The series is continued as the
Analysis and Differential Equations Seminar.
The slot was shared
Applied Analysis Reading Group (AARG),
which met in the weeks indicated below.
| Date | Speaker | Title/Abstract |
|---|---|---|
| 1 Oct | Jonathan Ben-Artzi Imperial College London |
Can we compute everything? It is often desirable to solve mathematical problems as a limit of simpler problems. However, are such techniques always guaranteed to work? For instance, the problem of finding roots of polynomials of degree higher than three was only solved in the 1980s (Newton's method isn't guaranteed to converge)! Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one. They called such structures "Towers of Algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis, kinetic theory), show that Towers of Algorithms are a necessary tool, and introduce the Solvability Complexity Index -- a measurement of the complexity of a given problem. An important consequence is that solutions to some problems can never be obtained as a limit of finite dimensional approximations (and hence can never be solved numerically). I will discuss the implications of this in particular for applied mathematicians, and offer some potential remedies. Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau). |
| 8 Oct | AARG | |
| 15 Oct | Shane Cooper University of Bath |
Asymptotic analysis of stratified elastic media in the space of functions with bounded deformation We consider a heterogeneous elastic structure which is stratified in some direction (say x_1). We derive the limit problem under the assumption that the Lam\'e coefficients and their inverses weakly* converge to Radon measures. Our method applies also to linear second-order elliptic systems of partial differential equations, and in particular to the fully anisotropic heat equation. This is joint work with Michel Bellieud (Universit\'e de Montpellier 2). |
| 22 Oct | AARG | |
| 29 Oct | Claude Warnick Imperial College/University of Warwick |
Scattering resonances and black holes A feature of many open systems describing wavelike phenomena is that their late-time behaviour is characterised by time-harmonic components with a complex frequency, exhibiting oscillation and decay. Much as the spectrum of a manifold carries information about its geometry, these scattering resonances carry information about the system. I shall discuss recent progress in understanding scattering resonances using methods and ideas coming from the study of general relativity, in particular black holes. I shall not assume any familiarity with relativity. |
| 5 Nov | AARG | |
| 12 Nov | Carlo Mercuri University of Swansea |
On a class of nonlinear Schroedinger-Poisson-Slater equations I will present some recent results on a class of nonlinear Schroedinger-Poisson-Slater equations, discuss the variational formulation, and highlight the role of some new critical exponents related to several (lack of) compactness issues. This is a joint work with Vitaly Moroz (Swansea University) and Jean Van Schaftingen (UCL-Louvan la Neuve). |
| 19 Nov | AARG | |
| 26 Nov | Antoine Choffrut University of Warwick |
Rayleigh-Benard Convection: physically relevant a priori estimates A fluid contained between two horizontal plates is heated from below and cooled from above. Heat transfer is effected via two mechanisms: (1) thermal conduction, at the microscopic level; and (2) thermal convection, where lighter, warmer particles carry their internal energy to the top. On the other hand, fluid motion is resisted by inner friction due to viscosity. The governing equations are those of the Boussinesq approximation. The average upward heat flux relative to pure conduction is measured by the Nusselt number (Nu). The temperature gradient is measured by the Rayleigh number (Ra). The relative strength of viscosity over inertia is measured by the Prandtl number (Pr). In this talk I will present near optimal scaling laws for Nu as a function of Ra for two regimes of Pr, whereas previous work, pioneered by Constantin and Doering, with contributions from many others, assumed infinite Pr. The proof relies on maximal regularity estimates for the (linear) Stokes system in $L^\infty$- and $L^1$-type spaces, the latter with a borderline failing Muckenhoupt weight. This is joint work with Camilla Nobili and Felix Otto. |
| 3 Dec | AARG | |
| 10 Dec | Sergey Zelik University of Surrey |
Infinite energy solutions for damped Navier-Stokes equations in R^2
The damped Navier-Stokes equations in the whole 2D space will be discussed. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces will be verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field ${u_0 \in L^\infty(\mathbb{R}^2)}$ is allowed and no assumptions on the spatial decay of solutions as ${|x| \to \infty}$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in ${\mathbb{R}^2}$ , we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity. |
The series is continued as the Analysis and Differential Equations Seminar.