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 three broad headings:
Related areas of our research include:
- Algebra and Geometry
- Continuum Mechanics of Solids and Fluids
- Dynamical systems and complexity
- Industrial Applied Mathematics
- Mathematical Biology
- Mathematical Control Theory
- Numerical Analysis
Further details of specific research projects are given below, organised under each of the three broad headings above.
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.