PhD projects in Analysis of PDEs
The research in Analysis of PDEs uses a wide variety of techniques such as energy and ladder methods, dynamical systems techniques, geometry, and the calculus of variations. The methods are used to analyse PDEs in various contexts such as nonlinear elastostatics, reaction-diffusion systems, dispersive wave equations, Navier-Stokes equations, delay equations, and both dissipative and Hamiltonian equations. Results include advances in regularity theory, estimates for universal attractors, length scales, stability of travelling waves, fronts and solitary waves. More details, including staff members working in this area, can be found on theAnalysis of PDEs research group pages.
Below are some examples of PhD projects in Analysis of PDEs. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Estimating the cost of phase nucleation in shape-memory crystals (Supervisor: Dr Jon Bevan)
Certain shape-memory crystals rearrange themselves `internally' in response to a change in temperature. As a result, you can bend and twist them into all sorts of shapes at room temperature and, on heating in a cup of hot tea, for example, they spring back into their original shape. They are the subject of a huge amount of research and their applications are numerous.
There is a widely accepted mathematical model, due to Ball and James, which explains why this change happens as the temperature decreases: roughly speaking, the material minimizes its free energy differently at high temperatures compared to low. But this theory does not explain all features. At low temperatures in two space dimensions, for example, the material prefers to deform according to a certain map u such that
∇u ∈ SO(2)A∪SO(2)B
where A and B are fixed 2x2 matrices dictated by the particular material under consideration. (SO(2) is the group of rotations in the 2x2 matrices; ∇u is the matrix of partial derivatives of the map u: R2 → R2.) The patterns typically observed show material where bands of ∇u = A alternate with bands of ∇u = B. The Ball-James model correctly predicts the ratio of A to B, but it does not limit the fineness of these bands. The `minimum energy' state does not exist. However, the model can be improved by adding a so-called surface energy term. The aim of this project is to extend the mathematical analysis of these enhanced material models using tools from the Calculus of Variations. In particular, we would focus on finding the activation energy of one (new) material phase in another, i.e. the least energy required to transform the material from one phase to another. No background in materials science is needed for this project.
Stress-induced twinned microstructure - image courtesy of R D James
Embedding (Interpolation) Inequalities and their Applications to
the Analysis of Solutions of Dissipative PDEs (Supervisor: Michele Bartuccelli)
This project is concerned with the very important problem of interpolation inequalities of functional analysis, such as the celebrated Gagliardo-Nirenberg inequality. These inequalities are crucial for the analysis and estimates of solutions of dissipative partial differential equations (DPDEs); in particular they have recently been used for establishing positivity of solutions and for the calculation of the attractor dimension of some important DPDEs arising in population dynamics.
Some important progress has also recently been made on finding explicitly the values of sharp constants appearing in these inequalities.
The goals of this research will be to calculate the attractor dimension of a class of DPDEs by using the sharpest available interpolation inequalities. In particular we wish to (hopefully) consider the Navier-Stokes equations of fluid-dynamics; having a sharp estimate of the attractor of the Navier-Stokes flow is fundamental in the study of numerous geophysical phenomena such as large scale atmospheric dynamics.
Linear and Nonlinear Stability, Length Scales and Patterns in Solutions of Dissipative PDEs Modelling Real World Phenomena (Supervisor: Michele Bartuccelli)
The purpose of this project is to investigate the solutions of some dissipative partial differential equations modelling real world phenomena. Analytical and computational methods will be used for determining instability regions for the solutions as a function of the parameters which appear in the equations.
Another fundamental question in the dynamics of solutions of these equations is the study of their length scales. Length scales provide information about the smallest features in the flow, and therefore are crucial for its detailed and accurate numerical investigation. Of particular interest for this project will be to investigate the so-called multiscale chaos, namely the situation where a solution can support the co-existence of two or more length scales; in this regime the dynamics of the solution can be chaotic and its accurate description needs in general an appropriate separate analysis. Length scales give also essential information on the patterns involved in the dynamical flows of these differential equations, and they are arguably one of the most important dynamical concepts for properly understanding the spatio-temporal patterns of dissipative flows.
Attractors for hydrodynamical problems in unbounded domains (Supervisor: Dr Sergey Zelik)
The concept of a global attractor plays a very important role in the modern theory of dissipative systems generated by PDEs. For the case of equations in bounded domains (like a square, disk or ball), this attractor is usually finite-dimensional. Thus, despite the infinite-dimensionality of the initial phase space, the limit dynamics reduced to the attractor occurs finite-dimensional and can be studied by the methods of the classical dynamical systems theory. The situation is more difficult for the case where the underlying domain is unbounded (like R2 or R3) since the dimension of the attractor is typically infinite and the finite-dimensional reduction is no more possible. Nevertheless, a reasonable theory has been recently developed for large class of such systems. The aim of this project is to extend/apply this theory to hydrodynamical problems in unbounded (cylindrical) domains. Such extension was problematic during a long time since even the well-posedness of the Navier-Stokes equation in 2D unbounded domains in the proper classes of spatially non-decaying solutions was not known. The situation is changed now due to recent results on the solvability and dissipativity of 2D and 3D Navier-Stokes equations in cylindrical domains.
Soliton Switching in Fibres (Supervisor: Dr Gianne Derks)
For the optical transmission of data across a cable, one can use two (or more) coupled fibre cables. Experiments have shown that if a certain type of signal is put at one end of the cable, it will go to the other end of this cable and hardly anything happens in the other cable. However, if one puts other types of signals on the cable, the signal will switch to the other cable. This gives a convenient way of sending data consisting of zeros and ones. In this project we will aim for a better understanding of this experimentally observed process by investigating the family of soliton-like solutions, especially issues like existence, stability, bifurcations and invariant manifolds will be investigated.
Methods for Obtaining Discrete Symmetries (Supervisor: Prof Peter Hydon)
Differential equations are widely used as models of real systems. To understand the behaviour of a system, it is usually necessary to find the symmetries of the model differential equation. Recent work at Surrey has led to the first systematic technique for obtaining discrete symmetries. The purpose of this project is to simplify and extend the technique, and to create a Maple package that will systematically obtain the discrete symmetries of a given differential equation. This project combines differential equation techniques, linear algebra, group theory and Maple programming. It is expected that the resulting package will be distributed with future releases of Maple, and will be used by those working in many areas of mathematics and theoretical physics.
Spiral waves and defect interaction (Supervisor: David Lloyd and Bjorn Sandstede)
Spiral waves have been found in many experiments such as the famous Belousov-Zhabotinsky reaction and the oxidation of carbon-monoxide on platinum surfaces. In the former reaction, period doubling of spiral waves has been observed, and a numerical simulation of a model problem is shown to the left. This bifurcation has been analysed in 2D. Of interest would be an analytic and numerical study of this phenomenon in 3D where the line defect that connects the core to the bottom boundary becomes a two-dimensional surface, which itself may exhibit a interested dynamics. More generally, the various line defects visible in the simulation may interact with each other, and it would be of interest to study the time and length scales involved in this interaction from a analytical and numerical perspective.
Multi-symplectic structures and nonlinear PDEs (Supervisor: Prof Thomas J. Bridges)
An understanding of nonlinear PDEs is one of the great challenges for the twenty-first century. In this project the geometry of nonlinear PDEs is studied. Multi-symplectic structure, which generalises classical symplectic geometry, is used as a backbone for the analysis. The project has three parts:
- Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and construct Dirac operators on this bundle as a basis for canonical hyperbolic and elliptic PDEs.
- Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
- Apply numerical methods to either part (I) or part (II).