# Partial Differential Equations and their Applications 2016-17

### Meetings are held on Wednesdays at 15.00-16.00 in B3.03.

##### Organisers: Charlie Elliott & Jose Rodrigo

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 or restaurant 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 3

3rd May ** Mi-Ho Giga** (Tokyo)

**Title:** *On planar anisotropic curvature flow and its approximation by*

*deterministic games*

## Term 1

19th October **Ana Djurdjevac **(F.U. Berlin)

**Title:** *Parabolic PDEs with random coefficients on moving hypersurfaces
*

**Abstract:**Sometimes partial differential equations with random coefficients can be better formulated on moving domains, especially in biological applications. We will introduce and analyse the advection-diffusion equations with random coefficients on moving hypersurfaces. We will consider both cases, uniform and log-normal distributions of coefficients. In the uniform case, under suitable regularity assumptions, using Banach-Necas-Babuska theorem, we will prove existence and uniqueness of the weak solution and also we will give some regularity results about the solution. For log-normal case, we will prove the measurability and p-integrability of the path-wise solution. For discretization in space, we will apply

the evolving surface finite element method. In order to deal with uncertainty, we will use Monte Carlo method. This is a joint work with Charles M. Elliott (University of Warwick, UK), Ralf Kornhuber (Free University Berlin, Germany) and Thomas Ranner (University of Leeds, UK).

26th October **Joana Terra
**

**Title:** *A coupled bulk-surface reaction-diffusion system
*

**Abstract:**In this talk I will address a coupled bulk-surface reaction-diffusion system set in a moving domain, which we have been studying in a joint work with Amal Alphonse and Charles Elliott. I will first explain how to derive the problem and what is the correct mathematical formulation. Then I will discuss the results we have obtained on existence, regularity and convergence to equilibrium. The proofs are rather technical, so an outline will be presented instead, emphasizing the key steps.

2nd November **Hans Fritz **(Regensburg)

**Title:** * On the computation of harmonic maps by unconstrained algorithms based on totally geodesic embeddings
*

**Abstract:**We present an algorithm for the computation of harmonic maps, and respectively, of the harmonic map heat flow between two closed Riemannian manifolds. Our approach is based on the totally geodesic embedding of the target manifold into some Euclidean space. Totally geodesic embeddings allow to reformulate the harmonic map heat flow in a neighbourhood of the embedded target manifold. The reformulation has the advantage that the problem becomes unconstrained: Instead of assuming a priori that the solution to the flow maps into the target manifold this fact becomes a property of the solution to the extended flow for special initial data. This simplifies the discretization of the problem. Based on this observation, we propose algorithms for the computation of the harmonic map heat flow and of harmonic maps. We prove error estimates in the stationary case and present some numerical tests.

9th November ** Tom Ranner **(Leeds)

**Title:** * Mathematical modelling of C. elegans locomotion
*

**Abstract:**In this talk, I will present results from an new interdisciplinary approach for the imaging, modelling and simulation of the locomotion of C. elegans using geometric partial differential equations.The nematode Caenorhabditis elegans is a microscopic roundworm found in soil in many temperate regions. It is a popular model organism in many fields, including neuroscience, due to its relatively simple nervous system and anatomy.

16th November

**Title:** *TBA
*

**Abstract:**TBA

23rd November Thomas Hudson (Warwick)

**Title:** *Thermally-driven motion of screw dislocations
*

**Abstract:**Dislocations are topological line defects found in crystals, and their motion governs the plastic behaviour of such materials. Due to the long-range stress fields they induce, their collective behaviour is complex, and so developing a deeper understanding of it would allow us to develop improved predictive models of plasticity. The first part of this talk presents a series of results related to this aim, concerning a simple stochastic model for dislocation motion. In a certain parameter regime, we show that this model satisfies a Large Deviations Principle, converging to a gradient flow of the so-called "renormalised energy" for Coulomb particles. This result provides a first principles justification of Discrete Dislocation Dynamics, a simulation technique commonly used in Material Science to model the ensemble behaviour of dislocations. In the latter part of the talk, I will discuss recent work with Marco Morandotti in which we prove some qualitative and quantitative properties of this evolution, which requires the development of precise boundary asymptotics for the gradient of the Dirichlet Green's function.

30th November **John Barrett
**

**Title:** *Existence of Weak Solutions to a Compressible Oldroyd-B Model
*

**Abstract:**A compressible Oldroyd--B type model with stress diffusion is derived from a compressible Navier--Stokes--Fokker--Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a isentropic, compressible, isothermal, viscous Newtonian solvent, are idealized as pairs of massless beads connected with Hookean springs. We develop a priori bounds for the model, including a logarithmic bound,

which guarantee the nonnegativity of the elastic extra stress tensor, and we prove the existence of global-in-time weak solutions in two space dimensions.

