Analysis of PDEs
Existence, regularity, and blow-up of solutions for elliptic and parabolic equations
Modern methods to construct solutions of PDEs generally depend to a high degree on abstract functional analysis and produce generalised solutions in the first instance. Often one can then prove that these solutions are in fact smooth and solve the PDE in the classical sense, but for nonlinear problems in particular, it is also possible that they form singularities, often referred to as blow-up for time-dependent problems.
Problems studied in the department include second and higher order PDEs coming from a range of backgrounds such as combustion, mathematical biology, fluid mechanics, nonlinear optics, liquid crystals, elasticity, or differential geometry.
Nonlocal energies arise naturally as the continuum limit of discrete, pairwise interactions, as the number of particles becomes increasingly large. The nature and size of the particles can be very different, ranging from atoms, molecules, droplets, charges, defects, to insects, birds, people, cars. The variety of the systems that can be broadly thought of as particle systems makes this field very relevant across disciplines - in particular in materials science and in biology.
Global bifurcation theory
For many nonlinear PDEs, the main applied interest is in solutions which are "large" in a certain sense, rather than the "small" solutions accessible with perturbative methods. Global bifurcation theory studies curves or other connected sets of solutions to such problems, and in particular how large solutions can be constructed from small ones via continuation arguments.
Calculus of variations
Asymptotic analysis of variational problems
When a problem depends on a parameter that typically takes very small (or very large) values, then some insight can often be gained by studying the asymptotic behaviour as the parameter tends to 0 (or ∞). For variational problems, one of the most successful tools for this purpose is Γ-convergence, which often allows to identify an effective (limiting) variational problem that captures the essential features of the underlying system.
Geometric variational problems
In differential geometry, the most interesting objects often satisfy variational principles and are described by differential equations. For example, on a Riemannian manifold, the shortest connection between two points is a geodesic and satisfies a certain ODE. At Bath, research in geometric analysis concentrates on elliptic and parabolic problems. The problems studied include harmonic and biharmonic maps, curvature flows, and equations from conformal geometry.
Variational principles in continuum mechanics
Variational principles are ubiquitous in mechanics. In the context of a continuum theory for a solid or fluid, they typically give rise to partial differential equations, for example in the form of Euler-Lagrange equations. Our interests include (but are not limited to) variational problems in elasticity, fracture and damage theories, liquid crystals, micromagnetics, and plasticity.
Variational problems in L∞
Most of the known methods in the calculus of variations are designed for maximising or minimising quantities given in terms of the integral of a function. In contrast, when working with variational problems in L∞, we minimise the maximum of a function instead. Not much is known for problems in L∞ that involve vector-valued functions or derivatives of second or higher order, and we aim to develop new tools for the study of these problems.
Infinite-dimensional control systems
The control of infinite-dimensional, or so-called distributed-parameter, systems, such as dynamical systems specified by controlled and observed partial or delay differential equations is motivated by:
- their prevalence in applications, and;
- the mathematical challenges caused by infinite-dimensionality.
Of particular interest to our group is the development of aspects of the other listed research areas in infinite-dimensional settings.
It is often desirable to replace an accurate but complex model for a physical biological or engineering system by a perhaps slightly less accurate but simpler model. The process to extract the simpler model from the more complex one is called model reduction. Our research on model reduction mainly focuses on two cases:
- the case where the original complex model is given by partial differential equations;
- the case where the original model has a structure (such as positivity or passivity) that must be retained in the simpler model.
Pattern formation and infinite-dimensional dynamical systems
Pattern formation is the study of the spontaneous appearance of coherent spatial structure, often with a characteristic wavelength, in externally driven, nonlinear PDEs and spatially discrete coupled systems, which all have Infinite-dimensional phase space. One important example are travelling waves in lattice dynamical systems. 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.
Multi-scale analysis and scale bridging
Homogenisation and asymptotic analysis
The mathematical theory of homogenisation is a branch of analysis that has been in active development since the late 1960s, motivated by the need for analytical tools for qualitative and quantitative analysis of continuous media with multiple length-scales and by the problem of scale bridging, i.e. understanding how specific geometric arrangements and interaction rules at a small scale result in certain kinds of material response at a large scale, i.e. when the number of microscopic constituents is large.
Discrete systems with many degrees of freedom occur when physical systems are resolved up to the atomic level. Particles then form lattices or gases. 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). For particle gases the research aims to explain how equations on different scales can have fundamentally different behaviour by providing rigorous convergence results for the correct scaling in the many particle limit.
Operator theory and applications
Applications in systems and control
Many aspects of operator theory play a role in systems and control. Our work includes the application of and the further development of operator theory related to:
1.semigroups of linear operators; 2.Hankel operators; 3.operator-valued holomorphic functions; 4.optimization in Hilbert spaces; 5.positive operators; 6.operator Lyapunov and Riccati equations; 7.dissipative operators in both definite and indefinite inner-product spaces.
Applications to problems in materials science
Methods of functional analysis and abstract operator theory provide powerful tools in understanding the behaviour of existing materials and in predicting the material response of new composite media obtained by combining known materials according to specific geometric rules. Asymptotic analysis with respect to the size of the microstructure then yields new asymptotically equivalent models in the contexts of diffusion, elasticity, electromagnetism etc., shedding light on the behaviour of the original complex system.
The development of spectral theory of operators has been closely linked to the needs of scattering theory, especially in the context of quantum-mechanical scattering. Using as a starting point the interpretation of a system, e.g. a composite medium, in operator-theoretic terms, allows one to obtain a new point of view on problems of wave propagation in the context of materials science.
Pseudo-differential operators and semiclassical analysis
A pseudo-differential operator is an extension of the concept of differential operator, and its definition relies on the Fourier transform. The theory of pseudo-differential operators emerged in the mid 1960s from the the study of singular integral operators in harmonic analysis, and pseudo-differential operators are used extensively in the analysis of partial differential equations and quantum field theory.
Semiclassical analysis of wave propagation problems
In the context of wave problems involving acoustic and electromagnetic waves, semiclassical analysis seeks to understand the behaviour of the waves in the difficult high-frequency limit. Our particular interest is in working at the interface between semiclassical analysis (SCA) and numerical analysis (NA) of wave propagation problems.