ON DEMAND International PDE Conference 2022 - DAY ONE
Prof. John Ball, Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh
Image comparison via nonlinear elasticity
The talk will describe how models based on nonlinear elasticity can be used to compare images and parts of images. The requirement that the corresponding minimization algorithm delivers a linear map between linearly related images leads to a new condition of quasiconvexity. This is joint work with Chris Horner
Prof. Pierre-Louis Lions - Collège de France in Paris / École Polytechnique
The Master Equation for Mean Field Games
Prof. Elia Bruè - Institute for Advanced Study in Princeton
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this talk, we exhibit two distinct Leray solutions with zero initial velocity and identical body force. Building on a recent work of Vishik, we construct a linear unstable self-similar solution to the 3D Navier-Stokes with force. We employ the linear instability of the latter to build the second solution, which is a trajectory on the unstable manifold, in accordance with the predictions of Jia and Šverák.
Further Research:
https://annals.math.princeton.edu/2022/196-1/p03
Prof. Eduard Feireisl - Speaker - Institute of Mathematics, Czech Academy of Sciences
On singular limits for the Rayleigh-Benard problem
We consider the full Navier-Stokes-Fourier system in the Rayleigh-Benard setting and its singular limit in the low Mach/low Froude number. The limit problems is identified as the Oberbeck-Boussinesq system with non-local boundary conditions.
Prof. Charles M. Elliott - Mathematics Institute, University of Warwick
PDEs in Evolving Domains
Prof. Peter Topping - Mathematics Institute, University of Warwick
Hamilton's pinching conjecture
*No presentations slides published.
Over recent decades an extraordinary number of long-standing open problems in differential geometry and topology have been solved by the development of new technology for nonlinear partial differential equations. In this talk, intended to be accessible to specialists in PDEs of all flavours, I will discuss the topic of pinching problems where one assumes some local properties of curvature and tries to deduce global geometric and topological properties. In particular, I will explain how developments this year in Ricci flow theory have finally led to a resolution of Hamilton's pinching conjecture. Joint work with Man Chun Lee.
Prof. Philip Maini - Mathematical Institute, University of Oxford
Modelling collective cell migration in repair and disease
Collective cell movement is very common in biology, occurring in normal development, repair and disease. Here, I will consider two examples: angiogenesis – the process by which new blood vessels are formed in response to wound healing or tumour growth. By using a coarse-graining approach, we derive a partial differential equation (PDE) model for this phenomenon that is a generalisation of the classical phenomenological snail-trail model in the literature, and we show under what conditions the latter is not valid. We then consider a model for cancer cell invasion that involves a coupled system of PDEs with a degenerate cross-diffusion term, and we analyse the travelling wave behaviour of the model.
Further Research:
- S. Pillay, H.M. Byrne, P.K. Maini, Modeling angiogenesis: a discrete to continuum description, Phys. Rev. E., 95, 012410 (2017) link
- M.R.A. Strobl, A.L. Krause, M.Damaghi, R. Gillies, A.R.A. Anderson, P.K. Maini, Mix and Match: phenotypic coexistence as a key facilitator of cancer invasion, Bull. Math. Biol., 82 (2020) link
- C. Colson , F. Sánchez-Garduño, H.M. Byrne, P. K. Maini, T. Lorenzi, Travellingwave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion, Proc. R. Soc. A 477, 20210593 (2021) supp[pdf] link