Dr Anna Lisa Vari will talk about: 'The orbital structure of the Hill's problem'

Hill’s problem is a limiting case of the circular restricted gravitational three-body problem in which the mass ratio between the two massive bodies tends to zero, leaving a small region surrounding the secondary in which it remains gravitationally dominant. Originally formulated in terms of point masses, Hill’s problem may be modified to include a secondary of finite extent, thus providing a more realistic description of the dynamics internal to a stellar cluster orbiting within a host galaxy. By considering stellar energies above the cluster escape energy, we may investigate the dynamics that underpin the process of stellar escape from star clusters -- a topical issue in contemporary astrophysics. Specifically, we construct a self-consistent formulation of Hill’s problem using a tidally perturbed cluster model for the secondary body. The behaviour of energetically unbound stellar orbits within such a self-consistent problem, as characterised using Poincaré surfaces of section, is then numerically explored via a structure-preserving integrator, revealing a previously unknown bifurcation in the orbital structure.

We talk about the global-in-time well-posedness of classical solutions to the vacuum free boundary problem of the 1D viscous Saint-Venant system for laminar shallow water with large data. Since the depth of the fluid vanishes on the moving boundary, the momentum equations become degenerate both in the time evolution and spatial dissipation, which may lead to singularities for the derivatives of the velocity of the fluid and then makes it challenging to study classical solutions. By exploiting the intrinsic degenerate-singular structures of the viscous Saint-Venant system, we are able to identify two classes of admissible initial depth profile and obtain the global well-posedness theory here: the first class of the initial depth profile satisfies the well-known BD entropy condition; the second class of the initial depth profile satisfies the well-known physical vacuum boundary condition, but violates the BD entropy condition. One of the key ingredients of the analysis here is to establish some new degenerate weighted estimates for the effective velocity via its transport properties, which do not require the initial BD entropy condition or the physical vacuum boundary condition. These new estimates enable one to obtain the upper bound for the first order spatial derivative of the flow map. Then the global-in-time regularity uniformly up to the vacuum boundary can be obtained by carrying out a series of singular or degenerate weighted energy estimates carefully designed for this system.