# Analysis Seminar 2015-16

### The seminars are held on Thursdays at 16:00 in Room B3.02 - Mathematics Institute unless noted otherwise

### Term 1 2015/16

Organiser: Jose Rodrigo

15th October **Kai Zheng** (Warwick)

**Title:** *Asymptotic behaviour of global solutions to the geometric flows
*

**Abstract:**In this talk, we will present a general approach, from the geometric analysis point of view, to study the asymptotic behaviour of global solutions to the geometric flows, with emphasis on the Kahler-Ricci flow, the Calabi flow, etc. Key aspects of this approach include: the geometry of infinite-dimensional manifolds, the uniqueness of the critical metrics, the backward estimates and the description of the singularities.

22nd October **Tomasz Cieslak** (Warsaw)

**Title:** *Spirals of vorticity, a measure theory point of view
*

**Abstract:**In my talk I will treat a special type of vortex sheets, spirals of vorticity, as time-dependent Borel measures $\mu_t$ satisying

a condition $\mu_t(B(0,r))=C(t)r^{\alpha}$ for some $C(t), \alpha>0$. It turns out that Prandtl spirals satisfy the above condition,

I will also discuss physical arguments suggesting that Kaden spirals should also satisfy it. Next I will show that such measures

have locally finite kinetic energy, I will discuss the relation of such measures with the so-called Morrey measures.

At the end of my talk I would like to address the question of conservation of the energy by such objects. The talk will be based on

the results in T. Cieslak, M.Szumanska JFA 2014, G. Jamroz CRAS 2015 as well as recent computation done in collaboration with M.Preisner.

29th October **Witold Sadowski** (Warsaw)

**Title:** *On the long time behaviour of solutions of the dissipative Euler equation
*

**Abstract:**(joint work with L. Berselli - Pisa).

5th November **Jim Wright** (Edinburgh)

**Title:** *Lebesgue Constants: connections with pointwise ergodic theorems
*

**Abstract:**The classical Lebesgue constant for continuous periodic functions is useful in the study of pointwise and uniformly convergent fourier series. We examine variants for functions with a sparse spectrum and in particular we look at extensions to functions of several variables. Interestingly there are some connections with extensions and generalisations of Bourgain’s work on pointwise ergodic theorems along sparse subsets of integers.

12th November **Judith Campos** (Augsburg)

**Title:** *Full regularity and uniqueness for a class of minimizers in the quasiconvex setting
*

**Abstract:**Under suitably small boundary conditions, global minimizers of strongly quasiconvex integrands are smooth up to the boundary and, even more, they can be shown to be unique. These results appear in contrast to the classical partial regularity results for minimizers, as well as to the non-uniqueness previously established by Spadaro (based on a counterexample due to Lawson and Osserman). This is joint work with Jan Kristensen.

19th November **Ulisse Stefanelli **(Vienna)

**Title:** *Carbon geometries as energy minimizers
*

**Abstract:**Graphene, nanotubes, and fullerenes are locally planar carbon nanostrcuctures: each atom forms three covalents bonds which (ideally) create bonding-angles of $2\pi/3$. In distinguished regimes, this phenomenology can be modeled by minimizing specific atomic-interaction potentials including three-body interaction terms [T].

I intend to review some of the existing crystallization results for these of potentials [E,M,M2]. I will focus on the possibility of characterizing the geometry of ground states and local minimizers, especially in three-dimensions. Stability, fine geometry, emergence of Wulff shapes, and nanomechanics of classes of nanostructures will be discussed.

This is joint work with E. Davoli, M. Friedrich, E. Mainini, H. Murakawa, and P. Piovano.

[E] W. E, D. Li. On the crystallization of 2D hexagonal lattices. Comm. Math. Phys. 286 (2009) 1099-1140.

[M] E. Mainini, U. Stefanelli. Crystallization in carbon nanostructures. Comm. Math. Phys. 328 (2014) 545-571.

[M2] E. Mainini, P.Piovano, U. Stefanelli. Finite crystallization in the square lattice. Nonlinearity, 27 (2014) 717-737.

[T] J. Tersoff. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37 (1988) 6991-7000.

