hom})$ and a normal random variable by a constant times $L^{-d/2}\log^d L$, which we think is optimal. This is joint work with Jim Nolen (Duke).May 6, 2014Reza Pakzad, University of Pittsburgh

May 8, 2014

Marta Lewicka,University of PittsburghOn the biharmonic energy with Monge-Ampe`re constraintsMay 15, 2014

Miles Wheeler, BrownLarge-amplitude solitary water waves with vorticity

In this talk, we will construct exact solitary water waves of large amplitude and with an arbitrary distribution of vorticity. Starting from a shear flow with a flat free surface, we use a degree-theoretic continuation argument to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. We will also discuss solitary waves generated by a non-constant pressure on the free surface.May 22, 2014

Philip Rosneau, Tel-Aviv UniversityMulti-Dimensional Compact PatternsSolitons, kinks or breathers, are manifestations of weakly nonlinear excitations in, say, anharmonic mass-particle chains. In a stronglyanharmonic chains,the tails of the emerging patterns rather than exponentially, decay at adoubly-exponential rateand in the continuum limit collapse into a singular surface with the resulting wavesbecomingstrictly compact, hence their name: compactons. Using the Z-K, the Sub-linear Complex Klein-Gordon and the Sub-linear NLS equations as examples, we shall show how typical multidimensional compactons emergence and interact. In general, for compact- and hence non-analytical, structures to emerge, the underlying system has to admit a local loss of uniqueness due to, for instance, a degeneracy of the highest order operator or other, singularity inducing, mechanisms.June 12, 2014

Messoud EfendiyevDegenerate parabolic equations arising in the modelling of biofilms

** Fall 2013**

**September 12**

Laurent Thoman, Universite de Nantes

*Random weighted Sobolev inequalities
*

We extend a randomisation method, introduced by Burq-Lebeau on compact manifolds, to the case of the harmonic oscillator. We construct measures,

under concentration of measure type assumptions, on the support of which we prove optimal weighted Sobolev estimates on R^d. As an application we can prove almost sure global well posedness results for the nonlinear Schrödinger equation with harmonic potential. This is a joint work with Aurélien

Poiret and Didier Robert.

**September 19**

Zaher Hani, CIMS

*Out-of-equilibrium dynamics for the nonlinear Schroedinger
equation**: From energy cascades to weak turbulence
*

Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations on bounded domains. This is partly due to the fact that dispersion does not translate into decay in this setting (in contrast to the case of unbounded domains like $R^d$). In this talk, we will take the cubic nonlinear Schroedinger equation as our model and discuss some results and aspects of this out-of-equilibrium dynamics, from energy cascades (i.e. migration of energy from low to high frequencies) to weak turbulence.September 26

Vladimir Sverak, University of MinnesotaOn invariant measures for 2d Euler and some other models

We will discuss aspects of certain constructions of invariant measures on solutions with finite energy, some of their properties, and connections to the long-time behavior of solutions.October 3

Patrick Louis Combettes, Laboratoire Jacques-Louis Lions, Université Paris 6Splitting two monotone operators goes a long wayA basic problem in applied mathematics is to find a zero of the sum of two monotone operators. The main splitting algorithms for solving this problem were essentially developed in the late 1970s. We show that, by bringing into play duality tools and product formulations, these fundamental splitting principles can be exploited to solve considerably more complex multicomponent composite inclusion problems efficiently. Applications to machine learning, PDEs, and image recovery will be discussed.

**Oc****tober 10**

Christophe Lacave, Institut Mathematique de Jussieu

*On the motion of a small body immersed in a two
dimensional incompressible perfect fluid*

In this talk, we
consider the motion of a small solid body in a planar ideal
fluid, and the limit behavior of the system as the solid body
shrinks to a point. We study two regimes:* in the case of a
fixed mass, we prove that the solutions converge to a variant
of the vortex-wave system where the vortex, placed in the
point occupied by the shrunk body, is accelerated by a lift
force similar to the Kutta-Joukowski force of the irrotational
theory. * in the case of a fixed density, the limit behavior
is exactly the vortex-wave system. This work is in
collaboration with O. Glass (Paris Dauphine) and F. Sueur
(Paris 6-UPMC)

**Oct****ober
17**

Jianli Liu

Stability and existence of traveling wave solutions to
the timelike extremal surface in Minkowski space

In this talk, firstly we will give the background of global classical solutions to the equation of timelike extremal surface in Minkowski space. Under the appropriate small oscillation assumptions on initial traveling waves, we derive the stability result of the traveling wave solutions.October 24

Anne-Laure DalibardWell-posedness of the Stokes-Coriolis system in the half-space over a rough surface

