Annual General Meeting & Naylor Lecture

Location
Goodenough College, Mecklenburgh Square, London
Start date
-
Meeting Date
Speakers
Naylor Lecture: Endre Suli (Oxford). Accompanying Speaker: Josef Málek (Charles University, Prague)

LMS Annual General Meeting & Naylor Lecture

The lecture is aimed at a general mathematical audience. All interested, whether LMS members or not, are most welcome to attend this event.


Programme

3.00 pm Opening of the meeting and LMS Business (open to all but with voting for members only)

Agenda and Papers (to be announced in October 2022)

Election of the LMS Council and Nominating Committee in 2022

LMS Members' Book: Members of the London Mathematical Society can sign the Members' Book, which dates from 1865 when the Society was founded and contains signatures of members throughout the years, including Augustus De Morgan, Henri Poincaré, G. H. Hardy and Mary Cartwright.

4.00 pm  Supporting Lecture 2022: Josef Málek (Charles University, Prague)

Beyond the incompressible Navier-Stokes equations: mathematical foundations of models of non-Newtonian fluids

Abstract:  A century ago theoretical physicists and mathematicians changed the viewpoint regarding the meaning of solving problems associated with partial differential equations in general, and the incompressible Navier-Stokes equations in particular. Instead of considering a solution to a partial differential equation as a complicated mapping between two sets of variables, they began to view a solution as a point in a suitable infinite-dimensional space. In a landmark paper published in 1934, by paying careful attention on connections between mathematical approaches and the physical underpinnings of the problem,  Jean Leray, succeeded in developing a robust mathematical framework for the analysis of solutions to the Navier-Stokes equations.

The Navier-Stokes equations cannot, however, describe the physical responses of fluids that are endowed with complex microstructure or the extreme behaviour that certain fluids exhibit.

The study of non-Newtonian fluids, i.e. fluids whose flows cannot be described by the Navier-Stokes equations, is an area of fluid mechanics involving a vast array of various models, covering different physical phenomena, used in different application areas, in different physical contexts,  and developed on different, frequently rather ad hoc, intuitive bases. We provide a systematic classification of mathematical models of non-Newtonian fluids based on the phenomena they can exhibit, and identify three large classes of models. For two of them, namely implicitly-constituted viscous fluids and viscoelastic fluids with stress diffusion, we then report on recent work concerning  the development of robust mathematical theories, in the spirit of Leray’s work. These new theories precisely determine the mathematical objects that can be approximated by computational methods and have the potential to serve as correct analytical frameworks for the quantification of the error between exact and computed solutions in numerical approximations of these models. These developments build on several recent studies and have been finalized in recent joint papers with Michal Bathory, Miroslav Bulíček and Erika Maringová.

4.55 pm Election Results Announced.

5.00 pm  Naylor Lecture 2022: Endre Suli (Oxford)  

Hilbert's 19th problem and discrete De Giorgi--Nash--Moser theory: analysis and applications.

Abstract: Models of non-Newtonian fluids play an important role in science and engineering and their mathematical analysis and numerical  approximation have been active fields of research over the past decade. This lecture is concerned with the mathematical analysis of a large class of numerical methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically-reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra-filtrate of blood plasma that contains hyaluronic acid, whose concentration influences the shear-thinning property and helps to maintain a high viscosity; its function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power-law exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations one needs to derive a uniform Hölder norm bound on the sequence of approximations to the concentration in a setting where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely a bounded function with no additional regularity. This necessitates the development of a discrete counterpart of the De Giorgi--Nash--Moser theory, which is then used, via a combination of various weak compactness techniques, to deduce the convergence of the sequence of numerical approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration. The theoretical results are illustrated with numerical simulations.

6.00 pm Close of Meeting.


Registration: Registration will open in September 2022.

LMS Annual Dinner: Registration will open in September 2022.


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