几何分析与双曲方程会议日程安排

（2016年11月4-5日、中科院数学所）

会议地点：数学院南楼N913

报告题目及内容摘要

报告人：杨诗武（北京国际数学中心）

题目： Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms

摘要： It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein scalar-field system which are decaying at (null) infinity give rise to solutions which are global in the sense that they are future causally geodesically complete. In this talk, I will present a construction of a class of global solutions whose data are not required to decay towards infinity. As a consequence, there exist global solutions with arbitrarily large (and in fact infinite) bounded variation (BV) norms and initial Bondi masses. This construction is also extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, I will discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical Sobolev norms. This is jointed work with Jonathan Luk and Oh.

报告人：王成波（浙江大学）

题目: The radial Glassey conjecture with minimal regularity

摘要: In this talk, I will report our recent advances on the radial Glassey conjecture with minimal regularity assumption. More precisely, for semilinear wave equations of type $\Box u =|\partial_t u|^p$ with $p_c = 1 + 2/(n -1)<p<1+2/(n-2)$, we prove the global existence for small data in $H^s$ with $s>s_c=n/2+1-1/(p-1)$. It is known that $p>p_c$ is necessary for the problem to admit global solutions and $s\ge s_c$ is necessary for the problem to be local well-posed in $H^s$.

In the process, we exploit and prove a weighted fractional chain rule. We also show well-posedness for 3-D quadratic semi-linear wave equations with radial data in the almost scale-critical Sobolev space, which improves the earlier result of Klainerman and Machedon.

This is based on the joint work with Kunio Hidano, Jin-Cheng Jiang, Sanghyuk Lee.

报告人：黎俊彬（中山大学）

题目：Formation of trapped surfaces around a spherical singularity of a scalar field

摘要： We extend the famous formation of trapped surfaces result of Christodoulou in 2008, when a massless scalar field is coupled, to the case that the vertex of the initial incoming null cone, which is assumed to be spherically symmetric, is singular. No symmetries are imposed in the solution. The theorem we prove is also an extension to an earlier result of formation of black hole of Christodoulou in spherically symmetry, which is crucial in proving the weak cosmic censorship of the Einstein-scalar field equations in spherical symmetry.

报告人：王金花（厦门大学）

题目：Morawetz estimate for linear gravity on the Schwarzschild spacetime

摘要：

Concerning the linearized gravity for Schwarzschild, the extreme curvature scalars satisfy the Teukolsky equations. Remarkably, analysis of symmetry operators yields transformations between solutions of Regge-Wheeler and Teukolsky. We prove the pointwise decay for the Regge Wheeler equation, and this gives strong pointwise decay for Teukolsky. This is joint work with Steffen Aksteiner and Lars Andersson

报告人：马跃（西安交通大学）

题目：Non-linear stability of Minkowski space-time in f(R) theory and in general relativity with massive scalar field

摘要：In this talk we will present some recent work about the system of Einstein equation coupled with a massive scalar field and the system of f(R) field equation (partially published in [2]). More precisely, we will focus on the nonlinear global stability of the Minkowski space-time within these two similar contexts. In a PDE point of view, they are equivalent to the global existence of a special class of quasi-linear wave-Klein-Gordon system with small initial data.

To the author's knowledge there is not so much choice to deal with this kind of system (for a detailed explication of the major difficulty, see for example in [1] page 2), and we apply the hyperboloidal foliation method introduced by the author in [1] combined with some newly developed tools such as L1 estimates on Klein-Gordon equations in curved space-time and L1 estimates on wave equations based on the expression of spherical means. We also adapt some tools developed in classical framework for the analysis of Einstein equation into our hyperboloidal foliation framework, such as the estimates based on wave gauge conditions and the L1 estimates on wave equations based on integration along characteristics.

Reference:

[1] P. LeFloch and Y. Ma, The hyperboloidal foliation method, World Scientific, 2015

[2] P. LeFloch and Y. Ma, The nonlinear stability of Minkowski space for self-gravitating massive field, the wave-Klein-Gordon model.  Communications in Mathematical Physics pp 1-63. First online: 02 January 2016

报告人：黄守军（安徽师范大学）

题目：On smooth solutions to the relativistic string equations in Schwarzschild space-time and related problems

摘要：This talk mainly considers the motion of relativistic strings in Schwarzschild space-time. First, we discuss the basic equations for the motion of a p-dimensional extended object in a general enveloping space-time, and then investigate some interesting properties enjoyed by the equations for the motion of relativistic strings in Schwarzschild space-time. Particularly, the equations are in fact of totally linearly degenerate system of hyperbolic PDEs of first order in (1+1) dimensions. Based on this, under suitable assumptions we are able to prove a small-data global existence of smooth solutions to the corresponding Cauchy problem. This is a joint work with Professor De-Xing Kong and Dr. Chun-Lei He. In addition, some topics and progress on the relativistic membrane equations are also provided. 

报告人：王芳（上海交通大学）

题目： Limit of sharp Sobolev inequalities

摘要： Consider the sharp Sobolev inequalities on the n-sphere. By assuming the dimension constant to be a continuous parameter,  then the limit of sharp Sobolev inequalities gives the Moser-Trudinger inequality as n—>2. However, this is a fake proof of the Moser-Trudinger since the dimension constants can only be integers. In this talk, I will mainly introduce a new point of view to make the limit to be mathematically true, by taking advantage of the fractional GJMS operators and their energy extension to the hyperbolic space.