This talk is devoted to the well-posedness of the 3d stationary Stokes-Coriolis system set in the half-space over a rough Lipschitz surface. The main issue lies in the fact that we work with solutions of infinite energy: indeed, the Dirichlet data for the Stokes-Coriolis system on the rough boundary does not decrease at infinity. Moreover, we do not assume any kind of spatial structure on the system, i.e. there is no underlying periodicity or stationarity. Following an idea of Gerard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low horizontal frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space $H^{1/2}_{uloc}$. This is a joint work with Christophe Prange.October 31

Hatem ZaagBlow-up behavior for subconformal semilinear wave equationsNovember 7

Susan FriedlanderIll-posedness / Well-posedness Results for a Class of Active Scalar Equations

We discuss a class of active scalar equations where the transport velocities are more singular than the active scalar. There is a significant difference in the well-posedness properties of the problem depending on whether the Fourier multiplier symbol for the velocity is even or odd. The "even" symbol non-diffusive or weakly diffusive equations are ill-posed in Sobolev spaces. However the critically diffusive equations are globally well posed in both the odd and even cases.

Examples of "even" equations are the magnetogeostrophic equation that is a model for the geodynamo and the modified porous media equation.

This is joint work with Francisco Gancedo, Weiran Sun and Vlad Vicol.November 14

Jacob BedrossianA new proof of Landau damping in the nonlinear Vlasov equations

Landau damping is an important mechanism in kinetic descriptions of plasmas, however, mathematical works on the topic have been relatively scarce. We give a simpler proof of Landau damping in the Vlasov equations on the ND torus, originally due to Mouhot and Villani (2011). Moreover, we may take our initial data in Gevrey class smaller than 3, the regularity requirement conjectured in the original work. The proof combines ideas from the original proof with ideas from our recent work on inviscid damping in 2D Euler (joint with N. Masmoudi). The Newton iteration scheme and Lagrangian estimates of the original work are replaced by paraproduct decompositions and the treatment of the plasma echo resonances is simplified by controlled regularity losses in time. Joint work with N. Masmoudi and C. Mouhot.November 21Ronen EldanA Two-Sided Estimate for the Gaussian Noise Stability Deficit

The Gaussian Noise Stability of a set $A \subset \RR^n$ with parameter $0 <\rho < 1$ is defined as $$ S_\rho(A) = \PP(X,Y \in A) $$ where $X,Y$ are jointly Gaussian random vectors such that X and Y are standard Gaussian vectors and $E X_i Y_j = \delta_{ij} \rho$. Borel's celebrated noise stability inequality states that if $H$ is a half-space whose Gaussian measure is equal to that of $A$, then $S_\rho(H) \geq S_\rho (A)$ for all $0 < \rho < 1$. We will present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set A and its corresponding half-space H(namely the distance between the two centroids), we show that the deficit between the noise stability of A and H can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors.As a consequence, we also manage to get the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variationdistance as a metric. Moreover, in the limit $\rho \to 1$, we get an improved dimension free robustness bound for the Gaussian isoperimetricinequality.

Luis Vega** this talk is at 11:55 directly following the 11:00 talkThe vortex filament equation for a regular polygon

I shall present some recent results obtained in collaboration with V. Banica and F. de la Hoz about the evolution of vortex filaments according to the so-called binormal law. After reviewing the results for filaments with one corner, we will look at the case of general polygons. The dynamics turn out to be quite complex. Among other things numerical evidence on the appearance of multifractals will be given. This multifractal structure turns out to be connected to the one proved by S. Jaffard on Riemann's non-differentiable function.December 4

Benoit Pausader, Princeton UniversityAsymptotic behavior for the nonlinear Schrodinger equation with partially periodic data

We consider the NLS equation on quotients of R^d, focusing on the case of RxT^{^2}. The question is to explore the asymptotic behavior of solutions in a more ``compact'' setting. We show how the scattering theory in the quintic case (the equivalent of the mass-critical case) is affected by the ``smaller'' volume and how, in the cubic case theasymptotic behavior is strongly modified by the presence of a secondary dynamics in logarithmic time. In the case of R or RxT (completely integrable case), this secondary dynamics can be explicitely integrated and only causes a phase correction. In the case of RxT^d, d>=2, this dynamics is more complicated and leads to new regimes. In particular, in this case, one can find global solutions which start arbitrarily small in H^s and grow unboundedly with time. This is a joint work with Z. Hani as well as (for the cubic case) N.Tzvetkov and N. Visciglia.December 12

Remi SchweyerOn the different rates of blow-up for the 1-corotational harmonic heat flowMarch 27, 2014

Scott ArmstrongMarch 20, 2014

Valeria Banica, Universite d'Evry

April 4, 2014

Bob Strichartz, CornellSpectral asymptotics for "April Fools" wave equations on compact > spacetimes that pretend to be elliptic.

