LMS Annual General Meeting and Naylor Lecture 2024

Welcome and Society Business

• Agenda
• Minutes of the General Meeting on 28 June 2024 (to be published in October 2024)
• Trustees Report for 2023-24
• Announcement of LMS Election Results
• Signing the Members’ book
• Presentation of the LMS Prizes 2024.

Supporting Lecture:

Professor Marco Fontelos (Madrid)

Singularities in Partial Differential Equations

Starting from smooth initial data, solutions to evolutionary nonlinear Partial Differential Equations may develop singularities in finite time. These singularities can be such that solutions diverge (blow-up), or loose continuity (shocks), etc. A fundamental concept in the context of singularities is that of selfsimilarity. We use a similarity transformation of the original equation with respect to the singular point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterizing the singularity, in that the dynamics frequently reduces to very few dimensions.

Once the problem reduces to a finite number of degrees of freedom, one can use ideas from finite dimensional dynamical systems: attractors, center manifolds, bifurcations, etc. Finally, once a singularity develops, the issue fn how one can continue the solution to a PDE after the singularity arises. We will present these problems with examples of well-known Partial Differential Equations and systems and point to the many open mathematical problems, as well as to the current state of the art in the attempts to solve them. We put special emphasis on problems arising in connection to the Euler and Navier-Stokes equations of Fluid Mechanics.

Naylor Lecture 2024:

Professor Jens Eggers (Bristol).

The role of singularities in hydrodynamics

Some of the most interesting structures observed in hydrodynamics are best understood as singularities of the equations of fluid mechanics. Examples are drop formation in free-surface flow, shock waves in compressible gas flow, or vortices in potential flow. These examples show that singularities are characteristic for the tendency of the hydrodynamic equations to develop small-scale features spontaneously, starting from smooth initial conditions. As a result, new structures are created, which form the building blocks of more complicated flows.

The mathematical structure of singularities is self-similar, and their characteristics are fixed by universal properties: they are the fingerprints of a partial differential equation. We discuss how a more complicated (and possibly chaotic) temporal evolution emerges through Hopf bifurcations. By a generic mechanism of "unfolding", these one-dimensional structures can be translated into higher dimensions, creating complex spatio-temporal patterns.