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.

# Past PDE CDT Lunchtime Seminar

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.

In this talk I will illustrate how solutions of nonlinear nonlocal Fokker-Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined outside such domain. Two different approaches are analysed, making crucial use of uniform estimates for energy and entropy functionals respectively. In both cases we prove that the problem in a bounded domain can be seen as a limit problem in the whole space involving a suitably chosen sequence of large confining potentials.

This is joint work with Maria Bruna and José Antonio Carrillo.

This talk will be an introduction to the use of conformal methods in asymptotic analysis in general relativity. We shall consider the explicit example of flat spacetime (Minkowski spacetime). The full conformal compactification will be constructed. For a simple example of a conformally invariant equation (we'll take the wave equation), we shall see how the compactification allows to infer precise informations on the asymptotic behaviour of the solution in all directions, for a certain class of data at any rate. Then, depending on time and questions, I will either describe how a scattering theory can be constructed using the same method or, explain how conformal methods can be used on other asymptotically flat geometries.

Harmonic map equations are an elliptic PDE system arising from the

minimisation problem of Dirichlet energies between two manifolds. In

this talk we present some some recent works concerning the symmetry

and stability of harmonic maps. We construct a new family of

''twisting'' examples of harmonic maps and discuss the existence,

uniqueness and regularity issues. In particular, we characterise of

singularities of minimising general axially symmetric harmonic maps,

and construct non-minimising general axially symmetric harmonic maps

with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.

Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$

to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for

$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and

Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)