7th December

**Title:**

**Abstract:**

## Term 2

18th January **Grzegorz Jamroz** (Warwick)

**Title:** Uniqueness of dissipative solutions of the Camassa-Holm equation

**Abstract:** TBA

25th January

**Title:** *TBA
*

**Abstract:**TBA

1 February **Charalambos Makridakis** (Sussex)

**Title:** *Energy/entropy consistency and computational methods
*

**Abstract:**The computation of singular phenomena (shocks, defects, dislocations, interfaces, cracks) arises in many complex systems. In order to simulate such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. Often weak solutions of PDEs related to these problems are not unique. Since numerical methods perturb the mathematical model, mathematical analysis emerges as a necessary tool providing mathematical guarantees ensuring that our computational methods approximate physically relevant solutions. Our purpose in this talk is to review results and discuss related computational and analytical challenges for such nonlinear problems modelled by PDEs giving particular emphasis on the energy/entropy consistency of the approximations. In addition we shall discuss related issues emerging in adaptive modelling across scales.

8th February ** Klaus Deckelnick **(Magdeburg)

**Title:** * An obstacle problem for elastic graphs
*

**Abstract:**We consider an obstacle problem for elastic curves given by graphs with fixed ends. We give conditions on the obstacles that ensures existence of solutions and discuss their regularity. At the end we present a numerical method to obtain approximate solutions and show examples of test calculations. This is joint work with Anna Dall'Acqua (University of Ulm).

15th Febuary ** Tatsu-Hiko Miura **(Tokyo)

**Title:** * Singular limit problems for the heat equation and the Navier-Stokes equations in curved moving thin domains
*

**Abstract:**We consider the heat equation and the incompressible Navier-Stokes equations in a moving thin domain with very small width. Our problem is to find limit equations of the bulk equations as the moving thin domain shrinks to a closed moving surface. For the heat equation, we take the weighted average of a variational solution to the heat equation and show its weak convergence in a function space on the moving surface. Then we derive the limit equation, which is a linear parabolic surface equation including the outward normal velocity and the mean curvature of the surface, as a partial dierential equation that the weak limit satises. We also show a formal result on the limit equation of the Navier-Stokes equations and give the idea for the formal derivation.

22nd February **TBA
**

**Title:** *TBA
*

**Abstract:**TBA

1st March** Stefano Spiritio** (L'Aquila)

**Title:** *Suitable Weak Solutions of Navier-Stokes Equations obtained by Navier-Stokes-Voight model in bounded domains
*

**Abstract:**This talk will discuss the problem of the approximation of suitable weak solutions of Navier-Stokes equations. It is well known that suitable weak solutions enjoy the partial regularity theorem of Caffarelli-Kohn-Nirenberg, hence they are more regular than Leray weak solutions. However, since the uniqueness of both Leray weak solutions and suitable weak solutions of Navier-Stokes is not known it is not possible to establish a priori whether an approximation method for Leray weak solutions leads also to suitable weak solutions. I will present a result obtained with L. C. Berselli (University of Pisa) where we proved that solutions obtained by a type of Large Eddy Simulation approximation, precisely the Navier-Stokes-Voight, are suitable. The novelty is that the problem is considered in a bounded domain with Dirichlet boundary conditions where estimates for the pressure are difficult to obtain. Moreover, other approximation methods leading to Suitable Weak Solutions will be discussed as well as some open problems.

8 March **Mark Wilkinson **(Heriot Watt)

**Title:** *Non-uniqueness of Classical Solutions to Euler’s Equations of Rigid Body Mechanics
*

**Abstract:**In this talk, we consider Euler’s equations of motion which describe the physical evolution of smooth, compact subsets of Euclidean space. We show that in the case of analytic sets, for almost every given initial datum there exist infinitelymany distinct classical solutions of Euler’s equations which conserve total (i) linear momentum, (ii) angular momentum, and (iii) kinetic energy of the initial datum.

We argue that this observation falls in a similar vein to those results of Scheer and Shnirelman – and, more recently, those of De Lellis and Sz´ekelyhidi and those of Buckmaster, Shkoller and Vicol – on non-uniqueness of solutions of equations of fluid mechanics. In particular, in light of a non-uniqueness result, one might seek a selection criterion for solutions in the hope of recovering a uniqueness result.

In order to prove our main result, we analyse the multiplicity of solutions of a class of Monge-Amp'ere equations on the whole space.

15th March **Daniel Coutand **(Heriot Watt)

**Title:** *Finite time singularity formation for Euler interface problem with surface tension
*

**Abstract:**

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