L3

In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.  

Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

When nematic liquid crystal droplets are produced in the form or tori (or such is the shapes of confining cavities), they may be called toroidal nematics, for short. When subject to degenerate planar anchoring on the boundary of a torus, the nematic director acquires a natural equilibrium configuration within the torus, irrespective of the values of Frank's elastic constants. That is the pure bend arrangement whose integral lines run along the parallels of all inner deflated tori. This lecture is concerned with the stability of such a universal equilibrium configuration. Whenever its stability is lost, new equilibrium configurations arise in pairs, the members of which are symmetric and exhibit opposite chirality. Previous work has shown that a rescaled saddle-splay constant may be held responsible for such a chiral symmetry breaking. We shall show that that is not the only possible instability mechanism and, perhaps more importantly, we shall attempt to describe the qualitative properties of the equilibrium nematic textures that prevail when the chiral symmetry is broken.

Many symptoms of Parkinson’s disease are connected with abnormally high levels of synchrony in neural activity. A successful and established treatment for a drug-resistant form of the disease involves electrical stimulation of brain areas affected by the disease, which has been shown to desynchronize neural activity. Recently, a closed-loop deep brain stimulation has been developed, in which the provided stimulation depends on the amplitude or phase of oscillations that are monitored in patient’s brain. The aim of this work was to develop a mathematical model that can capture experimentally observed effects of closed-loop deep brain stimulation, and suggest how the stimulation should be delivered on the basis of the ongoing activity to best desynchronize the neurons. We studied a simple model, in which individual neurons were described as coupled oscillators. Analysis of the model reveals how the therapeutic effect of the stimulation should depend on the current level of synchrony in the network. Predictions of the model are compared with experimental data.

The derivation of so-called `effective descriptions' that explicitly incorporate microscale physics into a macroscopic model has garnered much attention, with popular applications in poroelasticity, and models of the subsurface in particular. More recently, such approaches have been applied to describe the physics of biological tissue. In such applications, a key feature is that the material is active, undergoing both elastic deformation and growth in response to local biophysical/chemical cues.

Here, two new macroscale descriptions of drug/nutrient-limited tissue growth are introduced, obtained by means of two-scale asymptotics. First, a multiphase viscous fluid model is employed to describe the dynamics of a growing tissue within a porous scaffold (of the kind employed in tissue engineering applications) at the microscale. Secondly, the coupling between growth and elastic deformation is considered, employing a morpho-elastic description of a growing poroelastic medium. Importantly, in this work, the restrictive assumptions typically made on the underlying model to permit a more straightforward multiscale analysis are relaxed, by considering finite growth and deformation at the pore scale.

In each case, a multiple scales analysis provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics provided by prototypical `unit cell-problems'. Importantly, due to the complexity that we accommodate, and in contrast to many other similar studies, these microscale unit cell problems are themselves parameterised by the macroscale dynamics.

In the first case, the resulting model comprises a Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. Stokes-type cell problems retain multiscale dependence, incorporating active cell motion [1]. Example numerical simulations indicate the influence of microstructure and cell dynamics on predicted macroscale tissue evolution. In the morpho-elastic model, the effective macroscale dynamics are described by a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection--reaction--diffusion equation [2].

[1] HOLDEN, COLLIS, BROOK and O'DEA. (2018). A multiphase multiscale model for nutrient limited tissue growth, ANZIAM (In press)

[2] COLLIS, BROWN, HUBBARD and O'DEA. (2017). Effective Equations Governing an Active Poroelastic Medium, Proceedings of the Royal Society A. 473, 20160755