Time: Monday 4 pm

Location: Hodson 313

Organizer(s): Liming Sun and Yi Wang

Date |
Speaker |
Title |
Abstract |

(TBA) |
TBA | TBA | |

(TBA) |
TBA | TBA | |

(Princeton) |
Some conformally invariant gap theorems for Bach-flat 4-manifolds | A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as model case more than ten years ago. In this talk, we extend this result to prove conformally invariant gap theorems for Bach-flat 4-manifolds with Fubini-Study metric on complex projective space and product metric on the product of spheres as model cases. An iteration argument plays an important role in the case of complex projective space and the convergence theory of Bach-flat metrics is the main analytic tool in the case of the product of spheres. If time permits, I shall also discuss some joint work with Chang and Gursky. | |

(Princeton) |
The asymptotically self-similar regime for the Einstein vacuum equations | We will dynamically construct singular solutions to the Einstein vacuum equations which are asymptotically self-similar in that successive rescalings around the singularity converge to a self-similar solution. Connections both to Christodoulou’s bounded variation solutions of the spherically symmetric Einstein-scalar field system and to the ambient metric construction of Fefferman and Graham will be elaborated on. This is joint work with Igor Rodnianski. | |

(University of Oklahoma) |
Liouville theorems on the upper half space | In this talk I will describe some new Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a \in (0, 1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a \nabla u)=0$ on the upper half space {(x_1, ..., x_n) : x_n>0}. This is a joint work with Lei Wang. | |

(MIT) |
The linear stability of the Schwarzschild spacetime in the harmonic gauge | We study the solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of the Regge-Wheeler and Zerilli equations, we estimate the Lichnerowicz d'Alembertian equation. In particular, we show that up to a one dimensional stationary mode, the solution decays to a linearlized Kerr solution. This is ongoig joint work with S. Brendle. | |

(Cambridge UK) |
The weak null condition and the p-weighted energy method | The Einstein equations in wave coordinates are an example of a system which does not obey the "null condition", and this leads to many difficulties, most famously when attempting to prove global existence, otherwise known as the "nonlinear stability of Minkowski space". Traditional approaches to overcoming these problems suffer from a lack of generalisability - among other things, they make the a priori assumption that the space is approximately scale-invariant. Given the current interest in studying the stability of black holes and related problems, removing this assumption is of great importance. The p-weighted energy method of Dafermos and Rodnianski promises to overcome this difficulty by providing a flexible and robust tool to prove decay. However, so far it has mainly been used to treat linear equations. In this talk I will explain how to modify this method to nonlinear systems which only obey the "weak null condition", a large class of systems including the Einstein equations. This involves adapting the p-weighted energy method, and combining it with the many of the geometric methods originally used by Christodoulou and Klainerman. Among other things, this allows us to enlarge the class of wave equations which are known to admit small-data global solutions, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by these systems: it is due to the formation of "shocks at infinity". | |

(UCLA) |
Almost sure scattering for the energy critical nonlinear wave equation | We will discuss the energy-critical nonlinear wave equation in four dimensions. For deterministic and smooth initial data, it is widely known that the solutions scatter, i.e., they asymptotically behave like solutions to the linear wave equation. In this talk, we will show that the scattering behaviour persists under random and rough perturbations of the initial data. As part of the argument, we will discuss techniques from restriction theory, such as wave packet decompositions and Bourgains bush argument. | |

(Princeton) |
Partial regularity for a class of nonlocal stationary Navier-Stokes | Weak solutions of Navier-Stokes are known to exist for all times, but whether they are in fact regular given smooth initial data is an important open question. I will discuss a Caffarelli-Kohn-Nirenberg type partial regularity result for time-independent Navier-Stokes with a large fractional Laplacian dissipation, which bounds from above the Hausdorff dimension of the singular set of a weak solution depending on the order of the dissipation. The Caffarelli-Silvestre type extension theory for higher order fractional differential operators of R. Yang plays a major role in allowing us to use local blowup arguments. | |

(MIT) |
Bubbling Dynamics for Corotational Wave Maps with Prescribed Radiation | Abstract | |

(UMass Amherst) |
Convex functional and the stratification of the singular set of their stationary points. | In this talk, I discuss partial regularity of stationary solutions and minimizers u from a set \Omega\subset \R^n to a Riemannian manifold N, for the functional \int_\Omega F(x,u,|\nabla u|^2) dx. The integrand F is convex and satisfies some ellipticity, boundedness and integrability assumptions. Using the idea of quantitative stratification I show that the k-th strata of the singular set of such solutions are k-rectifiable. | |

(Rutgers) |
Monge-Ampere equation with bounded periodic data | We consider the Monge-Ampere equation det(D^2u) = f in R^n, where f is a positive bounded periodic function. We prove that u must be the sum of a quadratic polynomial and a periodic function. For f ≡ 1, this is the classic result by Jorgens, Calabi and Pogorelov. For f \in C^\alpha, this was proved by Caffarelli and Y.Y. Li. This is a joint work with Y.Y. Li. | |

(University of Florida) |
Non-simple bubbling solutions and degree counting theorems for Liouville equations and Liouville systems. | For singular Liouville equations, bubbling solutions may appear to have “nonsimple” blowup phenomenon if singular sources are multiples of 4π. Such a phenomenon may also occur for more general Liouville systems. In this talk I would present some recent works with J. C. Wei and Yi. Gu, respectively on non-simple bubbling solutions. Among other applications of our new discovery, we derive some degree counting theorems for Liouville systems defined on Riemann surface, which not only establish existence results, but also link the global, topological information of the manifold with local, detailed blowup analysis. |

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