OxPDE has been offering summer projects for mid-term undergraduate students since 2009; successful applicants have the chance to work on a project in a supportive environment with each project offering:

- Support from a specific supervisor(s)
- Computer access
- Access to desk space
- Other department resources where appropriate to complete the project
- Feedback from your supervisor regarding your performance

The projects aren't eligible for credit award and you will be required to submit a report to your supervisor and the Centre's administrator by the end of the project period to outline the activities undertaken during the project, changes made to the project as it progressed and the main outcomes from the project, with details of possible future work. To apply you must be:

- Registered at a recognized higher education institution
- In the middle years of your undergraduate degree
- Able to commit to undertaking a research project over the summer vacation
- And be based in Oxford for its duration (a minimum of 4 weeks and maximum of 10 weeks)
- On track to receive at least a second-class honours degree in a discipline relevant to the project you will undertake

This year we offer four projects with different levels of support, Therefore details of relevant support are given for each project.

## Project 1 - Constrained problems in the calculus of variations

**Outline**: In several variable calculus, if you try to minimise a smooth function the first step is usually to set the derivative to zero. In the calculus of variations the aim is to minimise a function of a function, and a similar method can be used which typically gives a differential equation, called the Euler-Lagrange equation, rather than an algebraic one. In several variable calculus however, if you have a constraint (for example, you are minimising over a compact set), there is a possibility that the minimum could be attained at the boundary, at which point setting the derivative to zero is no longer helpful. This too is seen in the calculus of variations where there are often constraints imposed by the physical system that is being modelled. This means that without knowing properties of the minimiser beforehand, solving the Euler-Lagrange equation may not give the correct solution.

The project will be concerned with problems in the calculus of variations that model liquid crystal systems which have physically imposed constraints, and the task of finding conditions that ensure that the constraints are satisfied in a strong sense by minimisers.

**Prerequisites**: The project will require a strong understanding in analysis, several variable calculus and linear algebra. Any background in convex analysis, functional analysis, calculus of variations and measure theory will facilitate the project, but this is not required and any relevant material will be covered. Whilst the problems are motivated by problems in liquid crystals, the problems can be introduced and dealt with entirely mathematically so that there is no requirement for a background in physics/materials science.

**Keywords**: Young measures, elliptic PDEs, differential inclusions, lower-semicontinuity.

**Supervisors**: Jamie Taylor & Dr Angkana Rüland

**References**: n/a

**Duration**: Between 1 July to 7 September, 7 weeks duration.

**Financial Support:** College accommodation for the duration of the project only.

## Project 2 - Efficient theoretical/algorithmic approaches to the computation of microstructure with application to Earth’s science.

**Outline**: Solid-to-solid phase transformations are often accompanied by appearance of complicated reorganization of crystal phases at a microscopic level, the so-called microstructure. Formation of microstructure is often accompanied by defects in the crystal lattice, such as dislocations and disclinations. For instance, in the case of orthorombic olivine [1], which constitutes the dominant mineral phase of the Earth’s upper mantle, rotational defects (disclinations) may be caused by lack of slip systems to accommodate mechanical deformations. This can be viewed as a natural mechanism to rule out fracture (hence quakes) by storing a localized stress. In other situations related to the literature of metal alloys, disclinations, dislocations and cavities may occur as the result of large strains in the crystal lattice caused by quenching.

Non-convexity is the keyword to label many models of phase-transforming solids such as polymineral aggregates, martensitic steels, elastic crystals and biological structures. In the framework of calculus of variations, these systems are modelled with multi-well energies whose minimizers tend to develop spatially highly modulated gradients. The problem proposed for this summer project is the study of simple energy models for the formation of microstructure in the scenario of elastoplasticity. An important question to be addressed is the investigation of lattice incompatibilities that arise from a microstructure containing topological defects in terms of the defect density tensor. An alternative approach would be the atomistic modelling of lattice incompatibilities possibly with the language of spin-systems, thus shedding some light on the underlying mechanisms that drive formation of material defects. An application of this research work would be particularly important in those situations, such as in the Earth’s mantle, in which direct observation is challenging or not feasible.

Depending on the student’s background and preferences, the project may be developed on either the analytical or computational side.

**Prerequisites**: Some courses in calculus of variations and numerical/computational mathematics would be advantageous but not mandatory.

**Key words:** Young measures, elliptic PDEs, differential inclusions, lower-semicontinuity.

Supervisor: Dr Pierliugi Cesana

**References**:

1) P. Cordier et al. , “Disclinations provide the missing mechanism for deforming olivine-rich rocks in the mantle“, Nature, 507 (2014)

2) S. Patching, P. Cesana, A. Rueland, “Modelling of martensitic disclinations“, in preparation

3) K. Bhattacharya, “Microstructure of Martensite” Oxford series

**Duration**: Between 22 June to 31 August, 6 weeks duration.

**Financial Support:** This project is supported by the Oxford-Met Office Academic Partnership. A bursary of £250 per week for the duration of the project.

## Project 3 - Twinning in solid crystals

**Outline**: During cooling metallic alloys such as NiTi, NiMn or CuZn undergo a solid-to-solid phase transformation that results in very finely twinned structures. A mathematical model based on non-linear elasticity (see [1]) explains the observed features as those arising from energy minimizing deformations. However computing the set of all energy minimizing deformations, known as the quasiconvex hull, is a highly non-trivial problem and is only known for a few specifc cases (e.g. [2], [3]). In the known cases, the problem was solved by constructing laminates of sufficiently high order "twins of twins" and show that they exhaust the so called polyconvex hull, which is an outer bound on the set of minimizing deformation gradients. The aim of this summer research project is to follow a similar agenda and construct higher order laminates for a specific case and thus provide inner bounds on the set of energy minimizing deformation gradients. Mostly analytical methods will be employed but there is some potential for numerical implementation if desired.