The Weyl asymptotic law tells you about the spectrum of an elliptic pde on a compact manifold. You would not ordinarily think it would tell you anything about wave equations. Nevertheless, for certain compact spacetimes, such as the product of a circle and a 2-sphere, there is an asymptotic law for the spectrum (separately for the positive and negative part). However, "April Fools!", it gets the dimension wrong. The gist of the argument is elementary number theory (distictions > between even and odd numbers). This is joint work with Jonathan Fox

Erik Wahlen, University of Lund

Antoine Gloria

**Spring 2013**

January
24

Yin, Z.Y., Sun Yat Sen University

February
7

Phil Isett, Princeton University ** This
seminar is at 10am in room 312

Frederic Rousset,

Construction of
multi-solitons solutions for the water waves

February
14

Kenji Nakanishi, Kyoto University

Center-stable manifold of
the ground state in the energy space for the critcal wave
equation

February
21

No seminar today, Abel Day

February
22

Walter Craig

Vortex filament interactions

February 28

Vlad Vicol, Princeton University

On the
inviscid limit for the stochastic Navier-Stokes equations

We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. We prove that the limiting measures are supported on bounded vorticity solutions of the 2D Euler equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow. Motivated by 2D turbulence considerations we are lead to the problem of well-posedness for the stochastic 2D Euler equations. This is joint work with Nathan Glatt-Holtz and Vladimir Sverak.

March 5 ** This seminar is at 11am in room 1302

Tak Kwong Wong

Local existence and uniqueness of Prandtl equations

The Prandtl equations, which describe the boundary layer behavior of a viscous incompressible fluid near the physical wall, play an important role in the zero-viscosity limit of Navier-Stokes equations. In this talk we will discuss the local-in-time existence and uniqueness for the Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity assumption. The proof is based on weighted energy estimates, which come from a new type of nonlinear cancellations between velocity and vorticity.

March
7

Emmanuel Hebey

Recent
advances on Klein-Gordon-Maxwell-Proca systems

Mihai Tohaneanu, Johns Hopkins University ** This seminar is at
12pm

The
Strauss conjecture on black holes

```
The Strauss conjecture for the Minkowski spacetime in three dimensions states that the semilinear equation wave (u) = u^p has a global solution for compactly supported and sufficiently small data if $p> 1+\sqrt 2$. We prove a similar result in the context of Schwarzschild and
Kerr with small angular momentum black holes. This is joint work with H. Lindblad, J. Metcalfe, C. Sogge, and C. Wang.
```

March 14

Camillo De Lellis, University of Zurich
/ Princeton University

Regularity theory for
area-minimizing currents

```
It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. To reach this goal Almgren developed an entirely new regularity theory, which occupies a very large monograph, published posthumously.
This talk is based on a series of joint works with Emanuele Spadaro, where we give alternative proofs to all Almgren's main steps, resulting into a much more manageable approach to his entire theory.
```

March 21

Yannick Sire, University of Marseille

Fractional
Ginzburg-Landau equations and boundary harmonic maps

March 28

Thomas Sideris, U.C. Santa Barbara

Almost global
existence of small solutions for 2d incompressible Hookean
elastodynamics

We will examine the IVP for
equations of 2d incompressible isotropic elastodynamics.

We show that classical
solutions exist “almost globally" in time, for small initial
perturbations of the identity.

We use the energy method
with a ghost weight. This is combined with decay estimates,
obtained via generalized Sobolev inequalities and weighted
L^2 bounds.
The argument exploits the inherent null structure of the
nonlinearities.

This is joint work with Zhen Lei
and Yi Zhou of Fudan University.

April 4

Marius Beceanu, IAS

Strichartz Estimates
for Equations with Time-Dependent Potentials

In this talk I present some inequalities valid for Schroedinger's equation and for the wave equation, which hold for a more general class of time-dependent potentials than previously known estimates.

April 18

Yan Yan Li, Rutgers University

A compactness
theorem for a fully nonlinear Yamabe problem under
a lower Ricci curvature bound

April 23

Fabio Pusateri, Princeton University

Global existence
for two-dimensional water waves

April 25

Yoshikazu Giga, University of Tokyo

Analyticity of
the Stokes semigroup in space of bounded functions

The Stokes
system is a linearized system of the Navier-Stokes equations
describing the motion of incompressible viscous fluids. It is
believed that the nonstationary problem is very close to the
heat equation. (In fact, if one considers the Stokes system in a
whole space R^n, the problem is reduced to the heat equation.)
The solution operator S(t) of the Stokes system is called the
Stokes semigroup. It is well-known that S(t) is analytic in the
L^p setting for a large class of domains including bounded and
exterior domains with smooth boundaries provided that p is
finite and larger than 1. This property is the same as the heat
semigroup. Moreover, for the heat semigroup it is analytic even
when p equals the infinity.

