3:00pm Mahir Hadzic (King’s College)
Stability of the FLRW Solutions to the Dust-Einstein System with a Positive Cosmological Constant.
Abstract: The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstroem’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.
4.30pm Diego Cordoba (ICMAT, Madrid)
Finite time singularities in incompressible fluid interfaces abstract.
Abstract: We consider the evolution of an interface generated between two immiscible, incompressible fluids. Specifically we study the Surface Quasi-geostrophic equation (a sharp front of cold and hot air), the Muskat equation (the interface between oil and water in sand) and the water wave equation (interface between water and vacuum). For both Muskat and water wave equations we show the existence of smooth initial data for which the smoothness of the interface breaks down in finite time.