Marco Di Francesco
Analysis and Differential Equations
Analysis is concerned with quantifying change, continuity and approximation. It is the branch of mathematics that studies the notionof limit, and the infinite and the infinitesimal.
Differential equations provide much of the mathematical language used to describe the physical world around us; they emerge naturally in many mathematical contexts as well, for example in differential geometry.
Research in analysis and differential equations at Bath covers a wide range of topics: the theory of nonlinear partial differential equations underlies much of the research and provides links.
There is a lively research environment including the Theory of PDE seminar series, the Applied Analysis Reading Group and events organised by the Centre for Nonlinear Mechanics, see the Events page for details of these and other seminars.
Mathematical geology allows us to explain the patterns that form in folded rocks.
Our research in analysis and differential equations can be arranged under four broad headings:
- Dynamical systems
- Geometric and harmonic analysis
- Nonlinear partial differential equations
- Variational problems
Related areas of our research include:
- Algebra and Geometry
- Mathematical Control Theory
- Industrial Applied Mathematics
- Mathematical Biology
- Numerical Analysis
- Continuum Mechanics of Solids and Fluids
Further details of specific research projects are given below, organised under each of the four broad headings above.
Understanding the temporal behaviour of dynamical systems defined by differential equations or maps is key to many questions in applications as well as in Pure Mathematics. Our work studies a number of fundamental problems and phenomena.
Research staff: Anthony Dooley
Research in Ergodic Theory on non-singular measurable dynamical systems has included development of the theory of G-measures: it can now be shown that they all have the structure of Bratteli-Vershik systems with Markov odometer actions. This opens up new questions on the structure and classification of dynamical systems. The theory of critical dimension for non-singular systems controls how quickly the Radon-Nikodým derivative sums grow along orbits, and is linked to a notion of non-singular entropy.
Systems of completely positive entropy with amenable group exist in abundance, and continue to be objects of investigation. Current interests also include probability on groups and the theory of random dynamical systems.
Research staff: Johannes Zimmer
Nonequilibrium problems and the derivation of thermodynamic quantities, such as entropy, from microscopic models are of considerable current interest. A starting point is here the analysis of the over-damped limit of an evolution, that is, a context where the damping is so strong that inertial effects can be neglected. Then the behaviour of many materials can be described by the desire to reduce their energy. We call an evolution a gradient flow when, loosely speaking, a dynamical system moves at any time in the direction of the steepest descent of the energy. Here a fascinating problem is to derive rigorously the effective energy and evolution equation for a given microscopic model. For example, it is known that the diffusion equation can be interpreted as gradient flow of the entropy in the Wasserstein metric. How can we link this description directly to an underlying model of Brownian motion? This link shows that entropy is indeed macroscopically the driving force, and explains why the Wasserstein metric occurs. The argument combines probabilistic and analytic techniques, namely Large Deviation Principles and Gamma-convergence.
Many particle systems and lattice dynamical systems
Dynamical systems with many degrees of freedom occur when physical systems are resolved up to the atomic level. Particles form then lattices or gases. Recent research studies the connection and differences of such discrete systems and the corresponding continuum descriptions via partial differential equations.
Many physical problems are well described on a microscopic scale by atoms which are linked by nonlinear springs. Such models appear in solid state physics (e.g., in models for the elastic and plastic behaviour of crystals). The governing equations are so-called lattice dynamical systems; these systems arise as well in statistical mechanics and biology, for example as idealisations of DNA molecules in the biological sciences.
Fermi, Pasta and Ulam discovered in 1953 the complexity of these systems in a now-famous computer simulation of a chain of 64 atoms. The discovery that the system does not behave ergodically strongly influenced the development of KAM theory, chaos and solitons. Many challenges remain; one important problem is the existence theory for travelling waves in lattice dynamical systems. A variety of tools from the calculus of variations and PDEs can be applied, such as mountain pass arguments or compensated compactness. Yet, many tools from PDEs cannot be easily adapted, and the existence theory for lattice dynamical systems is far from satisfactory. One challenge is to prove the existence of waves for a number of lattice dynamical systems; another challenge is to prove qualitative statements, such as stability.
The dynamics of atoms and molecules pose challenging mathematical problems. Many problems can be described as a potential energy landscape with many potential wells separated by barriers. One then often wants to find a trajectory joining a given initial configuration with a given final one. For example, the two configurations could be different conformational states of a molecule. The dynamics is complicated: Typically, the trajectories will jostle around in the well belonging to the initial configuration, before a rare spontaneous fluctuation occurs and the trajectory crosses the barrier and reaches the next valley of the energy landscape. How can we compute these rare events?