The corresponding (p=infinity) result for the Stokes semigroup
S(t) has been open for more than thirty years even if the domain
is bounded. Using a blowup-argument, we have now solved this
long-standing problem for a large class of domains, including
bounded and exterior domains. A key step is to derive a harmonic
pressure gradient estimate by a velocity gradient. We give a
sketch of the proof as well as a few possible applications to
the Navier-Stokes equations. This is a joint work of my student
Ken Abe and the main paper is going to appear in Acta
Mathematica.

April 30

Isabelle Gallagher, University
Paris 7

The
diffusion limit from a hard spheres system, via the linear
Boltzmann equation

In this talk we report on a recent result with Laure Saint-Raymond and Thierry Bodineau in which we derive the linear Boltzmann equation from a Newtonian system of hard spheres, following Lanford's proof. This convergence holds for a very long time and enables us to obtain the heat equation in the diffusion limit.

May 3

Edris Titi, UC Irvine

May 9

Philip Rosenau, Tel-Aviv University

On a Well-Tempered Diffusion

The classical transport theory as expressed by, say, the Fokker-Planck equation, lives in an analytical paradise but, in sin. Not only its response to initial datum spreads at once everywhere oblivious of the basic tenets of physics, but it also induces an infinite flux across a sharp interface. Attempting to overcome these difficulties one notices that the moment expansion of any of the micro ensembles of the kind that beget the equations of the classical mathematical physics, say the Chapman-Enskog expansion of Boltzmann Eq., if extended beyond the second moment, yields an ill posed PDE (the Pawla Paradox)!

We shall describe mathematical strategies to overcome these generic difficulties. The resulting flux-limited transport equations are well posed and capture some of the crucial effects of the original ensemble lost in moment expansion. For instance, initial discontinuities do not dissolve at once but persist for a while. There is a critical transition from analytical to discontinuous states with embedded sub-shock(s).

May 16

Neston Guillen, UCLA

Global
well-posedness for the homogeneous Landau equation

In joint work with Maria Gualdani we consider the homogeneous Landau equation from plasma physics. Both global well-posedness and exponential decay to equilibrium are proved assuming only boundedness and spatial decay of the initial distribution. In particular, we can handle discontinuous initial conditions that might be far from equilibrium. Despite the equation not having a maximum principle the key steps of the proof rely on barrier arguments and parabolic regularity theory.

June 13

Anne-Laure Dalibard, ENS Paris ***Please note this seminar is in room 512

Mathematical study of a degenerate boundary layer

The goal of this talk is to analyze asymptotically an equation stemming from oceanographic models describing the motion of large scale currents. This equation is known to give rise to boundary layers on the east and west coasts of the domain. One of the major issues of our study lies in the fact that the size of these lateral boundary layers becomes very large as one approaches the north and south end points of the domain. In a neighbourhood of these zones, the classical construction of boundary layers must therefore be completely changed. We prove that the north and south boundary layers are the solutions of some evolutionary equation, and that their profile is thus non-intrinsic. We also exhibit discontinuity boundary layers, which penetrate the interior of the domain when the latter has islands, for instance. This is a joint work with Laure Saint-Raymond.

September
6

Edriss Titi, University of California - Irvine

On the Loss of Regularity
for the Three-Dimensional Euler Equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to C1;. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very diferent from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. In addition, we will use this shear flow to provide a nontrivial example for the use of vanishing viscosity limit, of the Navier-Stokes solutions, as a selection principle for uniqueness of weak solutions of the 3D Euler equations.

This is a joint work with Claude Bardos.

September 20

Philippe LeFloch, University of Paris 6

Finite energy method for compressible fluids

I will discuss the initial value problem for the Euler system of compressible fluid flows governed by a general pressure law, when solutions enjoy a certain symmetry (for instance, plane symmetry) and have solely finite total energy. In a recent series of papers with Pierre Germain (Courant Institute), I have established an existence and compactness theory for this problem and analyzed the vanishing viscosity--capillarity limit in weak solutions with finite energy to the Navier-Stokes-Korteweg system. The proposed approach is referred to as the Finite Energy Method for compressible fluid flows and leads to a generalization of DiPerna's theorem (bounded solutions) and LeFloch-Westdickenberg's theorem (finite energy solutions) to the Euler system of polytropic fluids.

September 27

Andrew Comech, Texas A&M

Linear instability of solitary waves in nonlinear Dirac equation

We study the linear instability of solitary wave solutions to the nonlinear Dirac equation (known to physicists as the Soler model). That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part.

We show that the linear instability of the small amplitude solitary waves is described by the Vakhitov-Kolokolov stability

criterion which was obtained in the context of the nonlinear Schroedinger equation: small solitary waves are linearly

unstable in dimensions 3, and generically linearly stable in 1D.

A particular question is on the possibility of bifurcations of eigenvalues from the continuous spectrum; we address it

using the limiting absorption principle and the Hardy-type estimates.

