# 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).