# Analysis of PDEs

# Existence, regularity, and blow-up of solutions for elliptic and parabolic equations

### Research staff: Victor Galaktionov, Roger Moser, Monica Musso, Manuel del Pino, Hartmut Schwetlick

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 PDEs

### Research staff: Monica Musso, Manuel del Pino, Johannes Zimmer

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.

## Pseudo-differential operators

### Research staff: Veronique Fischer

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

### Research staff: Euan Spence

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.

# Calculus of variations

## Asymptotic analysis of variational problems

### Research staff: Kirill Cherednichenko, Roger Moser

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

### Research staff: Roger Moser, Hartmut Schwetlick

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

### Research staff: Roger Moser, Jey Sivaloganathan, Johannes Zimmer

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∞

### Research staff: Roger Moser

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.

# Control theory

## Control of positive systems including applications to mathematical biology

### Research staff: Chris Guiver, Hartmut Logemann

Positive dynamical systems are dynamical systems where the evolution map leaves a positive cone invariant, such as the nonnegative orthant of Euclidean space, equipped with the partial order of component wise inequality. The study of positive systems is motivated by numerous applications across a diverse range of scientific and engineering contexts where invariance of a positive cone capture the essential property that state-variables of positive systems must take nonnegative values to be meaningful.

## Infinite-dimensional control systems

### Research staff: Chris Guiver, Hartmut Logemann, Mark Opmeer

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.

## Model reduction

Research staff: Chris Guiver, Mark Opmeer

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.

# Nonlinear control systems and input-to-state stability

### Research staff: Chris Guiver, Hartmut Logemann

The concept of input-to-state stability (ISS) relates to stability properties of forced nonlinear systems (the forcing is frequently called input). Since its inception in 1989, this concept has generated a rich body of results, in particular, extending classical Lyapunov theory to systems with inputs. Our work in this area includes ISS and classical absolute stability theory, ISS in the context of differential inclusions, ISS results for hysteretic feedback systems, and ISS methods in the context of population dynamics.

## Sampled data feedback systems

### Research staff: Hartmut Logemann

A sampled-data feedback system consists of a continuous-time system a discrete-time controller and certain continuous-to-discrete (sample) and discrete-to-continuous (hold) operations. Sampled-data control theory forms the mathematical foundation of digital control which addresses the problem of implementing control strategies using digital computers.

# Dynamical systems

## Pattern formation and infinite-dimensional dynamical systems

### Research staff: Karsten Matthies, Hartmut Schwetlick, Johannes Zimmer

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.