| Date | Speaker | Title/Abstract |
|---|---|---|
| 10 Feb | Karsten Matthies University of Bath |
A semigroup approach to the Justification of Kinetic Theory A method is presented to show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. Considering a simplified case of hard-sphere dynamics, where particles are removed after the first collision, we characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods. Joint work with Florian Theil. |
| 24 Feb | Nikolaos Bournaveas University of Edinburgh |
Global existence vs blow up for nonlinear kinetic models of chemotaxis Chemotaxis is the directed motion of cells towards higher concentrations of chemoattractants. It is modeled at the macroscopic level by nonlinear parabolic equations and at the microscopic level by nonlinear kinetic equations. We shall discuss some recent global existence and blow up results. For the existence results we shall use Strichartz and dispersion estimates. (Joint work with Benoit Perthame, Vincent Calvez and Susana Gutierrez). |
| 9 Mar | John D Gibbon Imperial College London |
Conditional regularity of solutions of the three dimensional Navier-Stokes equations & implications for intermittency |
| 23 Mar | Isaac Chenchiah University of Bristol |
Quasi-Static Brittle Damage Evolution with Multiple Damaged Elastic States We present a variational model for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. We allow for multiple damaged states. Moreover, unlike current formulations, the materials are allowed to be anisotropic and the deformations are not restricted to anti-plane shear. The model can be formulated either energetically or through a strain threshold. We explore the relationship between these formulations. This is joint work with Christopher Larsen, Worcester Polytechnic Institute. |
| 20 Apr | Andre Neves Imperial College London |
Min-Max Theory and the Willmore conjecture In 1965, Willmore conjectured that for every torus in space, the integral of the mean curvature squared is bigger or equal to $2\pi^2$. I will talk about how to prove this conjecture using min-max methods. This is joint work with Fernando Marques. |
| 11 May, 2:45 5W 2.4 |
Jan Kristensen University of Oxford |
From Ornstein's non-inequalities to rank-one convexity Questions about sharp integral estimates for partial derivatives of mappings can often be recast as questions about quasiconvexity of associated integrands. Quasiconvexity was introduced by Morrey in his work on weak lower semicontinuity in the Calculus of Variations. It is by now recognized as a fundamental concept, but it remains somewhat mysterious. A closely related notion is that of rank-one convexity. Rank-one convexity is a necessary condition for quasiconvexity, and it is easy to check whether or not a given integrand is rank-one convex. Unfortunately, rank-one convexity is not equivalent to quasiconvexity. An example of Sverak shows that, in high dimensions, there exists a quartic polynomial that is rank-one convex but not quasiconvex. However, it is still plausible that rank-one convexity could be equivalent to quasiconvexity within more restricted classes of integrands. An interesting class being the positively one homogeneous integrands. Their quasiconvexity properties correspond to $L^1$-estimates. In this talk I briefly review the convexity notions from the Calculus of Variations. I then show how the above viewpoint can be used to give a proof, and a generalization, of Ornstein's non-$L^1$-inequalities. |
| 11 May, 4:15 5W 2.4 |
Norman Dancer University of Sydney |
Nonlinear elliptic and parabolic problems with large interactions |
| 15 June | Friedemann Brock American University of Beirut |
Continuous rearrangements, and symmetry and monotonicity of solutions to some elliptic problems |