26th November **Manuel del Pino** (Chile)

**Title:** Bubbling blow-up in critical parabolic problems

**Abstract:** We construct solutions with finite and infinite time type-II blow-up (and analyze their stability) in two related parabolic problems:

the standard semilinear heat equation with a power nonlinearity at the critical exponent in a bounded domain in $\mathbb{R}^N$, and the harmonic map flow from a two-dimensional domain into the sphere $\mathbb{S}^2$. Both problems have stationary states with energy scaling-invariance in entire space which are the building blocks of the bubbling patterns.

3rd December** Emil Wiedemann** (Bonn)

**Title:*** Weak-Strong Uniqueness in Fluid Dynamics
*

**Abstract:**Various concepts of weak solution have been suggested for the fundamental equations of inviscid fluid flow over the last few decades. A common problem is the vast degree of non-uniqueness that all these types of solution exhibit. Nevertheless, a conditional notion of uniqueness, the so-called weak-strong uniqueness, can be established in various situations. We present some recent results, both positive and negative, on weak-strong uniqueness in the realm of incompressible and compressible Euler flows, and for a related model of avalanche motion.

10th December **Lucia Scardia** (Bath)

**Title:** *Convergence of Interaction-driven Evolutions of Dislocations
*

**Abstract:**TBA

### Term 2 2015/16

Organiser: Peter Topping

28 January** Juraj Főldes** (Université Libre de Bruxelles)

*Convergence of invariant measures in randomly forced Boussinesq system*

In this talk we investigate properties of invariant measures for the Boussinesq equations in the presence of a degenerate random forcing acting only in the temperature component. The main goal is to prove convergence of invariant measures in singular limits when Prandtl numbers approach infinity. As an application we recover estimates for the Nusselt number in a stochastic setting.

More precisely, we show a general framework for converting the problem of convergence of measures to the question of finite time convergence of solutions. Then we analyze singular limit problems in a stochastic setting.

This is a joint work with S. Friedlander (U. of Southern California), N. Glatt-Holtz (Virginia Tech), G. Richards (U. of Rochester), E. Thomann (Oregon State), and J. Whitehead (Brigham Young U.).

4 February **Daniele Valtorta** (EPFL)

*Singularities of harmonic maps*

In this talk we present some new regularity results proved for the singular sets of minimizing and stationary harmonic maps in collaboration with Aaron Naber (see arXiv:1504.02043).

We prove that the singular set of a minimizing harmonic map is rectifiable with effective *n*-3 volume estimates. The results are based on an improved quantitative stratification technique, which consists in a detailed analysis of the symmetries and almost symmetries of the map u and its blow-ups at different scales, and rely on a new* W ^{1,p}* version of Reifenberg's topological disk theorem. The application of this theorem in the situation of harmonic maps hinges on the monotonicity formula for the normalized energy.

Similar results are available for minimizing and stationary currents (see arXiv:1505.03428)

11 February** Otis Chodosh** (Cambridge/Princeton)

* Area minimizing surfaces in asymptotically flat three manifolds*

I will discuss recent work with M. Eichmair. We show that an asymptotically flat three-manifold with non-negative scalar curvature cannot admit an unbounded area minimizing surface unless the ambient space is flat.

18 February **Yong Wei** (UCL)

*The Laplacian flow for closed G _{2 }structures*

In this talk, we discuss the Laplacian flow for closed G_{2} structures. This flow was introduced by R. Bryant in 1992 to study the geometry of G_{2} structures, inspired by Hamilton's Ricci flow in studying the generic Riemannian structures and the Kahler-Ricci flow in studying Kahler structures. The primary motivation is to understand the conditions under which the Laplacian flow can converge to a torsion free G_{2} structures, and thus Ricci flat metric with holonomy G_{2}. I will start with the background of G_{2} structure and the motivation of introducing the Laplacian flow, and then describe recent progress on this flow (Joint work with Jason D. Lotay).

25 February **Tobias Huxol** (Warwick)

*Limiting behaviour of the Teichmuller harmonic map flow*

3 March **Matteo Novaga** (Pisa)

*Singularities of timelike extremal surfaces*

10 March **Marco Romito** (Pisa)

*Regularity for logarithmically hyperdissipative Navier-Stokes equations and related models*

We give some hints on the evolution and interaction of energy in models for the motion of incompressible fluids.