The method is applicable to other systems, such as the Dirac-Maxwell system.

Some of the results are obtained in collaboration with Nabile Boussaid, Université de Franche-Comté, and

Stephen Gustafson, University of British Columbia.

October 4

Tej-eddin Gouhl, CIMS

October 11

Arie Israel, CIMS

The Whitney Extension Problem in Sobolev Spaces

October 18

Christophe Lacave (Paris 7)

Large time behavior for the two-dimensional motion of a disk in a viscous fluid

October 25

Wang Changyou, University of Kentucky

Well-posedness of the nematic liquid crystal flow in $L

In this talk, I will consider a simplified Ericksen-Leslie system modeling the incompressible nematic liquid crystal flow. For any initial data $(u_0,d_0)$, with $(u_0,\nabla d_0)$ belonging to $L

November 1

Jeremie Szeftel, ENS, France

Bounded L2 curvature conjecture

November 8

Gilles Francfort, University of Paris 13

Elasto-plasticity: heterogeneity and homogenization

November 15

Sylvia Serfaty, CIMS

Large vorticity local minimizers to the Ginzburg-Landau functional

November 29

Andrew Lawrie, University of Chicago ** This seminar is at 11am

Classification of large energy equivariant wave maps

I will discuss some recent joint work with Raphael Cote, Carlos Kenig, and Wilhelm Schlag. We consider energy critical 1-equivariant wave maps, R1+2 ! S2. This problem admits a unique (up to scaling) harmonic map, Q, given by stereographic projection. We show that every topologically trivial (degree 0) solution with energy less than twice the energy of Q exists globally in time and scatters. Next we establish a classication, in the spirit of recent work by Duyckaerts, Kenig, and Merle, of all degree 1 solutions with energy below three times the energy of Q.

Vedran Sohinger, University of Pennsylvania. **This seminar is at 11:55am

The Boltzmann equation, Besov spaces, and optimal decay rates in R^n

In this talk we will study the large-time convergence to the global Maxwellian of perturbative classical solutions to the Boltzmann equation on R^n, for n geq 3, without the angular cut-off assumption. We prove convergence of the k-th order derivatives in the norm L^r_x (L^2_v), for any 2 leq r leq infty, with optimal decay rates, in the sense that they are equal to the rates which one obtains for the corresponding linear equation. The initial data is assumed to lie in a mixed norm space involving the negative homogeneous Besov space of order geq -n/2 in the space variable, without a smallness assumption on the appropriate norm. The space for the initial data is physically relevant since it contains spaces of the type L^p_x (L^2_x), by the use of Besov-Lipschitz space embeddings. Due to the nature of the vector valued spaces, we need to use a vector analogue of the Calderon-Zygmund theory to prove the necessary nonlinear energy estimates. These results hold both in the hard and soft potential case. Furthermore, in the hard potential case, we prove additional optimal decay results if the order of the Besov space belongs to [-(n+2)/2,-n/2). The latter result requires a closer study of the spectrum of the linearized Boltzmann operator for small frequencies dual to the spatial variable. This is a joint work with Robert Strain.

December 4

Frederic Bernicot, University of Nantes **Please note special day and place of this event 10am room 1314

Transport of BMO-type spaces by a measure-preserving map and applications to 2D Euler equation

```
We will present some sharp estimates about the transport of BMO-type spaces, via a bi-Lipschitz measure preserving map
in the Euclidean space. More precisely, we are interested in inequalities of the following type: $$ \| f(\phi) \|_X \lesssim C(\phi) \|f\|_X $$
where $X$ is a space like BMO, Lipschitz space, Hardy space, Carleson measure spaces .... and $\phi$ is a bi-Lipschitz measure preserving map.
The aim is to prove such inequalities with a sharp constant $C(\phi)$. We want to emphasize how the "measure preserving" property allows us to
get some improved inequalities. Then, we will explain how we can use this argument to describe a new framework for 2D Euler equations. We can define a space strictly containing $L^\infty$ where global well-posedness results can be proved with the vorticity living in this new space.
```

December 6

Jonathan Luk, Princeton University **Please note this seminar is at 11am

Impulsive Gravitational Waves

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We study the problem of the propagation and nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. For a single impulsive gravitational wave, we show that in the resulting spacetime, the delta singularity propagates along a 3-dimensional characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. We also construct spacetimes representing interaction of two impulsive gravitational waves in which the curvature delta singularities propagate along two 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away from the continuing propagating individual waves. This is joint work with Igor Rodnianski.
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Remi Schweyer, Toulouse**Please note this seminar is at 11:55am

Stable blow-up dynamics for the parabolic-elliptic Keller-Segel model of chemotaxis

December 13

Clement Mouhot, Cambridge

Spring 2012

Tianling Jin (Rutgers)

In this talk, we introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(S^n, [g_{S^n}])$, it converges to the standard sphere up to a Mobius diffeomorphism. This result allows us to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the remainder term. This is joint work with Jingang Xiong.

