# PDE Geometric Analysis seminar

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

# Seminar Schedule Spring 2013

date | speaker | title | host(s) |
---|---|---|---|

February 4 | Myoungjean Bae (POSTECH) |
Transonic shocks for Euler-Poisson system and related problems |
Feldman |

February 18 | Mike Cullen (Met. Office, UK) |
Modelling the uncertainty in predicting large-scale atmospheric circulations. |
Feldman |

March 18 | Mohammad Ghomi(Math. Georgia Tech) | Angenent | |

April 8 | Wei Xiang (Oxford) |
Shock Diffraction Problem to the Two Dimensional Nonlinear Wave System and Potential Flow Equation. |
Feldman |

Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room) | Adrian Tudorascu (West Virginia University) | One-dimensional pressureless | Feldman |

May 6 | Diego Cordoba (Madrid) | Kiselev |

# Abstracts

### Myoungjean Bae (POSTECH)

*Transonic shocks for Euler-Poisson system and related problems*

Abstract: Euler-Poisson system models various physical phenomena including the propagation of electrons in submicron semiconductor devices and plasmas, and the biological transport of ions for channel proteins. I will explain difference between Euler system and Euler-Poisson system and mathematical difficulties arising due to this difference. And, recent results about subsonic flow and transonic flow for Euler-Poisson system will be presented. This talk is based on collaboration with Ben Duan and Chunjing Xie.

### Mike Cullen (Met. Office, UK)

*Modelling the uncertainty in predicting large-scale atmospheric circulations*

Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.

### Mohammad Ghomi(Math. Georgia Tech)

>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".

### Wei Xiang (Oxford)

Abstract: The vertical shock which initially separates two piecewise constant Riemann data, passes the wedge from left to right, then shock diffraction phenomena will occur and the incident shock becomes a transonic shock. Here we study this problem on nonlinear wave system as well as on potential flow equations. The existence and the optimal regularity across sonic circle of the solutions to this problem is established. The comparison of these two systems is discussed, and some related open problems are proposed.

### Adrian Tudorascu (West Virginia University)

Abstract: This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).

### Diego Cordoba (Madrid)

Abstract: We consider the evolution of an interface generated between two immiscible, incompressible and irrotational fluids. Specifically we study the Muskat equation (the interface between oil and water in sand) and water wave equation (interface between water and vacuum). For both equations we will study well-posedness and the existence of smooth initial data for which the smoothness of the interface breaks down in finite time. We will also discuss some open problems.