Difference between revisions of "PDE Geometric Analysis seminar"
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(→PDE GA Seminar Schedule Fall 2018-Spring 2019) |
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|December 10, Colloquium, '''Time: 4:00''' | |December 10, Colloquium, '''Time: 4:00''' | ||
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT) | | [https://math.mit.edu/~maxe/ Max Engelstein] (MIT) | ||
− | |[[# Max Engelstein| The role of Energy in Regularity ]] | + | |[[#Max Engelstein| The role of Energy in Regularity ]] |
| Feldman | | Feldman | ||
|- | |- | ||
|January 28, | |January 28, | ||
| Ru-Yu Lai (Minnesota) | | Ru-Yu Lai (Minnesota) | ||
− | |[[# Ru-Yu Lai | | + | |[[#Ru-Yu Lai | Inverse transport theory and related applications ]] |
| Li and Kim | | Li and Kim | ||
− | |||
− | |||
− | |||
− | |||
− | |||
|- | |- | ||
| February 4, | | February 4, | ||
| | | | ||
− | |[[# | No seminar (several relevant colloquiums in | + | |[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]] |
| | | | ||
|- | |- | ||
| February 11, | | February 11, | ||
| Seokbae Yun (SKKU, long term visitor of UW-Madison) | | Seokbae Yun (SKKU, long term visitor of UW-Madison) | ||
− | |[[# Seokbae Yun | | + | |[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]] |
| Kim | | Kim | ||
+ | |- | ||
+ | | February 13 '''4PM''', | ||
+ | | Dean Baskin (Texas A&M) | ||
+ | |[[#Dean Baskin | Radiation fields for wave equations]] | ||
+ | | Colloquium | ||
|- | |- | ||
− | | February 18, '''Room: VV B239''' | + | | February 18, '''3:30PM, Room: VV B239''' |
| Daniel Tataru (Berkeley) | | Daniel Tataru (Berkeley) | ||
− | |[[# Daniel Tataru | | + | |[[#Daniel Tataru | A Morawetz inequality for water waves ]] |
| Ifrim | | Ifrim | ||
|- | |- | ||
− | | February 19, | + | | February 19, '''Time: 4-5PM, Room: VV B139''' |
| Wenjia Jing (Tsinghua University) | | Wenjia Jing (Tsinghua University) | ||
− | |[[#Wenjia Jing | | + | |[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]] |
| Tran | | Tran | ||
|- | |- | ||
|February 25, | |February 25, | ||
| Xiaoqin Guo (UW) | | Xiaoqin Guo (UW) | ||
− | |[[#Xiaoqin Guo | | + | |[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]] |
| Kim and Tran | | Kim and Tran | ||
|- | |- | ||
Line 134: | Line 134: | ||
|[[#Jonathan Luk | TBA ]] | |[[#Jonathan Luk | TBA ]] | ||
| Kim | | Kim | ||
+ | |- | ||
+ | |March 12, '''4:00 p.m. in VV B139''' | ||
+ | | Trevor Leslie (UW-Madison) | ||
+ | |[[# Trevor Leslie| TBA ]] | ||
+ | | Analysis seminar | ||
|- | |- | ||
|March 18, | |March 18, | ||
Line 139: | Line 144: | ||
|[[# | ]] | |[[# | ]] | ||
| | | | ||
+ | |- | ||
+ | |March 25 | ||
+ | | Jiaxin Jin | ||
+ | |[[# Jiaxin Jin |TBA ]] | ||
+ | | local speaker | ||
|- | |- | ||
|April 1 | |April 1 | ||
Line 144: | Line 154: | ||
|[[#Zaher Hani | TBA ]] | |[[#Zaher Hani | TBA ]] | ||
| Ifrim | | Ifrim | ||
+ | |- | ||
+ | |April 8 | ||
+ | | Jingrui Cheng (Stony Brook) | ||
+ | |[[#Jingrui Cheng | TBA ]] | ||
+ | | Feldman | ||
|- | |- | ||
|April 15, | |April 15, | ||
Line 157: | Line 172: | ||
|April 29, | |April 29, | ||
| Beomjun Choi (Columbia) | | Beomjun Choi (Columbia) | ||
− | |[[# | | + | |[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]] |
| Angenent | | Angenent | ||
|} | |} | ||
Line 241: | Line 256: | ||
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur. | We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur. | ||
+ | |||
+ | |||
+ | ===Ru-Yu Lai=== | ||
+ | Title: Inverse transport theory and related applications. | ||
+ | |||
+ | Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed. | ||
+ | |||
+ | ===Seokbae Yun=== | ||
+ | Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation | ||
+ | |||
+ | Abstract: In this talk, we consider the propagation of the uniform upper bounds | ||
+ | for the spatially homogenous relativistic Boltzmann equation. For this, we establish two | ||
