Fourth Scottish Partial Differential

Equations Colloquium

Equations Colloquium

9-10 June 2016, University of Dundee

Tower Building, Baxter Suite 1.36

Tower Building, Baxter Suite 1.36

Agissilaos Athanassoulis | Generalised wavepackets for the semiclassical nonlinear Schrödinger equation.Generalised wavepackets for the semiclassical nonlinear Schrödinger equation.We consider semiclassical Schrödinger equations with power nonlinearities, and show that for a broad, non-parametric class of wavepacket initial data, the solution remains a wavepacket for any O(1) timescale, and we compute the Wigner measure under appropriate conditions. This is the first positive result on Wigner measures for a Schrödinger equation with power nonlinearity, following several high-profile negative results. Some preliminary results towards the extension to general data are discussed; these are based on recent phase-space dispersive estimates. |

John Ball | Interfaces and metastability in solid and liquid crystals.Interfaces and metastability in solid and liquid crystals.When a new phase is nucleated in a martensitic solid phase transformation, it has to fit geometrically onto the parent phase, forming interfaces between the phases accompanied by possibly complex microstructure. The talk will describe some mathematical issues involved in understanding such questions of compatibility and their influence on metastability, as illustrated by recent experimental discoveries. For liquid crystals planar (as opposed to point and line) defects are not usually considered, but there are some situations in which they seem to be relevant, such as for smectic A thin films where compatibility issues not unlike those for martensitic materials arise. |

Gabriel Barrenechea | Nonlinear diffusion and the discrete maximum principle.Nonlinear diffusion and the discrete maximum principle.In this talk I will review some recent advances in numerical methods satisfying the discrete maximum principle. This property, which is not satisfied by the standard finite element method unless some very restrictive hypothesis are made, is of capital importance in some applications. The main tool to impose the satisfaction of this property is to propose methods that add numerical diffusion to the formulation. This diffusion can be either based on the elements, or on the edges of the triangulation. In this talk I will present two recent examples of discretisations that add diffusion, different in nature, but with similar outcomes. |

David Bourne | Energy-driven pattern formation in nonlinear elasticity.Energy-driven pattern formation in nonlinear elasticity.In this talk I will discuss self-similar folding patterns in a compressed elastic film that has partially delaminated from a substrate. Such patterns were observed in recent experiments to fabricate nanoscale structures from semiconductor films. We model the system with a suitable energy functional and prove rigorous upper and lower bounds on the minimum value of the energy. Some of the upper bound constructions match the patterns observed in experiments. This is joint work with Sergio Conti and Stefan Mueller. |

Dominic Breit | Martingale solutions to the stochastic compressible Navier-Stokes system.Martingale solutions to the stochastic compressible Navier-Stokes system.I will present new results on stochastic Navier-Stokes equations for compressible fluids. In particular, I will introduce the concept of "finite energy weak martingale solutions". These solutions are week in the analytical sense and in the probabilistic sense as well. In addition, they allow to control the evolution of the total energy. |

Raluca Eftimie | Analytical and numerical investigation of nonlocal kinetic and macroscopic models for collective behavior in animals.Analytical and numerical investigation of nonlocal kinetic and macroscopic models for collective behavior in animals.Collective movement of animals (e.g., birds, fish, insects, ungulates, etc.) has attracted scientists’ attention for over 2000 years. The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. Here, we start with a class of nonlocal hyperbolic models for animal behavior that exhibit a large variety of spatio-temporal patterns, and investigate how these patterns are preserved following different scaling approaches. |

