## Seminars and Colloquia by Series

### Global existence for quasilinear wave equations close to Schwarzschild

Series
PDE Seminar
Time
Tuesday, November 8, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Mihai TohaneanuUniversity of Kentucky
We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

### The relativistic dynamics of an electron coupled with a classical nucleus

Series
PDE Seminar
Time
Tuesday, October 25, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Anne-Sophie de SuzzoniUniversity Paris XIII
This talk is about the Dirac equation. We consider an electron modeled by awave function and evolving in the Coulomb field generated by a nucleus. Ina very rough way, this should be an equation of the form$$i\partial_t u = -\Delta u + V( \cdot - q(t)) u$$where $u$ represents the electron while $q(t)$ is the position of thenucleus. When one considers relativitic corrections on the dynamics of anelectron, one should replace the Laplacian in the equation by the Diracoperator. Because of limiting processes in the chemistry model from whichthis is derived, there is also a cubic term in $u$ as a correction in theequation. What is more, the position of the nucleus is also influenced bythe dynamics of the electron. Therefore, this equation should be coupledwith an equation on $q$ depending on $u$.I will present this model and give the first properties of the equation.Then, I will explain why it is well-posed on $H^2$ with a time of existencedepending only on the $H^1$ norm of the initial datum for $u$ and on theinitial datum for $q$. The linear analysis, namely the properties of thepropagator of the equation $i\partial_t u = D u + V( \cdot - q(t))$ where$D$ is the Dirac operator is based on works by Kato, while the non linearanalysis is based on a work by Cancès and Lebris.It is possible to have more than one nucleus. I will explain why.(Joint work with F. Cacciafesta, D. Noja and E. Séré)

### Some Properties of Effective Hamiltonians

Series
PDE Seminar
Time
Tuesday, October 18, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yifeng YuUniverstiy of California, Irvine
A major open problem in periodic homogenization of Hamilton-Jacobi equations is to understand deep properties of the effective Hamiltonian. In this talk, I will present some related works in both convex and non-convex situations. If time permits, relevant problems from applications in turbulent combustion and traffic flow will also be discussed.

### A min-max formula for Lipschitz operators that satisfy the global comparison principle.

Series
PDE Seminar
Time
Tuesday, September 20, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Russell SchwabMichigan State University
We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary. This is joint work with Nestor Guillen.

### Blowup for model equations of fluid mechanics

Series
PDE Seminar
Time
Tuesday, August 30, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vu HoangRice University
The incompressible three-dimensional Euler equations are a basic model of fluid mechanics. Although these equations are more than 200 years old, many fundamental questions remain unanswered, most notably if smooth solutions can form singularities in finite time. In this talk, I discuss recent progress towards proving a finite time blowup for the Euler equations, inspired numerical work by T. Hou and G. Luo and analytical results by A. Kiselev and V. Sverak. My main focus lies on various model equations of fluid mechanics that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner using the hyperbolic flow scenario in one and two space dimensions. This talk is based on joint work with B. Orcan-Ekmecki, M. Radosz, and H. Yang.

### Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space

Series
PDE Seminar
Time
Friday, April 29, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
SGSW is a third level specialization of Navier-Stokes (via Boussinesq, then Semi-Geostrophic), and it accurately describes large-scale, rotation-dominated atmospheric flow under the extra-assumption that the horizontal velocity of the fluid is independent of the vertical coordinate. The Cullen-Purser stability condition establishes a connection between SGSW and Optimal Transport by imposing semi-convexity on the pressure; this has led to results of existence of solutions in dual space (i.e., where the problem is transformed under a non-smooth change of variables). In this talk I will present recent results on existence and weak stability of solutions in physical space (i.e., in the original variables) for general initial data, the very first of their kind. This is based on joint work with M. Feldman (UW-Madison).

### Global well-posedness for the Cubic Dirac equation in the critical space

Series
PDE Seminar
Time
Wednesday, April 27, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Ioan BejenaruUniversity of California, San Diego
We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space. The theory we develop is the Klein-Gordon counterpart of the Wave Maps / Schroedinger Maps theory. This is joint work with Sebastian Herr.

### Euler sprays and Wasserstein geometry of the space of shapes

Series
PDE Seminar
Time
Tuesday, April 19, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dejan SlepcevCarnegie Mellon University
We will discuss a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesic equations for this shape distance are Euler equations for incompressible, inviscid flow of fluid with zero pressure and surface tension on the free boundary. We will discuss the instability that the minimization problem develops and the resulting connections to optimal transportation. The talk is based on joint work with Jian-Guo Liu (Duke) and Bob Pego (CMU).

### Recent progress on geometric wave equations

Series
PDE Seminar
Time
Wednesday, April 6, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Sung-Jin OhUniversity of California, Berkeley
The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own. I will discuss the recent progress on the problem of global regularity and asymptotic behavior of solutions to these PDEs.

### A deterministic optimal design problem for the heat equation

Series
PDE Seminar
Time
Wednesday, March 30, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Alden WatersCNRS Ecole Normale Superieure
In everyday language, this talk addresses the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using wave packet decompositions from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated by solving a sequence of finite-dimensional optimization problems. This talk will also address future applications to inverse problems.