January 25 ** Special Wednesday Seminar**

Frederic Bernicot

Differential inclusions describing unilateral constraints

We will present existing and new results about differential inclusions, involving proximal normal cone. This tool aims to encode unilateral constraints in some differential equations of order 1 and 2. Using some convex analysis, we presnet results of global existence for solutions of such equations and we will discuss about the uniqueness of them

January 26

Lazhar Tayeb (University of Tunis)

Approximation by diffusion and homogenization for Fermi-Dirac-Poisson Statistics

We study the diffusion approximation of a Boltzmann-Poisson system dealing with Fermi-Dirac statistics in the presence of an extra external oscillating electrostatic potential. The relative entropy disspation and the two-scale Young measures are used to prove a two-scale strong convergence leading to a nonlinear Drift-diffusion with a effective potential and coupled to Poisson equation.

February 2

Hideyuki Miura

On fundamental solutions for fractional diffusion equations with divergence free drift

We are concerned with fractional diffusion equations in the presence of a divergence free drift term. By using the Nash approach, we show the existence of fundamental solutions, together with the continuity estimates, under weak regularity assumptions on the drift. Our results give the alternative proof of Caffarelli-Vasseur's theorem on the regularity for the critical 2D dissipative quasi-geostrophic equations. This is a joint work with Yasunori Maekawa.

February 9

Alessio Figalli, UT Austin ** this talk is at 10am**

Di Perna-Lions theory, with application to semiclassical limits for the Schrodinger equation

At the beginning of the '90, DiPerna and Lions studied in detail the connection between transport equations and ordinary differential equations. In particular, by proving an existence and uniqueness result at the level of the transport equation, they obtained (roughly speaking)

existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend such a result to BV vector fields. In some recent works we have investigated this theory in a more general setting, which allows us to show the semiclassical convergence of the quantum dynamics to the Liouville dynamics for the linear Schrodinger equations, under very weak regularity assumptions on the potential. In analogy to the classical DiPerna-Lions' theory, the price to pay for allowing singular potential is that the convergence result holds true only for "a.e. initial data", where "a.e." is with respect to a suitable family of reference measures in the space of the initial data. The aim of this talk is to give an overview of these results.

Mahir Hadzic (MIT)

The Classical Stefan problem and the vanishing surface tension limit

We develop a new unified framework for the treatment of well-posedness for the Stefan problem with and without surface tension. We provide new estimates for the regularity of the moving surface in the absence of surface tension. We conclude by proving that solutions of the Stefan problem with positive surface tension converge to solutions of the

Stefan problem without surface tension. This is joint work with S. Shkoller.

February 16

Paul Feehan (Rutgers)

Degenrate Obstacle Problems

Degenerate elliptic and parabolic obstacle problems arise in mathematical finance when valuing American-style options on an underlying asset modeled by a degenerate diffusion process. We will describe our work on existence, uniqueness, and regularity of solutions to stationaryand evolutionary variational inequalities and associated obstacle problemswhen the underlying asset is modeled by a degenerate diffusionprocess.This is joint work with Panagiota Daskalopoulos (Department ofMathematics, Columbia University) and Camelia Pop (Department ofMathematics, Rutgers University).

February 23

Vitaly Moroz (Swansea, UK)

Existence and concentration for nonlinear Schroedinger equations with fast decaying potentials

We discuss the existence of positive stationary solutions for a class of nonlinear Schrödinger equations. Amongst other results, we prove the existence of semi-classical solutions which concentrate around a positive local minimum of the potential. The novelty is that no restriction is imposed on the rate of decay of the potential at infinity. In particular, we cover the case where the potential is compactly supported. This is joint work with Jean Van Schaftingen(Louvain-la-Neuve, Belgium)

Erwan Faou

March 1

Frederic Rousset (Rennes)

Uniform regularity and inviscid limit for free surface Navier-Stokes

March 8

Masashi Aiki

March 9 ** this talk is at 1pm in WWH 512

Pengfei Guan (McGill University)

On a uniqueness problem in classical geometry and the maximum principle

March 15

P-E Jabin

March 22

Tuomas Hytonen

How much, or little, is needed for Harmonic Analysis?

One aspect of my recent work has been developing harmonic analysis under minimal assumptions on the space on which the considered functions are defined. Perhaps surprisingly, some classical methods, which at first sight seem to rely heavily on the structure and symmetries of the Euclidean space, can actually be extended to very general settings. On the other hand, some methods developed to tackle with abstract spaces, have shown to be instrumental for getting sharp results for classical inequalities on the Euclidean space.

