# Partial Differential Equations and their Applications Seminar 2015-16

### Meetings are held on Tuesdays at 16.00-17.00 in D1.07.

##### Organiser: Charlie Elliott

These meetings provide an opportunity for individuals to discuss in an informal manner progress on their current work, and describe interesting new problems related to PDEs and their applications and computation.

We have chosen this time to allow us to make a pub visit afterwards if we feel inclined.

Group emails can be sent via applied_maths_pde_workgroup at listserv dot warwick dot ac dot uk.

## Term 2

#### Boltzmann-type models in socio-economic applications

Boltzmann's kinetic theory was originally developed to describe the statistical behavior of systems not in equilibrium, for example the thermodynamics of dilute gases. The Boltzmann equation states the evolution of the probability distribution function of particles due to microscopic interactions, such as collisions. In socio-economic applications these collisions correspond to meetings between agents.

In this talk we present two different applications of Boltzmann type models. First a price formation model, in which collisions correspond to trading events. Second a Boltzmann mean-field game model for knowledge growth, in which individuals divide their time between producing goods with the knowledge they already have and exchanging ideas in meetings to enhance their knowledge level. We discuss the modeling assumptions and analysis in both applications, and illustrate the behavior of the models with numerical experiments.

#### Free Boundary Problems from a Model for Receptor-Ligand Dynamics

We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium.
We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry.

#### Existence for a fractional porous medium equation on an evolving surface

In this talk, which is based on recent joint work with Charlie Elliott, I will present an existence theory for a porous medium equation with a fractional diffusion on an evolving surface. The nonlocal nature of the fractional diffusion (which in our case is the square root of the Laplacian) in combination with the nonlinearity and the moving domain makes the problem interesting. After defining the fractional Laplacian and giving a Dirichlet-to-Neumann map characterisation of it in a general setting of closed Riemannian manifolds, I will define what we mean by a weak solution and then proceed with the proof of existence. This will involve harmonic extensions on semi-infinite and truncated cylinders, convergence/decay estimates and some technical results in order to deal with the time-evolving surface. I will finish by discussing several open issues.

#### Lane formation by side-stepping

In this talk we present a non-linear convection-diffusion model, which describes the evolution of two pedestrian groups walking in opposite
direction. We start from a two-dimensional lattice based model and formally derive the corresponding limiting equations using Taylor
expansion. We introduce an entropy functional, which allows us to obtain the necessary estimates to prove global existence of bounded weak
solutions. The proposed system exhibits nontrivial stationary states which correspond to the formation of directional lanes. Furthermore we
study the behaviour of the model for different ranges of parameters and illustrate the dynamics with various numerical simulations.

#### 8th March - John Mackenzie (Strathclyde) -

##### A Computational Method for the Coupled Solution of Reaction-Diffusion Equations on Evolving Domains and Manifolds: Application to a Model of Cell Migration and Chemotaxis

In this talk I will present details about a moving mesh finite element method for the approximate solution of partial differential equations on an evolving bulk domain in two dimensions, coupled to the solution of partial differential equations on the evolving domain boundary. Problems of this type occur frequently in the modeling of eukaryotic cell migration and chemotaxis - for these applications the bulk domain is either the interior or exterior of the cell and the domain boundary is the cell membrane. Fundamental to the success of the method is the robust generation of bulk and surface meshes for the evolving domains. For this purpose we use a moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known solutions which indicate second-order spatial and temporal accuracy. The method is then applied to a model of the two-way interaction of a migrating cell with an external chemotactic field.

## Term 1

#### 6th October - Charlie Elliott (Warwick) - PDEs on evolving domains

Many physical models give rise to the need to solve partial differential equations in time dependent regions. The complex morphology of biological membranes and cells coupled with biophysical mathematical models present significant computational challenges as evidenced within the Newton Institute programme "Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation". In this talk we discuss the mathematical issues associated with the formulation of PDEs in time dependent domains in both flat and curved space. Here we are thinking of problems posed on time dependent d-dimensional hypersurfaces $\Gamma(t)$ in $\mathbb{R}^{d+1}$. The surface $\Gamma(t)$ may be the boundary of the bounded open bulk region $\Omega(t)$. In this setting we may also view $\Omega(t)$ as $(d+1)-$dimensional sub-manifold in $\mathbb{R}^{d+2}$. Using this observation we may develop a theory applicable to both surface and bulk equations. We will present an abstract framework for treating the theory of well- posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces using generalised Bochner spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hyper-surfaces. Our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. We give some background to applications in cell biology. We describe how the theory may be used in the development and numerical analysis of evolving surface finite element spaces which unifies the discrtetisation methodology for evolving surface and bulk equations. We give some computational examples from cell biology involving the coupling of surface evolution to processes on the surface.