Ben Goddard | Photo-dissociation of molecules: mathematics meets quantum chemistry.Photo-dissociation of molecules: mathematics meets quantum chemistry.Photo-dissociation is the break down of molecules by light, e.g. during photosynthesis. Its study involves transitions in a two-level PDE. Given an initial wavepacket on the upper level, the challenge is to determine the wavepacket transmitted to the lower level at large times. This is typically very small with rapid oscillations, prohibiting accurate numerical calculations. Fortunately, there exists a small parameter $\epsilon$, the square root of the the ratio of the electron and nuclear masses. In the standard adiabatic representation, widely used in chemistry, the transmitted wavepacket is of order $\epsilon$ globally in time but exponentially small (~$\exp(-1/\epsilon)$) for large times. This strongly suggests that the adiabatic representation is not the right one for the problem. Using the more general superadiabatic representations, we obtain an explicit formula for the transmitted wavepacket. Our results agree extremely well with high precision ab-initio calculations, in particular for real-world NaI. |

Willi Jäger | Transmission Problems for Reactive Flow and Transport: Multiscale Analysis of the Interactions of Solutes with a Solid Phase. Transmission Problems for Reactive Flow and Transport:Multiscale Analysis of the Interactions of Solutes with a Solid Phase. Modelling reactive flows, diffusion, transport and mechanical interactions in media
consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is
giving rise to many open problems for multi-scale analysis, in particular at the
interfaces. So far, the interactions of the solvent with the solid phase are approximated
too roughly in many applications. In this lecture, we are discussing more detailed
mathematical representations of the processes in the solid phase on the micro-scale,
and we are going to sketch the analysis of the arising transmission problems.
As a particular example, the following specific transmission conditions on the
interface between the solid and the fluid phase in case of chemical reactions will be
considered: |

Tommaso Lorenzi | Asymptotic analysis of nonlocal parabolic equations for phenotypically-structured populations.Asymptotic analysis of nonlocal parabolic equations for phenotypically-structured populations.Nonlocal partial differential equations constitute a convenient apparatus to study in silico the dynamics of populations structured by physiological traits. These equations can be derived from stochastic individual based models in the limit of large number of individuals, and raise interesting mathematical questions. In this talk, I will present some recent results concerning the asymptotic analysis of nonlocal parabolic equations which arise in mathematical models of phenotypic evolution in cancer cell populations. |

Charalambos Makridakis | Computing physically relevant solutions in nonlinear phenomena.Computing physically relevant solutions in nonlinear phenomena.The computation of singular phenomena (shocks, defects, dislocations, interfaces, cracks) arises in many complex systems. In order to simulate such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. Often weak solutions of PDEs related to these problems are not unique. Since numerical methods perturb the mathematical model, mathematical analysis emerges as a necessary tool providing mathematical guarantees ensuring that our computational methods approximate physically relevant solutions. Our purpose in this talk is to review results and discuss related computational and analytical challenges for such nonlinear problems modelled by PDEs. In addition we shall discuss related issues emerging in adaptive modelling across scales. |

Barbara Niethammer | Self-similarity in Smoluchowski's coagulation equation.Self-similarity in Smoluchowski's coagulation equation.Smoluchowski's coagulation equation is a nonlocal integral equation that is used to describe such diverse aggregation phenomena as fog formation, polymerization, soot agglomeration or even the formation of stars. The rate at which aggregation takes place is determined by a rate kernel that subsumes the details of the aggregation process under consideration. We are particularly interested in the behaviour of solutions for large times that is conjectured to be described by a self-similar profile. While this issue is by now well-understood for the small class of solvable kernels there are still many open questions in the general case. I will give an overview of the results that have been obtained in the last decade and discuss some of the remaining main open problems. |

Michela Ottobre | Degenerate diffusions.Degenerate diffusions.The study of diffusion processes has a long history and the literature on the matter is vast. Most of the literature on the subject assumes that the Markov semigroup associated with the process is smoothing in every direction. More precisely, the generator of the process is usually assumed to be an operator of elliptic or hypoelliptic type. In this talk we will consider a class of degenerate diffusions (so-called UFG processes) introduced by Kusuoka and Stroock. The corresponding semigroup is smoothing only in certain directions. This type of diffusions emerges naturally in the study of a new class of algorithms, the cubature methods, which aim at approximating the law of the solution of SDEs. In particular, we will provide the first results on the long-time behaviour of this class of dynamics. This is a joint work with Dan Crisan (Imperial College). |