In the first part of the talk we prove global existence of smooth solutions for slightly supercritical hyperdissipative Navier–Stokes equations under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao in 2009. The proof is based on the idea that smoothness can be reduced to the smoothness of a suitable shell model, obtained by averaging the energy of a solution over dyadic shells in Fourier space.

In the second part of the talk we further prove that, at least at the level of a very simple dyadic model - a system of infinitely many differential equations that should capture the essential features of the energy of a fluids, global existence of smooth solutions can be proved in a full supercritical regime.

17 March** Dominik Inauen** (Uni- Zuerich)

*A Nash-Kuiper Theorem for C ^{1,⅕-ε} embeddings of surfaces in 3 dimensions*

* *

### Term 3 2015/16

Organiser: Jose Rodrigo

28th April **Manuel Friedrich** (Vienna)

**Title:** *Korn inequalities for special functions of bounded deformation
*

**Abstract:**The function spaces SBV (special functions of bounded variation) and SBD (special functions of bounded deformation) play a fundamental role in the variational analysis of damage and fracture models. For problems in a geometrically linear setting formulated in SBD major difficulties arise from the fact that in contrast to SBV for generic configurations only the symmetric part of the distributional derivative can be controlled. Therefore, it is desirable to gain a profound understanding of the relation between SBV and SBD functions. In this seminar I will present some results in this direction, which are based on the validity of certain Korn-type inequalities in these function spaces.

5th May **Mike Jolly** (Indiana)

**Title:** *Determining forms and data assimilation
*

**Abstract:**A determining form for a dissipative PDE is an ODE in a certain trajectory space where the solutions on the global attractor of the PDE are readily recognized. It is an ODE in the true sense of defining a vector field which is (globally) Lipschitz.

In this talk we focus on one type of determining form where solutions on the global attractor of the PDE are identified as steady states of the

form. This determining form is related to data assimilation by feedback nudging, which is one way to inject a coarse-grain time series into the model in order to recover the matching full solution. We show that for a given initial trajectory, the dynamics of this determining form reduces to that of a

one-dimensional ODE.

We prove lower bounds on the rates of convergence to trajectories in the attractor of the original system, and demonstrate numerically that they are achieved. Applications have been made to the 2D incompressible Navier-Stokes, damped-driven nonlinear Schrodinger, damped-driven Korteveg-de Vries and surface quasigeostrophic equations.

12th May CANCELLED (see 16 June)

19th May **Paul Bryan** (Warwick)

**Title:** *Ancient Solutions For Hypersurface Flows In The Sphere
*

**Abstract:**Ancient solutions of a parabolic PDE are those solutions existing on an interval $(-\infty,T)$. They are an important class of solutions,

often simpler in nature than arbitrary solutions, possibly exhibiting symmetry properties for example. In some circumstances they may also

be used as comparisons against general solutions, allowing for quantitative estimates. Moreover, importantly they arise as singularity

models for curvature flows. As such their study is important in understanding both intuitively and more rigorously the behaviour of

solutions, particularly singularity formation. In Euclidean space, ancient solutions, particularly of the Mean Curvature Flow have been

extensively, and are actively studied with partial classification results due to Daskalopoulus, Hamilton, Sesum and Huisken and Sinestrari. Working

in the sphere rather than Euclidean space,I will show that for just about any parabolic, geometric hypersurface flow, the only ancient solutions are

shrinking geodesic spheres. The methods used are largely geometric, more PDE-centric methods being unsuitable for the task.

26th May **Silvia Sastre Gomez** (University College Cork)

**Title:** *Steady periodic Water Waves with Discontinuous Vorticity
*

**Abstract:**In this work we study steady two-dimensional periodic water waves problems over a fixed depth with the vorticity discontinuous. We consider a modified height function, which explicitly introduces the mean depth into the rotational water wave problem. Since the vorticity is discontinuous, the equations are expressed in a weak form and the solutions are considered in the sense of distributions. We use Crandall-Rabinowitz local bifurcation to prove the existence of weak solutions.

