This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge

# Past PDE CDT Lunchtime Seminar

In this talk I will start with a brief overview of the Cauchy problem for the Einstein equations of general relativity, and in particular the nonlinear stability of the trivial Minkowski solution in wave gauge as shown by Lindblad and Rodnianski. I will then discuss the Kaluza Klein spacetime of the form $R^{1+3} \times K$ where $K$ is the $n-$torus with the flat metric. An interesting question to ask is whether this solution to the Einstein equations, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline how the proof of stability in a restricted class of perturbations in fact follows from the work of Lindblad and Rodnianski, and discuss the physical justification behind this restriction.

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.

In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

We consider the discrete Bilaplacian on a cube in two and three dimensions with zero boundary data and prove estimates for its Green's function that are sharp up to the boundary. The main tools in the proof are Caccioppoli estimates and a compactness argument which allows one to transfer estimate for continuous PDEs to the discrete setting. One application of these estimates is to understand the so-called membrane model from statistical physics, and we will outline how these estimates can be applied to understand the phenomenon of entropic repulsion. We will also describe some connections to numerical analysis, in particular another approach to these estimates based on convergence estimates for finite difference schemes.

We consider the symbiotic branching model, which describes a spatial population consisting of two types in terms of a coupled system of stochastic PDEs. One particularly important special case is Kimura's stepping stone model in evolutionary biology. Our main focus is a description of the interfaces between the types in the large scale limit of the system. As a new tool we will introduce a moment duality, which also holds for the limiting model. This also has implications for a classification of entrance laws of annihilating Brownian motions.

For energy functionals composed of competing short- and long-range interactions, minimizers are often conjectured to form essentially periodic patterns on some intermediate lengthscale. However, not many detailed structural results or proofs of periodicity are known in dimensions larger than 1. We study a functional composed of the attractive, local interfacial energy of charges concentrated on a hyperplane and the energy of the electric field generated by these charges in the full space, which can be interpreted as a repulsive, non-local functional of the charges. We follow the approach of Alberti-Choksi-Otto and prove that the energy of minimizers of this functional is uniformly distributed on cubes intersecting the hyperplane, which are sufficiently large with respect to the intrinsic lengthscale.

This is a joint work with A. Julia and F. Otto.

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.