March 27 ** Special Seminar 3/27 @ 11:30a.m.**

Zhen Lei

On Incompressible Euler and navier-Stokes Equations

In this talk I will report our recent results on finite time finite energy singularities of a 3D incompressible inviscid model of Euler and Navier-Stokes equations and a Liouville theorem for axi-symmetric navier-Stokes equations.

March 29

Victor Lie, Princeton University

Topics in the time-frequency analysis

This talk will be structured as follows: we start by discussing several facts about the history and evolution of the time-frequency area (Fourier Series, Calderon-Zygmund theory, wave-packet theory) and then refer to the general approach/tools for solving a time-frequency problem. Next we present two fundamental topics in this ¯eld: the pointwise con- vergence of the Fourier Series and the boundedness of the Bilinear Hilbert transform. Further we address several aspects of our work including the problem of the boundedness of the Bilinear Hilbert transform on smooth curves and the question regarding the boundedness of the Polynomial Car-leson operator

April 5

Cyrill Muratov

Asymptotic properties of ground states of scalar field equations with vanishing parameter

−Δu + εu−|u|

We study the leading order behavior of positive solutions of the equation

April 10I will discuss a non-local eigenvalue problem that arises as the Euler-Lagrange equation of Rayleigh quotients in the fractional Sobolev spaces. This can be seen as a non-local or fractional version of the eigenvalue problem for the p-Laplacian. In particular, I will talk about the limiting case when p goes to infinity for which the eigenvalues exhibit some strange behaviour that can be seen even in some one-dimensional examples.

Eric Lindgren (NTNU)

Fractional eigenvalues

April 12

David Gerard-Varet (Paris 7)

Dynamics of a rough body in a viscous fluid

April 19

Marcel Guardia, IAS **please note this seminar is at 10am**

Growth of Sobolev norms for the cubic defocusing nonlinear Schr\"odinger equation in polynomial time

We consider the cubic defocusing nonlinear Schr\"odinger equation in the two dimensional torus. Fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved existence of solutions with s-Sobolev norm growing in time by any given factor R. Refining their methods in several aspects we find solutions with s-Sobolev norm growing in polynomial time in R. This is a joint work with V. Kaloshin.

Dehua Wang (University of Pittsburgh)

April 26

Alberto Bressan (Penn State)

Nash equlibria for a odel of traffic flow

In connection with the Lighthill-Whitham model of traffic flow, a cost functional can be introduced depending on the departure

time and on the arrival time of each driver. Under natural assumptions, there exists a unique globally optimal solution, minimizing the total cost to all drivers. In a realistic situation, however, the actual traffic is better described by the Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution can be provided, establishing its existence and uniqueness. Extensions to the case of several groups of drivers on a network of roads will also be discussed, together with open problems.

May 1 **Please note special day**

Yifeng Yu (UC Irvine)

G-equations in the modeling of turbulent flame speeds

Predicting turbulent flame speed ($s_T$) is a fundamental problem in turbulent combustion theory. Several simplified models havebeen proposed to study $s_T$. The G-equation (A Hamilton-Jacobi level set equation) is a very popular model in turbulent combustion. Two important projects are (1) establish the theoretical existence of $s_T$ and (2)determine the dependence of turbulent flame speeds on the turbulence intensity (think of the relation between the spreading velocity of wild fire and strength of the wind). In this talk, I will present sometheoretical results under the G-equation model. If time permits, I will also compare it with predictions from a model introduced by Majda and

Souganidis. These are joint works with Jack Xin. May 3

Anne-Sophie de Suzzoni (CERGY)

On statistical description of the flow of dispersive PDEs

May 10

Ben Schweitzer (University of Dortmund)

Homogenization of Maxwell equations in complex geometries: on the counter-intuitive behavior of meta materials

Optically active meta-materials can nowadays be constructed as physical objects. They can have astonishing properties or lead to striking effects, the key-words are negative refraction, perfect imaging, and cloaking. I will present the effect of a negative magnetic permeability of the effective material and perfect light transmission through small holes in a metallic structure.

Mathematically, we analyze the time-harmonic Maxwell equations in a heterogeneous medium, the coefficients of the equation can oscillate on a small spatial scale and the oscillations of the values can be very large. The heterogeneity of the optical medium is prescribed by specifying the permittivity, which varies on a small length scale \eta. The electric and magnetic fields are determined by the time-harmonic Maxwell system. We analyze the weak limits of the electric and magnetic fields as \eta tends to zero, obtaining an "effective equation" that characterizes the limits. The coefficients of the effective equation describe the behavior of the metamaterial.

This is joint work with G. Bouchitte and with A. Lamacz.

May 17

Benjamin Texier

The onset of instability in quasi-linear systems

May 31

Messoud Efendiev

Finite and infinite dimensional attractors for porous medium
equations

June 7

Slim Ibrahim, University of Victoria ***Please note this talk is at
10:00am

Existence of a ground state
and scattering for a nonlinear Schroedinger equation with
critical growth

This talk concerns the focusing energy-critical nonlinear
Schroedinger equation with a mass-supercritical and
energy-subcritical perturbation. In particular, we consider the
existence of a ground state and the scattering problem in
the spirit of Kenig-Merle.

