# Frontiers in Mathematics Lecture Series

## Fall 2017

Date: | September 11, 2017 |

Time: | 4:00pm |

Location: | Blocker 117 |

Speaker: | Diego Cordoba, Consejo Superior de Investigaciones Científicas, Madrid, Spain |

Title: | Graduate Lecture: Splash and splat singularities for incompressible fluid interfaces |

Abstract: | The evolution of an interface between two immiscible incompressible fluids can develop singularities in finite time. In particular those contour dynamics that are given by basic fluid mechanics systems: Euler´s equations, Darcy´s law and the Quasi-geostrophic equation. These give rise to problems such as water wave, Muskat, and the evolution of sharp fronts of temperature. In this lecture we will present the main ideas and arguments of the formation in finite time of splash and splat singularities. A splash singularity is when the interfaceremains smooth but self-intersects at a point and a splat singularity is when it self-intersects along an arc. |

Date: | September 12, 2017 |

Time: | 4:00pm |

Location: | Blocker 117 |

Speaker: | Diego Cordoba, Consejo Superior de Investigaciones Científicas, Madrid, Spain |

Title: | Colloquium: Global existence results for the Surface Quasi-geostrophic equations (SQG) |

Abstract: | There has been high scientific interest to understand the behavior of the SQG equation because it is a possible model to explain the formation of fronts of hot and cold air. In a different direction P. Constantin, A. Majda and E. Tabak (1994) proposed this system as a 2D model for the 3D vorticity intensification and showed that there is a geometric and analytic analogy with 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities. In this lecture I will discuss some recent work, joint with Angel Castro, Javier Gomez-Serrano and Alex Ionescu, on the existence of non trivial families of global solutions of the inviscid surface quasi-geostrophic equation. |

Date: | September 14, 2017 |

Time: | 4:00pm |

Location: | Blocker 117 |

Speaker: | Diego Cordoba, Consejo Superior de Investigaciones Científicas, Madrid, Spain |

Title: | Colloquium: Shift of stability and mixing solutions for the Muskat problem |

Abstract: | The Muskat equation governs the motion of an interface separation of two incompressible fluids in a porous media. In this talk I will present the following recent results: (1) The existence of solutions which shift stability regimes in the following sense: they start stable, then become unstable, and finally return back to the stable regime before it breaks down (joint work with J. Gomez-Serrano and A. Zlatos). (2) The existence of mixing solutions of the incompressible porous media equation for all Muskat type H^5 initial data in the fully unstable regime (joint work with A. Castro and D. Faraco). |