| Date | Speaker | Title/Abstract |
|---|---|---|
| 15 Feb | Eugen Varvaruca University of Reading |
Singularities of axisymmetric free surface flows with gravity I will present some recent results, obtained in collaboration with Georg Weiss (University of Dusseldorf), on singularities of steady axisymmetric solutions of the Euler equations for a fluid, incompressible and with zero vorticity, having a free surface, and which is acted on only by gravity. We use geometric methods to analyze the asymptotics of the velocity field and of the free surface at stagnation points as well as at points on the axis of symmetry. At points on the axis of symmetry which are not stagnation points, constant velocity motion is the only blow-up profile consistent with the invariant scaling of the equation. This suggests the presence of downward pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blow-up profile consistent with the invariant scaling of the equation is the "Garabedian pointed bubble" solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A fine analysis of the blow-up velocity enables us to identify the correct scaling in that case under the additional assumption that the free surface is described by an injective curve. This last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by Delort; while the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application. |
| 22 Feb | Anthony Dooley University of Bath |
Lie symmetry methods for heat kernels and other evolution equations |
| 8 Mar | Patricia Bauman Purdue University |
Analysis of Energy Minimizers for Nematic Liquid Crystals with Disclination-Line Defects We investigate the structure of nematic liquid crystal thin films described by the Landau-de Gennes tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elasticity constant goes to zero in the energy, a limiting uniaxial nematic texture forms with a finite number of defects, all of degree 1/2 or −1/2, corresponding to vertical disclination lines at those locations. |
| 22 Mar 6E 2.1 |
Neshan Wickramasekera University of Cambridge |
Structure of branch sets of harmonic functions and minimal submanifolds I will present results from an ongoing project with Brian Krummel aimed at understanding fine properties (i.e. rectifiability, local finiteness of measure, uniqueness of blow-ups etc) of the singular sets of higher-multiplicity minimal submanifolds and associated multiple-valued harmonic functions. By way of providing context and motivation, some related old results will also be described. |
| 19 Apr | Jeyabal Sivaloganathan University of Bath |
Stability Criteria for Equilibria in Nonlinear Elasticity |
| 3 May | Laura Caravenna University of Oxford |
Lagrangian, Broad and Eulerian continuous solutions to a scalar conservation law with bounded source We consider a scalar conservation law with bounded source term: we discuss and introduce the correspondence between Lagrangian, Broadl and Eulerian points of view for solutions which are merely continuous. When the flux is quadratic, this equation is the most challenging one in a multi-D system describing intrinsic Lipschitz graphs in the Heisenberg groups. This description, expected from previous works on intrinsic regular graphs, was the first motivation for this investigation (joint work with F. Bigolin and F. Serra Cassano). For uniformly convex flux the setting is quite challenging, one should give proper definitions. Provided that, the correspondence among the various formulations works. In the general case instead we show that the notions can be different, and that more complicated behavior may occur. We give however an almost complete description of the equivalences (joint work with G. Alberti and S. Bianchini). |