+ | types of estimates for the the gain part of the collision operator: namely, a potential | ||
+ | type estimate and a relativistic hyper-surface integral estimate. We then combine them | ||
+ | using the relativistic counter-part of the Carlemann representation to derive a uniform | ||
+ | control of the gain part, which gives the desired propagation of the uniform bounds of | ||
+ | the solution. Some applications of the results are also considered. This is a joint work | ||
+ | with Jin Woo Jang and Robert M. Strain. | ||
+ | |||
+ | |||
+ | |||
+ | ===Daniel Tataru=== | ||
+ | |||
+ | Title: A Morawetz inequality for water waves. | ||
+ | |||
+ | Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru. | ||
+ | |||
+ | Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. | ||
+ | |||
+ | |||
+ | ===Wenjia Jing=== | ||
+ | |||
+ | Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof | ||
+ | |||
+ | Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results. | ||
+ | |||
+ | |||
+ | ===Xiaoqin Guo=== | ||
+ | |||
+ | Title: Quantitative homogenization in a balanced random environment | ||
+ | |||
+ | Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). | ||
+ | |||
+ | |||
+ | |||
+ | ===Beomjun Choi=== | ||
+ | In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. | ||
+ | |||
+ | The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces. |
Latest revision as of 06:12, 19 February 2019
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
Contents
- 1 Previous PDE/GA seminars
- 2 Tentative schedule for Fall 2019-Spring 2020
- 3 PDE GA Seminar Schedule Fall 2018-Spring 2019
- 4 Abstracts
- 4.1 Julian Lopez-Gomez
- 4.2 Hiroyoshi Mitake
- 4.3 Changyou Wang
- 4.4 Matthew Schrecker
- 4.5 Anna Mazzucato
- 4.6 Lei Wu
- 4.7 Annalaura Stingo
- 4.8 Yeon-Eung Kim
- 4.9 Albert Ai
- 4.10 Trevor Leslie
- 4.11 Serena Federico
- 4.12 Max Engelstein
- 4.13 Ru-Yu Lai
- 4.14 Seokbae Yun
- 4.15 Daniel Tataru
- 4.16 Wenjia Jing
- 4.17 Xiaoqin Guo
- 4.18 Beomjun Choi
Previous PDE/GA seminars
Tentative schedule for Fall 2019-Spring 2020
PDE GA Seminar Schedule Fall 2018-Spring 2019
date | speaker | title | host(s) |
---|---|---|---|
August 31 (FRIDAY), | Julian Lopez-Gomez (Complutense University of Madrid) | The theorem of characterization of the Strong Maximum Principle | Rabinowitz |
September 10, | Hiroyoshi Mitake (University of Tokyo) | On approximation of time-fractional fully nonlinear equations | Tran |
September 12 and September 14, | Gunther Uhlmann (UWash) | TBA | Li |
September 17, | Changyou Wang (Purdue) | Some recent results on mathematical analysis of Ericksen-Leslie System | Tran |
Sep 28, Colloquium | Gautam Iyer (CMU) | Stirring and Mixing | Thiffeault |
October 1, | Matthew Schrecker (UW) | Finite energy methods for the 1D isentropic Euler equations | Kim and Tran |
October 8, | Anna Mazzucato (PSU) | On the vanishing viscosity limit in incompressible flows | Li and Kim |
October 15, | Lei Wu (Lehigh) | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects | Kim |
October 22, | Annalaura Stingo (UCD) | Global existence of small solutions to a model wave-Klein-Gordon system in 2D | Mihaela Ifrim |
October 29, | Yeon-Eung Kim (UW) | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties | Kim and Tran |
November 5, | Albert Ai (UC Berkeley) | Low Regularity Solutions for Gravity Water Waves | Mihaela Ifrim |
Nov 7 (Wednesday), Colloquium | Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems | Feldman |
December 3, Time: 3:00, Room: B223 Van Vleck | Trevor Leslie (UW) | Flocking Models with Singular Interaction Kernels | Kim and Tran |
December 10, Time: 2:25, Room: B223 Van Vleck | Serena Federico (MIT) | Sufficient conditions for local solvability of some degenerate partial differential operators | Mihaela Ifrim |
December 10, Colloquium, Time: 4:00 | Max Engelstein (MIT) | The role of Energy in Regularity | Feldman |
January 28, | Ru-Yu Lai (Minnesota) | Inverse transport theory and related applications | Li and Kim |