**Prerequisites**: A basic understanding of the underlying theory and good knowledge of linear algebra and multivariate calculus should suffice to tackle most of the project. A background in numerical analysis is optional.

**Key words**: Calculus of Variations, Young measures, Non-linear elasticity

**Supervisor**: Anton Muehlemann and Dr. Andres Baldelli

**References**:

[1] J.M. Ball and R.D. James. Fine phase mixtures as minimizers of energy. In Analysis and Continuum Mechanics, pages 647-686. Springer, 1989.

[2] J.M. Ball and R.D. James. Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philosophical Transactions: Physical Sciences and Engineering, pages 389-450, 1992.

[3] J. Adams, S. Conti, and A. DeSimone. Soft elasticity and microstructure in smectic C elastomers. Continuum Mechanics and Thermodynamics, 18(6):319-334, 2007.

**Duration**: Between 1 July to 7 September, 7 weeks duration.

**Financial Support**: College accommodation for the duration of the project only.

## Project 4 - The real butterfly effect: dependence of the Navier Stokes equations on initial data

Outline:

For multi-scale fluid systems, the 'real butterfly effect' was a phrase coined to refer to the existence of an absolute finite-time predictability barrier, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times. Rather than considering the 3D primitive equations for planetary oceanic and atmospheric dynamics, our proposed investigation is concerned with the Navier Stokes equations, describing the flow of a viscous incompressible fluid in a domain $\Omega$. Namely, the velocity $\textbf{u}(\textbf{x},t)$ and pressure field $p(\textbf{x},t)$ satisfy \begin{equation}\label{navierstokes} \partial_{t}\textbf{u}(\textbf{x},t)+\textbf{u}(\textbf x,t).\nabla \textbf{u}(\textbf{x},t)-\nu\Delta\textbf{u}(\textbf{x},t)=-\nabla p, \,\,\,\,\,\,\,\, {\rm div}\,\textbf{u}(\textbf{x},t)= \textbf{0} \end{equation} for all $\textbf{x}\in\Omega$ and all instances of time $t > 0$, subject to the homogeneous Dirichlet boundary condition \begin{equation}\label{boundarycondition} \textbf{u}(\textbf{x},t)= \textbf{0} \end{equation} for all $\textbf{x}$ belonging to the boundary of the domain $\partial \Omega$ and for all $t >0$, and the initial condition \begin{equation}\label{initialcondition} \textbf{u}(\textbf{x},0)= \textbf{u}_{0}(\textbf{x}) \end{equation} for all $\textbf{x}\in \Omega$. It is supposed that $u_{0}$ is a given smooth divergence-free field vanishing on the boundary and the viscosity $\nu$ is a positive parameter. The question of the existence of a 'real butterfly effect' for the Navier Stokes equations is still open, rather we propose to investigate the following issue. We will consider a subclass of the so called 'weak Leray-Hopf solutions' for the Navier Stokes equations and specific subclasses of initial data, such that for solutions in this subclass there is continuous dependence on initial data for some finite time $T>0$ ($T$ may in fact be small). Suppose $\textbf{u}_{1}$ and $\textbf{u}_{2}$ are two solutions from this subclass, with suitable initial data $\textbf{u}_{0_{1}}$ and $\textbf{u}_{0_{2}}$. Our investigation centres on, for specific subclasses of weak Leray-Hopf solutions and initial data, examining the dependence constant $ C(\textbf{u}_{0_{1}},\textbf{u}_{0_{2}}, \textbf{u}_{1}, \textbf{u}_{2}, t, T, \Omega, \nu)>0$ such that (for $0$<$t$<$T$): \[ \int\limits_{\Omega}| \textbf{u}_{1}(\textbf{x},t)-\textbf{u}_{2}(\textbf{x},t)|^{2} d\textbf{x}\leqslant C(\textbf{u}_{0_{1}},\textbf{u}_{0_{2}}, \textbf{u}_{1}, \textbf{u}_{2}, t, T, \Omega, \nu). \] It is hoped that investigating suitable subclasses may provide insight into necessary conditions for the 'real butterfly effect' to be observed in the Navier Stokes equations. Additionally, the examination of the dependence constant might have applications for numerics, for example. Furthermore, we hope that this investigation may provide insight with regards to forecasting times in relation to the 3D primitive equations for planetary oceanic and atmospheric dynamics.

**Prerequisites**: Multivariable calculus; Real analysis and some functional analysis (including Lebesgue spaces). Some knowledge of Sobolev spaces and Elliptic PDE would facilitate the project.

**Key words**:

**Supervisor**: Tobias Barker, Prof G Seregin and Prof I Moroz

**References**:

[1] T N Palmer, A Doring and G Seregin. The real butterfly effect. Nonlinearity 27 R123, 2014

[2] Gregory Seregin. Lecture Notes on Regularity Theory for the Navier-Stokes Equations. World Scientific, 2014.

**Duration**: 6 July to 31 July, 4 weeks duration.

**Financial Support**: This project is supported by the Oxford- Met Office Academic Partnership. A bursary of £250 per week for the duration of the project.

Please note that projects may be offered to a second student in some circumstances with different financial support to that listed above.

Application for the projects is now closed.