The Einstein equations in wave coordinates are an example of a system

which does not obey the "null condition". This leads to many

difficulties, most famously when attempting to prove global existence,

otherwise known as the "nonlinear stability of Minkowski space".

Previous approaches to overcoming these problems suffer from a lack of

generalisability - among other things, they make the a priori assumption

that the space is approximately scale-invariant. Given the current

interest in studying the stability of black holes and other related

problems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises to

overcome this difficulty by providing a flexible and robust tool to

prove decay. However, so far it has mainly been used to treat linear

equations. In this talk I will explain how to modify this method so that

it can be applied to nonlinear systems which only obey the "weak null

condition" - a large class of systems that includes, as a special case,

the Einstein equations. This involves combining the p-weighted energy

method with many of the geometric methods originally used by

Christodoulou and Klainerman. Among other things, this allows us to

enlarge the class of wave equations which are known to admit small-data

global solutions, it gives a new proof of the stability of Minkowski

space, and it also yields detailed asymptotics. In particular, in some

situations we can understand the geometric origin of the slow decay

towards null infinity exhibited by some of these systems: it is due to

the formation of "shocks at infinity".

# Past PDE CDT Lunchtime Seminar

The finite element exterior calculus is a powerful approach to study many problems under the same lens. The canonical finite element spaces (see Arnold, Falk and Winther) are tied together with an exact sequence and have the required smoothness to define the exterior derivatives weakly. However, some applications require spaces that are more smooth (e.g. plate bending problems, incompressible flows). In this talk we will discuss some recent results in developing finite element spaceson simplicial triangulations with more smoothness, that also fit in an exact sequence. This is joint work with Guosheng Fu, Anna Lischke and Michael Neilan.

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.

In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.

The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).

Motivated by recent problems on mixing flows, it is useful to characterize Besov spaces via oscillation of functions (averages) and minimization problems for bounded variation functions (Bianchini-type norms). In this talk, we discuss various descriptions of Besov spaces in terms of different kinds of averages, as well as Bianchini-type norms. Our method relies on the K-functional of the theory of real interpolation. This is a joint work with S. Tikhonov (Barcelona).

We consider a non-linear PDE on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

In this talk, we demonstrate that formation of Type I singularities of suitable weak solutions of the Navier-Stokes equations occur if there exists non-zero mild bounded ancient solutions satisfying a 'Type I' decay condition. We will also discuss some new Liouville type Theorems. Joint work with Dallas Albritton (University of Minnesota).

In this talk, we study the long time behaviour of a cloud of weakly interacting Brownian particles, conditionally on the observation of their initial and final configuration. In particular, we connect this problem, which may be regarded as a nonlinear version of the Schrödinger problem, to the study of the long time behaviour of Mean Field Games. Combining tools from optimal transport and stochastic control we prove convergence towards the equilibrium configuration and establish convergence rates. A key ingredient to derive these results is a new functional inequality, which generalises Talagrand’s inequality to the entropic transportation cost.

Stein ($1981$) proved the borderline Sobolev embedding result which states that for $n \geq 2,$ $u \in L^{1}(\mathbb{R}^{n})$ and $\nabla u \in L^{(n,1)}(\mathbb{R}^{n}; \mathbb{R}^{n})$ implies $u$ is continuous. Coupled with standard Calderon-Zygmund estimates for Lorentz spaces, this implies $u \in C^{1}(\mathbb{R}^{n})$ if $\Delta u \in L^{(n,1)}(\mathbb{R}^{n}).$ The search for a nonlinear generalization of this result culminated in the work of Kuusi-Mingione ($2014$), which proves the same result for $p$-Laplacian type systems. \paragraph{} In this talk, we shall discuss how these results can be extended to differential forms. In particular, we can prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ $k$-differential form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous.

In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean–Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension. Our methods are motivated by the theory of stochastic viscosity solutions, which are applied after passing to the equation’s kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths. I will also mention the application of these methods to nonlinear diffusion equations with linear, multiplicative noise.

In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times.