Research staff: Chris Budd
Classical work on dynamical systems assumes that the underlying differential equations or maps are smooth. However, many problems in applications do not satisfy this assumption. These include problems with impact, friction, sliding and switching. In Bath there is significant interest in studying the basic principles of such non-smooth dynamical systems and in investigating and classifying the possible forms of the dynamics. This involves studying novel types of bifurcation (such as grazing, border-collision and sliding) and the subsequent behaviours, such as the period-adding, period incrementing and anharmonic routes to chaos. We also work closely with members of the departments of mechanical and electrical/electronic engineering to study applications in bearing dynamics and in switching networks.
Pattern formation and symmetric bifurcation theory
Pattern formation is the study of the spontaneous appearance of spatial structure, often with a characteristic wavelength, in externally driven, dissipative nonlinear PDEs and spatially discrete coupled cell systems.
Spatial structure appears through symmetry-breaking instabilities (bifurcations) and symmetry plays a central role in organising the new solution branches that appear. In some cases the system supports localised patches of pattern, and there is a rich bifurcation structure that explains aspects of their behaviour, at least in one spatial dimension. Research on these, and many other aspects of dissipative PDEs, typically involves a mixture of numerical and asymptotic analysis together with an appreciation of how bifurcation-theoretic ideas are augmented by methods involving group theory.
Spatial structures can also be forced by an underlying heterogeneous medium. A way to study particular elliptic equations on infinite strips is to treatthe unbounded direction as if it were "time", and to interpret the equations dynamically. The behaviour of travelling waves in inhomogeneous media can be understood much better by combining spatial dynamics with averaging and homogenisation. The effect known as "pinning" is of special interest.
Variational problems in differential geometry typically give rise to nonlinear PDEs, often with critical or supercritical nonlinearities that are difficult to control with analytic methods alone. But when combined with geometric insight, the tools from the theory of PDEs can provide powerful new methods, sometimes with applications outside of differential geometry as well. Current research interests include higher order geometric PDEs and curvature flows.
While many tools have been developed for geometric PDEs of order two, they often are not applicable to equations of higher order. The lack of a maximum principle is one of the main challenges, but it is also common to see more intricate relationships between higher and lower order terms. Our aim is to develop new approaches to come to terms with these difficulties.
The deformation of the metric on a manifold, or of a submanifold, by parabolic equations involving the curvature has proved a powerful tool to obtain information about the geometry of manifolds. The resulting curvature flows, however, typically give rise to singularities. It is then necessary to analyse these and to find suitable notions of weak solutions, so that the flow can be continued after the singularities.
Research staff: Anthony Dooley
Research in Harmonic Analysis of Lie groups covers a range of topics, including Lp-spaces on compact and semi-simple Lie groups, the theory of contractions, and the theory of representations. The theory of orbital convolutions, linked to the Kirillov character formula, together with the wrapping map, allows one to understand harmonic analysis on the group via Euclidean harmonic analysis on the Lie algebra; one can also transfer Brownian motion from the Lie algebra to the group. Lie symmetry groups of differential equations are studied from the point of view of non-commutative harmonic analysis. A new approach to analysis on semi-simple groups has been developed based on groups of Heisenberg type, leading to the resolution of the Lichnérowicz conjecture and to new estimates for the Knapp-Stein intertwining operators, and offers further interesting research applications.
Partial differential equations lie at the heart of many problems in physics, engineering, and mathematical biology. Central to our work is a study of the nature of these equations using a combination of rigorous techniques (such as comparison methods, bifurcation theory, homgenisation and the use of rescaling and energy arguments), formal asymptotic methods (such as matched asymptotic expansions and series solutions) and numerical techniques.
Finite time blow-up
Studies of blow-up include second, third and higher-order PDEs, the nonlinear Schrödinger equation and related systems, as well as problems in combustion, mathematical biology, fluid mechanics, and nonlinear optics. We are particularly interested in problems where the solutions of the equations develop singularities in a finite time. This behaviour is typical of models of combustion and burning which provide examples of problems with finite time blow-up, extinction or quenching, all of which are abstractions of observed physical phenomena. Singularities also arise in certain theories of the nature of turbulence. Describing these singularities requires the development of novel analytic and numerical tools and leads to many new mathematical problems. A further area of research involves the construction of new exact solutions of these equations by determining invariant manifolds (and invariant spaces) of the solution operator or by looking for self-similar solutions invariant under various group actions.
