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===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Fall 2016 | Tentative schedule for Spring 2017]]===
===[[Fall 2018 | Tentative schedule for Fall 2018]]===
 
 
 
== PDE GA Seminar Schedule Spring 2018 ==
 


= Analysis Seminar Schedule Fall 2016 =
{| cellpadding="8"
{| cellpadding="8"
!align="left" | date   
!style="width:20%" align="left" | date   
!align="left" | speaker
!align="left" | speaker
!align="left" | title
!align="left" | title
!align="left" | host(s)
!style="width:20%" align="left" | host(s)
 
|- 
|January 29, '''3-3:50PM,  B341 VV.'''
| Dan Knopf (UT Austin)
|[[#Dan Knopf |  Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons]]
| Angenent
|-
|February 5,  '''3-3:50PM,  B341 VV.'''
| Andreas Seeger (UW)
|[[#Andreas Seeger |  Singular integrals and  a problem on mixing flows ]]
| Kim & Tran
|-
|February 12
| Sam Krupa (UT-Austin)
|[[#Sam Krupa |  Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
| Lee
|-
|February 19
| Maja Taskovic (UPenn)
|[[#Maja Taskovic |  Exponential tails for the non-cutoff Boltzmann equation ]]
| Kim
|-
|February 26
|  Ashish Kumar Pandey (UIUC)
|[[#  |  Instabilities in shallow water wave models  ]]
| Kim & Lee
|-
|March 5
| Khai Nguyen (NCSU)
|[[#Khai Nguyen |  Burgers Equation with Some Nonlocal Sources ]]
| Tran
|-
|March 12
| Hongwei Gao (UCLA)
|[[#Hongwei Gao |  Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]
| Tran
|-
|March 19
| Huy Nguyen (Princeton)
|[[#Huy Nguyen |  Compressible fluids and active potentials ]]
| Lee
|-
|-
|Date
|March 26
| Person
|  
|[[#linktoabstract |   Title ]]
|[[#  | Spring recess (Mar 24-Apr 1, 2018) ]]
| Sponsor
|
|-
|-
|Date
|April 2
| Person
| In-Jee Jeong (Princeton)
|[[#linktoabstract |  Title ]]
|[[#In-Jee Jeong | Singularity formation for the 3D axisymmetric Euler equations ]]
| Sponsor
| Kim
|-
|-  
|Date
|April 9
| Person
| Jeff Calder (Minnesota)
|[[#linktoabstract |  Title ]]
|[[#Jeff Calder | Nonlinear PDE continuum limits in data science and machine learning ]]
| Sponsor
| Tran
|-
|-  
|Date
|April 21-22 (Saturday-Sunday)
| Person
| [https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]
|[[#linktoabstract |  Title ]]
|[[#Midwest PDE seminar | ]]
| Sponsor
| Angenent, Feldman, Kim, Tran.
|-
|-  
|Date
|April 25 (Wednesday)
| Person
| Hitoshi Ishii (Wasow lecture)
|[[#linktoabstract |  Title ]]
|[[#Hitoshi Ishii | Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory]]
| Sponsor
| Tran.
|-
|}
|Date
 
| Person
== Abstracts ==
|[[#linktoabstract  |  Title ]]
 
| Sponsor
===Dan Knopf===
|-
 
|Date
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
| Person
 
|[[#linktoabstract  |  Title ]]
Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.
| Sponsor
 
|-
===Andreas Seeger===
|Date
 
| Person
Title: Singular integrals and a problem on mixing flows
|[[#linktoabstract  |  Title ]]
 
| Sponsor
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
|-
 
}
===Sam Krupa===
 
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
 
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi.  Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.


=Abstracts=


===Daniel Spirn===
===Maja Taskovic===


Dipole Trajectories in Bose-Einstein Condensates
Title: Exponential tails for the non-cutoff Boltzmann equation


Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum stateOne interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasersI will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density functionThe effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernelFor a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.


===Donghyun Lee===
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.


The Boltzmann equation with specular reflection boundary condition in convex domains


I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.
===Ashish Kumar Pandey===


===Kevin Zumbrun===
Title: Instabilities in shallow water wave models


TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations  
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.


ABSTRACT:  We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class


of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and
===Khai Nguyen===


$D$ a bounded, symmetric bilinear map.  This is related to a number of other scenarios investigated recently for which the
Title: Burgers Equation with Some Nonlocal Sources


associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problemThe particular case studied here
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency.  This talk  will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equationsSome open questions will be discussed.


pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.
===Hongwei Gao=== 


Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations


Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.


===Will Feldman===
===Huy Nguyen===


Liquid Drops on a Rough Surface
Title : Compressible fluids and active potentials


I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness.  
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.


The talk is based on joint work with Inwon Kim.  A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks.
===In-Jee Jeong===


===Ryan Hynd===
Title: Singularity formation for the 3D axisymmetric Euler equations


Extremal functions for Morrey’s inequality in convex domains
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains  $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2  \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.


A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor.  We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.
===Jeff Calder===


===Gung-Min Gie ===
Title: Nonlinear PDE continuum limits in data science and machine learning


Boundary layer analysis of some incompressible flows
Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.


===Tau Shean Lim===
===Hitoshi Ishii===


Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory


We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.
Abstract:  In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.

Revision as of 00:34, 19 April 2018

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Tentative schedule for Fall 2018

PDE GA Seminar Schedule Spring 2018

date speaker title host(s)
January 29, 3-3:50PM, B341 VV. Dan Knopf (UT Austin) Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons Angenent
February 5, 3-3:50PM, B341 VV. Andreas Seeger (UW) Singular integrals and a problem on mixing flows Kim & Tran
February 12 Sam Krupa (UT-Austin) Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case Lee
February 19 Maja Taskovic (UPenn) Exponential tails for the non-cutoff Boltzmann equation Kim
February 26 Ashish Kumar Pandey (UIUC) Instabilities in shallow water wave models Kim & Lee
March 5 Khai Nguyen (NCSU) Burgers Equation with Some Nonlocal Sources Tran
March 12 Hongwei Gao (UCLA) Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations Tran
March 19 Huy Nguyen (Princeton) Compressible fluids and active potentials Lee
March 26 Spring recess (Mar 24-Apr 1, 2018)
April 2 In-Jee Jeong (Princeton) Singularity formation for the 3D axisymmetric Euler equations Kim
April 9 Jeff Calder (Minnesota) Nonlinear PDE continuum limits in data science and machine learning Tran
April 21-22 (Saturday-Sunday) Midwest PDE seminar Angenent, Feldman, Kim, Tran.
April 25 (Wednesday) Hitoshi Ishii (Wasow lecture) Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory Tran.

Abstracts

Dan Knopf

Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons

Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.

Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

Sam Krupa

Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case

Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.


Maja Taskovic

Title: Exponential tails for the non-cutoff Boltzmann equation

Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.

We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.


Ashish Kumar Pandey

Title: Instabilities in shallow water wave models

Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.


Khai Nguyen

Title: Burgers Equation with Some Nonlocal Sources

Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.

Hongwei Gao

Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations

Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.

Huy Nguyen

Title : Compressible fluids and active potentials

Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.

In-Jee Jeong

Title: Singularity formation for the 3D axisymmetric Euler equations

Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.

Jeff Calder

Title: Nonlinear PDE continuum limits in data science and machine learning

Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.

Hitoshi Ishii

Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory

Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.