2nd June **Niels Laustsen** (Lancaster)

**Title:** *Splittings of extensions of the algebra of bounded operators on a Banach space
*

**Abstract:**By an extension of a Banach algebra $B$, we understand a short-exact sequence of the form

\[ \{0\}\to\ker\varphi\to A\overset{\varphi}{\to} B\to \{0\}, \]

where $A$ is a Banach algebra and $\varphi: A\to B$ is a continuous, surjective algebra homomorphism. The extension splits algebraically (respectively, splits strongly) if $\varphi$ has a right inverse which is an algebra homomorphism (respectively, a continuous algebra homomorphism).

Bade, Dales and Lykova (Mem. Amer. Math. Soc. 1999) carried out a comprehensive study of extensions of Banach algebras, focusing in particular on the following automatic-continuity question: For which (classes of) Banach algebras $B$ is it true that every extension of $B$ which splits algebraically also splits strongly?

In the case where $B = B(E)$ is the algebra of bounded operators on a Banach space $E$, Bade, Dales and Lykova recorded some partial results, but they left the general question open. We shall show that the answer is negative, even if one strengthens the hypothesis to demand that $\ker\varphi$ is complemented in $A$ as a Banach space.

The Banach space $E$ that we use is a quotient of the $\ell_2$-direct sum of an infinite sequence of James-type quasi-reflexive Banach spaces; it was originally introduced by Read (J. London Math. Soc. 1989).

No specialist knowledge of extensions will be assumed.

The talk is based on joint work with Richard Skillicorn and Tomasz Kania; see

arxiv:1409.8203

arxiv:1602.08963

arxiv:1603.04275

9th June **Angkana Rüland** (Oxford)

**Title:** *The variable coefficient thin obstacle problem: A Carleman approach
*

**Abstract:**In this talk I present a new, very robust approach of proving optimal regularity for the thin obstacle problem. This relies on replacing the usual monotonicity argument by a Carleman inequality. Based on this it is possible to deduce doubling inequalities, upper semi-continuity of

the vanishing order and the existence of homogeneous blow-up sequences also in the setting of metrics with low regularity. These properties are then exploited to prove the optimal regularity of solutions and to deduce regularity of the regular free boundary. If time permits, I will also discuss higher regularity properties of the regular free boundary.

This is joint work with H. Koch and W. Shi.

16th June** (Note 2 seminar talks)**

**14:00 Room MS.05
**

**Dan Ketover**(Princeton and Imperial)

**Title:** *Sharp entropy bounds of closed surfaces and min-max theory*

**Abstract:** In 2010, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface in R^3 is at least that of the self-shrinking two-sphere. I will explain joint work with X. Zhou where we interpret this conjecture as a parabolic version of the Willmore problem and give a min-max proof of (most cases) of their conjecture

**16:00 Room B3.02
**

**Panagiota Daskalopoulos**(Columbia)

**Title:** *Ancient solutions to geometric flows
*

23rd June **(Note 2 seminar talks)**

**14:30 Room B3.02
Sigurd Angenent** (Wisconsin-Madison)

**Title: **Mean curvature flow from cones, and other sources of non uniqueness in MCF

**Abstract: **Self similar shrinking solutions to Mean Curvature Flow with conical ends are completely determined by their asymptotic cone at infinity, due to a result of Lu Wang. In contrast, there exist older examples of cones that admit more than one smooth expanding self similar evolution by MCF. In this talk I will exhibit examples of smooth cones that admit many more smooth solutions by MCF, most of which are not self similar.

**16:00 Room B3.02
Spyros Alexakis** (Toronto)

**Title:** Singularity formation in Black hole interiors.

**Abstract:**The prediction that solutions of the Einstein equations in the interior of black holes must always terminate at a singularity was originally conceived by Penrose in 1969, under the name of “strong cosmic censorship hypothesis”. The nature of this break-down (i.e. the asymptotic properties of the space-time metric as one approaches the terminal singularity) is not predicted, and remains a very hotly debated question to this day. One key question is the causal nature of the singularity (space-like, vs null for example). Another is the rate of blow-up of natural physical/geometric quantities at the singularity. Mutually contradicting predictions abound in this topic. Much work has been done under the assumption of spherical symmetry (for various matter models). We present recent developments (partly due to the speaker and G. Fournodavlos) which go well beyond this restrictive class. A key role is played by the axial symmetry reduction of the Einstein equations, where a wave map structure appears.