#### 13th October - Antonin Chambolle (Ecole Polytechnique) - Existence & uniqueness for a crystalline curvature flow

In this joint work with M. Morini and M. Ponsiglione, we propose a variant of the classical definitions for anisotropic mean curvature evolutions which allows to show easily existence and uniqueness for the simplest evolution laws (with "natural" mobility). The interesting point is that no regularity is required on the surface tension, so that these results cover also the crystalline case.

#### Stationary solutions and Linear stability of the Vlasov Poisson System with Boundary Condition

Under certain physical conditions (high temperature, low density) a plasma can be modeled by a system of equations known as the Vlasov-Poisson (VP) system. This VP system in the whole space has been well studied and by now many important issues have been addressed. However, VP system much less can be said for the analysis of the VP problem with boundary conditions. In the first part of the talk we will discuss the basic issues regarding the boundary problem and present some existence and non-existence results for the stationary VP system with absorbing and specular boundary conditions in one dimension. Then, well-posedness of the linearized problem will be discussed. Finally we will present on some recent results on the linear stability.

#### 27th October - Amal Alphonse (Warwick) - Function spaces and an abstract framework for parabolic PDEs on evolving Hilbert spaces

In this talk, I will first present a theory of function spaces on time-evolving Hilbert spaces; these spaces are generalisations of Bochner spaces and are useful for problems on moving domains and evolving hypersurfaces. In an abstract Hilbert space setting, an appropriate time derivative called the material derivative will be introduced and a corresponding weak time derivative will also be defined. After discussing all the necessary functional framework in order to properly formulate parabolic PDEs on evolving Hilbert spaces, we will look at some well-posedness and regularity results for linear parabolic PDEs on such evolving spaces.

As mentioned, this theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces, and in a later talk, I will go through an application of some aspects of this theory to a concrete problem on an evolving domain or surface. This is joint work with Charlie Elliott and Björn Stinner.

#### 10th November - Manh Hong Duong (Warwick) - Variational approach to coarse-graining of generalized gradient flows

In this talk, I will present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (A) a natural interaction between the duality structure and the coarse-graining, (B) application to systems with non-dissipative effects, and (C) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, I use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. If time permits, I will also discuss about how to use the technique to obtain quantitative results. The talk is based on joint work with Agnes Lamacz (Dortmund), Mark A. Peletier (Eindhoven), André Schlichting (Bonn) and Upanshu Sharma (Eindhoven).

#### A characterisation of local existence for semilinear heat equations in Lebesgue spaces

Joint work with Robert Laister (University of the West of England), Mikolaj Sierzega (Warwick), and Alejandro Vidal-Lopez (Xi'an Jiaotong-Liverpool University).
We consider the nonlinear heat equation $u_t-\Delta u=f(u)$ with $u(0)=u_0$, with Dirichlet boundary conditions on a bounded domain $\Omega\subset{\mathbb R}^d$.
We assume that $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing.
We show that if $q\in(1,\infty)$ then the equation has a local solution bounded in $L^q(\Omega)$ for all initial data in $L^q(\Omega)$ if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and that if in addition $f(s)/s$ is non-decreasing then the equation has a local solution bounded in $L^1(\Omega)$ for all $u_0\in L^1(\Omega)$ if and only if $\int_1^\infty s^{-(2+2/d)}f(s)\,\mathrm{d} s<\infty$.
Our proofs are also valid for the case $\Omega=\mathbb{R}^d$.

#### Large deviations and concentration properties for gradient and Laplacian interface models

We aim to give an overview how large deviation results are naturally connected with certain PDEs and their underlying stochastic model. In many cases scalings limits of microscopic stochastic models lead to variational problems and PDEs. We discuss these connections with two basic models, the gradient and the Laplacian models. These random fields are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. We discuss emerging free boundary value problems when the fields are perturbed by an attractive force towards certain subspaces of $\mathbb{R}^d$.