Oana Pocovnicu | A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line.A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line.In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse. The goal of the present work is to construct an asymptotic global-in-time modulated two-soliton solution of small mass, which exhibits the following two regimes: (i) a turbulent regime characterized by an explicit growth of high Sobolev norms on a finite time interval, followed by (ii) a stabilized regime in which the high Sobolev norms remain stationary large forever in time. This talk is based on joint work with P. Gerard (Orsay, France), E. Lenzmann (Basel, Switzerland), and P. Raphael (Nice, France). |

Nikola Popovic | A geometric analysis of fast-slow models for stochastic gene expression.A geometric analysis of fast-slow models for stochastic gene expression.Stochastic models for gene expression frequently exhibit dynamics on different time-scales due to significant differences in the lifetimes of mRNA and the protein it synthesises, which allows for the application of perturbation techniques. We develop a dynamical systems framework for the analysis of a family of "fast-slow" models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by characterising a two-stage model which assumes transcription, translation, and degradation to be first-order reactions. We develop a systematic expansion procedure for the resulting propagator probabilities that can be taken to any order in the perturbation parameter. We verify our asymptotics by numerical simulation, and we explore its practical applicability, as well as the effects of a variation in the system parameters and the scale separation. Finally, we discuss the generalisation of our framework to models for regulated gene expression that involve additional stages. |

Steven Roper | Centroidal power diagrams: Numerical analysis and applications.Centroidal power diagrams: Numerical analysis and applications.We present numerical analysis of a variant of Lloyd's algorithm for the computation of centroidal power diagrams. A power diagram (or Laguerre-Voronoi diagram) is a type of weighted Voronoi diagram that arises in, for example, consideration of certain problems in pattern formation in materials science. |

Lucia Scardia | Convergence of Interaction-driven Evolutions of Dislocations.Convergence of Interaction-driven Evolutions of Dislocations.It is well known that the plastic, or permanent, deformation of a metal is caused by the movement of curve-like defects in its crystal lattice. These defects are called dislocations. What is not yet clear is how to use this microscale information to make theoretical predictions at the continuum scale. Motivated by this, we considered systems of interacting dislocations and studied the convergence of the evolutions of the corresponding empirical measures in the limit of many dislocations. In the continuum limit we obtained evolution laws for the dislocation density. In this talk I will present these results and discuss their limitations and further extensions towards more realistic and complex systems. This is work in collaboration with M.G. Mora, M. Peletier, A. Garroni and P. van Meurs. |

David Siska | Nonlinear stochastic evolution equations of second order with damping.Nonlinear stochastic evolution equations of second order with damping.Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. |

Dumitru Trucu | Structured Models of Cell Migration Incorporating Membrane Reactions.Structured Models of Cell Migration Incorporating Membrane Reactions.The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes enables a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this work we establish a general spatio-temporal-structural framework that enables the description of surface- bound reaction processes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with two concrete examples arising from cancer invasion. |

Chandrasekhar Venkataraman | Free boundaries on cell boundaries: understanding asymptotic limits of a model for receptor-ligand dynamics.Free boundaries on cell boundaries: understanding asymptotic limits of a model for receptor-ligand dynamics.In this talk we investigate a simple model for receptor-ligand dynamics which consists of coupled bulk-surface systems of parabolic partial differential equations. Nondimensionalisation of the model using biologically relevant values for the various characteristic scales leads one to consider a number of biologically relevant asymptotic limits of the model. In this talk we develop a mathematical theory for the treatment of the original model together with a rigorous proof of convergence to a number of simplified limiting problems in the aforementioned limits. The theoretical results are supported by computations of the original and reduced problems on realistic geometries. One upshot of our analysis is that, in a suitable parameter regime, the original system converges to a simplified model which is considerably cheaper to approximate numerically. |

We are also grateful for financial support from the
Glasgow Mathematical Journal Learning and Research Support Fund