Past Partial Differential Equations Seminar

In this talk we discuss a new second order parabolic evolution equation

for hypersurfaces in space-time initial data sets, that generalizes mean

curvature flow (MCF). In particular, the 'null mean curvature' - a

space-time extrinsic curvature quantity - replaces the usual mean

curvature in the evolution equation defining MCF.  This flow is motivated

by the study of black holes and mass/energy inequalities in general

relativity. We present a theory of weak solutions using the level-set

method and  outline a natural application of the flow as a parabolic

approach to finding outermost marginally outer trapped surfaces (MOTS),

which play the role of quasi-local black hole boundaries in general

relativity. This is joint work with Kristen Moore.

We explore a specific system in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli 
[Phys Rev Lett 111, 174302, 2013].

There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, D\'{e}mery, and Davidovitch models this regime using a von-K\'{a}rm\'{a}n-like 
variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling. This is a joint work with  Robert V. Kohn in Courant Institute, NYU.

We give a new characterization of the property that the elliptic measure
belongs to the infinity weight Muckenhoupt class
in terms of a Carleson measure property of bounded solutions.
This is joint work with C.Kenig, J.Pipher and T.Toro

It has been shown that there are global solutions to 
Maxwell-Klein-Gordon equations in Minkowski space with finite energy 
data. However, very little is known about the asymptotic behavior of the 
solution. In this talk, I will present recent progress on the decay 
properties of the solutions. We show the quantitative energy flux decay 
of the solutions with data merely bounded in some weighted energy space. 
The results in particular hold in the presence of large total charge. 
This is the first result that gives a complete and precise description 
of the global behavior of large nonlinear fields.
 

In acoustic materials (non null sound velocity), there is a clear separation of scale between the relaxation to mechanical equilibrium, governed by Euler equations, and the slower relaxation to thermal equilibrium, governed by heat equation if thermal conductivity is finite. In one dimension in acoustic systems, thermal conductivity is diverging and the thermal equilibrium is reached by a superdiffusion governed by a fractional heat equation. In non-acoustic materials it seems that there is not such separation of scales, and thermal and mechanical equilibriums are reached at the same time scale, governed by a Euler-Bernoulli beam equation. We prove such macroscopic behaviors in chains of oscillators with dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. (Works in collaborations with T. Komorowski).

I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions and present a result about the existence of generalized “transition waves”. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.

While it is believed that many particle systems have periodic ground states, there are few rigorous crystallization results in two and more dimensions. In this talk I will show how results by the Hungarian geometer László Fejes Tóth can be used to prove that an idealised block copolymer energy is minimised by the triangular lattice. I will also discuss a numerical method for a broader class of optimal location problems and some conjectures about minimisers in three dimensions. This is joint work with Mark Peletier, Steven Roper and Florian Theil. 

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

For maps from surfaces there is a close connection between the area of the surface parametrised by the map and its Dirichlet energy and this translates also into a relation for the corresponding critical points. As such, when trying to find minimal surfaces, one route to take is to follow a suitable gradient flow of the Dirichlet energy. In this talk I will introduce such a flow which evolves both a map and a metric on the domain in a way that is designed to change the initial data into a minimal immersions and discuss some question concerning the existence of solutions and their asymptotic behaviour. This is joint work with Peter Topping.

I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of $\alpha$-harmonic maps.

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. 

We study the low-temperature limit in the Landau-de Gennes theory for liquid crystals. We prove that for minimizers for orientable Dirichlet data tend to be almost uniaxial but necessarily contain some biaxiality around the singularities of a limiting harmonic map. In particular we prove that around each defect there must necessarily exist a maximal biaxiality point, a point with a purely uniaxial configuration with a positive order parameter, and a point with a purely uniaxial configuration with a negative order parameter. Estimates for the size of the biaxial cores are also given.

This is joint work with Apala Majumdar and Adriano Pisante.

The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.

Leinster and Willerton have introduced the concept of the magnitude of a metric space, as a special case as that of an enriched category. It is a numerical invariant which is designed to capture the important geometric information about the space, but concrete examples of ts values on compact sets in euclidean space have hitherto been lacking. We discuss progress in some conjectures of Leinster and Willerton.

We study the adaptive finite element approximation of the Dirichlet problem $-\Delta u = f$ with zero boundary values using newest vertex bisection. Our approach is based on the minimization of the corresponding Dirichlet energy. We show that the maximums strategy attains every energy level with a number of degrees of freedom, which is proportional to the optimal number. As a consequence we achieve instance optimality of the error. This is a joint work with Christian Kreuzer (Bochum) and Rob Stevenson (Amsterdam).

We establish sharp Sobolev inequalities of order four on Euclidean $d$-balls for $d$ greater than or equal to four. When $d=4$, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls. This is joint work with Alice Chang. 

This will be an overview of Prof Stroffolini's research and precursor to the eight-hour mini-course Prof Stroffolini will be giving later in October.

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 in addition has singular initial data because of 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.

We consider the system of nonlinear differential equations
\begin{equation}
(1) \qquad
\begin{cases}
\dot u_n(t) + \lambda^{2n} u_n(t) 
- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\
u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.
\end{cases}
\end{equation}
In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.