This is a joint work with Akahori, Kikushi and Nawa

Adimurth Tifrcam, Bangalore **Please note
this talk is at 11:00 am

Structure Theorem for entropy
solutions of Conservation Law

Camil Muscalu, Cornell University

Zaher Hani, CIMS

Xavier Cabre, Universitat Politecnica de Catalunya

Benoit Pausader, CIMS

George Hagstrom, CIMS

Federica Sani, CIMS and Univ of Milan

Jacob Bedrossian, CIMS

Special Analysis Seminar

2 pm room 1302 WWH Nassif Ghoussoub

Vincent Duchene, Columbia University

Alex Ionescu, Princeton

I will discuss some recent work, joint with S. Klainerman, on the problem of extension of Killing vector-fields in manifolds that satisfy the Einstein vacuum equations. This problem is motivated by the black hole rigidity conjecture, concerning the uniqueness of the Kerr family among regular, stationary black hole solutions of the Einstein vacuum equations.

Comgming Li, University of Boulder

Christophe Lacave, Universite Paris-Diderot (Paris 7)

The
well-posedness of the Euler system has been of course the matter
of many works, but a common point in all the previous studies is
that the boundary is at least $C^{1,1}$. In a first part, we
will establish the existence of global weak solutions of the 2D
incompressible Euler equations for a large class of non-smooth
open sets. These open sets are the complements (in a simply
connected domain) of a finite number of connected compact sets
with positive capacity. Existence of weak solutions with $L^p$
vorticity is deduced from an approximation argument, that
relates to the so-called $\gamma$-convergence of domains. In a
second part, we will prove the uniqueness if the open set is the
interior or the exterior of a simply connected domain, where the
boundary has a finite number of corners. Although the velocity
blows up near these corners, we will get a similar theorem to
the Yudovich's result, in the case of an initial vorticity with
definite sign, bounded and compactly supported. The key point
for the uniqueness part is to prove by a Liapounov energy that
the vorticity never meets the boundary. The existence part is a
work in collaboration with David Gerard-Varet.

SPECIAL
ANALYSIS
SEMINAR, November 22,
11 a.m., room 1314

Shen Zhongwei, University of Kentucky

The Periodic Homogenization of
Green's and Neumann Functions

Oana Pocovnicu (Imperial College, London)

We consider
the non-linear wave equation on the real line iu_t-|D|u=|u|^2u.
Its resonant dynamics is given by the Szego equation, which is a
completely integrable non-dispersive non-linear equation. We
show that the solution of the wave equation can be approximated
by that of the resonant dynamics for a long time. The proof uses
the renormalization group method introduced by Chen, Goldenfeld,
and Oono in the context of theoretical physics. As a
consequence, we obtain growth of high Sobolev norms of certain
solutions of the non-linear wave equation, since this phenomenon
was already exhibited for the Szego equation.

December
8

Aynur Bulut (IAS)

The defocusing Cubic Nonlinear Wave Equation in the Energy
Super-critical Regime

In this talk, we will discuss a
series of recent works on the global well-posedness and
scattering conjecture for the defocusing cubic nonlinear wave
equation in the energy super-critical regime, that is dimensions
five and higher. More precisely, using a concentration
compactness approach we show that if a solution remains bounded
in the critical Sobolev space throughout its maximal interval of
existence then it is global and scatters.

Daniela Tonon (ICERM and SISSA)

We present
two results on the regularity of viscosity solutions of
Hamilton-Jacobi equations obtained in collaboration with
Professor Stefano Bianchini. When the Hamiltonian is strictly
convex viscosity solutions are semiconcave, hence their gradient
is BV. First we prove the SBV regularity of the gradient of a
viscosity solution of the Hamilton-Jacobi equation u_t+
H(t,x,D_x u)=0 in an open set of R^(n+1), under the hypothesis
of uniform convexity of the Hamiltonian H in the last variable.
Secondly we remove the uniform convexity hypothesis on the
Hamiltonian, considering a viscosity solution u of the
Hamilton-Jacobi equation u_t+ H(D_x u)=0 in an open
set of R^(n+1) where H is smooth and convex. In this case the
viscosity solution is only locally Lipschitz. However when the
vector field d(t,x):=H_p(D_xu(t,x)), here H_p is the gradient of
H, is BV for all t in [0,T] and suitable hypotheses on the
Lagrangian L hold, the divergence of d(t, ) can have
Cantor part only for a countable number of t's in [0,T]. These
results extend a result of Bianchini, De Lellis and Robyr for a
uniformly convex Hamiltonian which depends only on the
spatial gradient of the solution.

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