February 4, | No seminar (several relevant colloquiums in Feb/5 and Feb/8) | ||
February 11, | Seokbae Yun (SKKU, long term visitor of UW-Madison) | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation | Kim |
February 13 4PM, | Dean Baskin (Texas A&M) | Radiation fields for wave equations | Colloquium |
February 18, 3:30PM, Room: VV B239 | Daniel Tataru (Berkeley) | A Morawetz inequality for water waves | Ifrim |
February 19, Time: 4-5PM, Room: VV B139 | Wenjia Jing (Tsinghua University) | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof | Tran |
February 25, | Xiaoqin Guo (UW) | Quantitative homogenization in a balanced random environment | Kim and Tran |
March 4 | Vladimir Sverak (Minnesota) | TBA(Wasow lecture) | Kim |
March 11 | Jonathan Luk (Stanford) | TBA | Kim |
March 12, 4:00 p.m. in VV B139 | Trevor Leslie (UW-Madison) | TBA | Analysis seminar |
March 18, | Spring recess (Mar 16-24, 2019) | ||
March 25 | Jiaxin Jin | TBA | local speaker |
April 1 | Zaher Hani (Michigan) | TBA | Ifrim |
April 8 | Jingrui Cheng (Stony Brook) | TBA | Feldman |
April 15, | Yao Yao (Gatech) | TBA | Tran |
April 22, | Jessica Lin (McGill University) | TBA | Tran |
April 29, | Beomjun Choi (Columbia) | Evolution of non-compact hypersurfaces by inverse mean curvature | Angenent |
Abstracts
Julian Lopez-Gomez
Title: The theorem of characterization of the Strong Maximum Principle
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.
Hiroyoshi Mitake
Title: On approximation of time-fractional fully nonlinear equations
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)
Changyou Wang
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.
Matthew Schrecker
Title: Finite energy methods for the 1D isentropic Euler equations
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.
Anna Mazzucato
Title: On the vanishing viscosity limit in incompressible flows
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.
Lei Wu
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.
Annalaura Stingo
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.
Yeon-Eung Kim
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.
Albert Ai
Title: Low Regularity Solutions for Gravity Water Waves
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.
Trevor Leslie
Title: Flocking Models with Singular Interaction Kernels
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.
Serena Federico
Title: Sufficient conditions for local solvability of some degenerate partial differential operators
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.
Max Engelstein
Title: The role of Energy in Regularity
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.
Ru-Yu Lai
Title: Inverse transport theory and related applications.
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.
Seokbae Yun
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation
Abstract: In this talk, we consider the propagation of the uniform upper bounds for the spatially homogenous relativistic Boltzmann equation. For this, we establish two types of estimates for the the gain part of the collision operator: namely, a potential type estimate and a relativistic hyper-surface integral estimate. We then combine them using the relativistic counter-part of the Carlemann representation to derive a uniform control of the gain part, which gives the desired propagation of the uniform bounds of the solution. Some applications of the results are also considered. This is a joint work with Jin Woo Jang and Robert M. Strain.
Daniel Tataru
Title: A Morawetz inequality for water waves.
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit.
Wenjia Jing
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.
Xiaoqin Guo
Title: Quantitative homogenization in a balanced random environment
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).
Beomjun Choi
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed.
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.