Monge-Ampere equations, optimal transport and fully nonlinear PDEs
Optimal transport is a fascinating topic in the calculus of variations, but it is also linked to many other mathematical subjects. One popular connection is the link to gradient flows (for example diffusion), but optimal transport can be used to understand, for example, Hamiltonian systems or the Schroedinger equation. One aspect is the derivation of macroscopic (effective) equations from particle models, which involves a derivation of the underlying optimal transport structure.
The behaviour of the solutions of fully nonlinear Monge-Ampere type equations is of special interest and links such diverse topics as differential geometry, image processing, meteorology (where the singularities manifest themselves as weather fronts) and mesh generation.
Many partial differential equations and dynamical systems involve multiple scales in time and space. Averaging, homogenisation and other asymptotic techniques aim at providing effective descriptions of the behaviour of the solutions without explicit dependence on these fast scales. One area of interest is to derive higher (or exponential) order descriptions to obtain accurate and rigorous estimates of the multiscale effects.
An increasingly important area where these asymptotic techniques find application is in the design and analysis of numerical methods for multiscale problems arising from physics and modern technology. Activity here in Bath in this area includes investigation of high frequency wave scattering problems in collaboration with the numerical analysis group.
Variational problems arise in many guises in pure and applied mathematics. In the simplest case, one seeks to minimise a given integral functional on a set of admissible functions. At Bath we work on the rigorous analysis of a range of variational problems from geometry and optimal transport to problems motivated by continuum mechanics. We study fundamental questions such as existence, regularity and qualitative properties such as the multiplicity of solutions. Intriguing new phenomena arise which are not amenable to classical approaches and hence require new theories and approaches. In these studies there is often a rich interplay between mathematical analysis, geometric intuition and the underlying physics of the situation being modelled and bring together aspects of functional analysis, differential equations, geometry and topology.
Geometric variational problems
In differential geometry, objects of interest are often described by differential equations. For example, on a Riemannian manifold, the shortest connection between two points is a geodesic and satisfies a certain ODE. Typically, the equations obtained from such problems are nonlinear PDE due to the curvature of the underlying spaces and typically give rise to nonlinear PDEs, often with critical or supercritical nonlinearities that are difficult to control with analytic methods alone. However, when combined with geometric insight, the tools from the theory of PDEs can provide powerful new methods, sometimes with applications outside of differential geometry as well.
At Bath research in geometric analysis concentrates on elliptic and parabolic problems. Equations of higher order are of particular interest. The problems studied include harmonic and biharmonic maps, curvature flows, equations from conformal geometry, and applications in materials science.
Rearrangements of functions and vortices in fluids
Research staff: Geoffrey Burton
Problems involving rearrangements arise when studying steady flows of an ideal fluid in two dimensions and stem from the requirement that initial vorticity distribution moves to a rearrangement of itself under the flow. Monotone rearrangement and symmetrisation of real-valued functions play a role in overcoming loss of compactness in variational problems for integral functionals defined on unbounded domains. The set of all equimeasurable rearrangements of a single function has some interesting properties viewed as a subset of a Banach space and also forms a natural constraint set for a novel variational problem governing steady vortices in a planar ideal fluid. The foregoing work gives rise to solutions of semilinear elliptic PDE. Recent developments include applications to stability of steady vortices, monotone rearrangement of vector-valued functions and variational problems for vortices in a fluid with a free surface.
Variational problems motivated by nonlinear elasticity
Research staff: Jey Sivaloganathan
In nonlinear hyperelasticity one seeks equilibrium states of an elastic body by minimising the total energy that is stored in the deformed body over possible configurations of the body. We study fundamental questions in the Calculus of Variations and partial differential equations relating to important notions of existence, multiplicity and qualitative properties of minimisers.
Research staff: Jey Sivaloganathan
Here one seeks to prove the symmetry of energy minimising deformations under symmetric boundary conditions by constructing a vector symmetrisation procedure that does not increase the total energy of any given deformation. To date this approach has successfully been applied to equilibrium configurations of cylinders and shells.
Research staff: Jey Sivaloganathan
An intriguing aspect of the variational approach to nonlinearelasticity is that it is possible to start with an initially perfect body and to find (mathematically) that, if a sufficiently large boundary displacement or load is imposed, then the configurations which minimise the energy stored in the body must develop singularities and these can be interpreted as "fractures" forming in the initially perfect material. These minimisers correspond to singular weak solutions of the equilibrium equations. We study existence and qualitative properties of these singular minimisers and their connections with fracture mechanisms observed in polymers.