Lecture series I
Hongjie Dong (Brown)
Title: Global well-posedness for the one-phase Muskat problem with or without surface tension
Abstract: We study the free boundary problem for 2D and 3D incompressible flow in porous media, known as the one-phase Muskat problem. In the first part of my talk, I will discuss a global well-posedness result: in the absence of surface tension, when the initial interface is given by a Lipschitz graph, there exists a unique global Lipschitz strong solution. In the second part of my talk, I will present a more recent small-data global well-posedness result in both the whole-space and periodic settings when surface tension is included.
This talk is based on joint work with Francisco Gancedo (Universidad de Sevilla), Huy Q. Nguyen (University of Maryland), and Hyunwoo (Will) Kwon (Brown University).
Juhi Jang (USC)
Title : Long Time Dynamics of Compressible Fluids
Abstract : I will discuss recent progress on long time dynamics of compressible fluids with or without gravity. The focus will be on scaling symmetry, self-similar solutions, and stability analysis of solutions, describing expansion and implosion phenomena, to the compressible Euler and gravitational Euler-Poisson system. The talk is based on joint works with Yan Guo, Mahir Hadzic and Matthew Schrecker.
Nader Masmoudi (NYU)
Title : TBA
Abstract : TBA
Robert Strain (University of Pennsylvania)
Title : Future global stability of Maxwell-Jüttner equilibria for the massless Boltzmann equation on FLRW spacetimes
Abstract : The Einstein-massless Boltzmann system is the Einstein equations coupled to the massless Boltzmann equation as a matter model. The Einstein-Boltzmann system is a fundamental physical model in astrophysics and cosmology, for example in systems of galaxies, in supernova explosions, a model of the early universe, and as a model for hot gasses and plasmas. In particular, the Einstein-massless Boltzmann system describes the radiation dominated phase of the early universe after the Big Bang. The Robertson-Walker metric, or Friedmann–Lemaître–Robertson–Walker metric (FLRW), describes the universe evolving from a big bang singularity and expanding indefinitely at rate $t^{q}$ for different values of $q>0$, typically for $0 < q \leq 1$. In this talk I will present an overview of this non-linear PDE system. As a model, we consider the general relativistic massless Boltzmann equation on FLRW spacetimes with spatial topology of the torus $T^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^q$ with $0 < q \leq 1$. The massless general relativistic Boltzmann equation with the Robertson-Walker metric admits a family of non-stationary Maxwell-Jüttner equilibria of the form $J(t^{2q} p) = \exp(- (t^q)^2|p|)$. I will describe our recent work on the global-in-time existence and uniqueness of small perturbations of these equilibria in the case of hard ball interaction without any symmetry assumptions. For $0< q < 1/3$ we prove that the perturbation --- measured in a suitable $L^2_p$ based energy norm --- has the superpolynomial large time-decay rate of $\exp(-t^{1-3q})$, whereas for $1/3< q \leq 1$ we obtain the polynomial time-decay rate of $t^{-3q}$. In the borderline case $q=1/3$, we prove the time-decay rate of \(t^{-3q -c}\) with a uniform constant $c>0$ depending upon the linearized collision operator. We also consider the non-expanding case $q=0$ for which stability near equilibrium is shown with an exponentially decaying perturbation. Finally, for $1/3< q \leq 1$, we obtain a future global-in-time a priori estimate in the near vacuum regime on $T^3$. This appears to be the first global-in-time a priori estimate in the near vacuum regime for the Boltzmann equation on the torus $T^3$. (Previous works crucially rely on the dispersion of the transport operator in the whole space $R^3$.) This is a joint work with Martin Taylor and Renato Velozo Ruiz (both of Imperial College in London).
Renjun Duan (CUHK)
Title : The Boltzmann equation for kinetic shear flow
Abstract : The first part of the talk is to survey recent results on the kinetic shear flow governed by the nonlinear Boltzmann equation in both spatially homogeneous and inhomogeneous settings. In the second part, we focus on a specific problem on the 3D kinetic Couette flow in the diffusive limit.
Lecture series II
Yan Guo (Brown)
Title 1 : Gravitational Collapse for Gaseous Stars
Abstract : We review recent developments in the study of gravitational collapse of gaseous stars in both Newtonian and relativistic contexts.
Title 2 : Inflow and Viscous Inflows for Channel Flows
Abstract : We review recent developments in the study of inflow type of boundary conditions for viscous or inviscid fluids
Nader Masmoudi (NYU)
Title : TBA
Abstract : TBA
Zhenfu Wang (Peking)
Title : Quantitative Mean-Field Limits for Singular Interactions: Entropy, Modulated Energies, and Recent Results
Abstract : Mean-field limits and propagation of chaos for interacting particle systems with singular interaction forces have recently seen major advances, driven by new quantitative stability techniques beyond the classical Lipschitz regime. This talk surveys a work-centered line of developments based on relative entropy, modulated energy / modulated free energy, and duality methods, highlighting how they deliver explicit quantitative control for singular kernels. I will then present our recent progress on particle approximations of the Landau equation, non-exchangeable particle systems, and selected links to PDE numerical algorithms. The talk concludes with several open problems and future directions.
Junhee Ryu (KIAS, Junior talk)
Title: $L_p$-estimates for nonlocal equations with general L\'evy measures
Abstract: In this talk, we consider time-dependent nonlocal operators associated with general L\'evy measures of order $\sigma \in(0,2)$. We allow the class of L\'evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.
ms and future directions.
Ian Tice (CMU)
Title : Solitary and bore-like traveling wave solutions to the free boundary incompressible Navier-Stokes system
Abstract : One of the oldest endeavors in mathematical fluid mechanics is the construction of traveling wave solutions. However, until recently this focused entirely on inviscid solutions. In this talk we will detail recent developments in the corresponding theory of viscous traveling wave solutions. We will describe a general well-posedness theory for solitary solutions in Sobolev-type spaces as well as the construction on non-Sobolev bore wave solutions.
Youngheon Kim (UBC)
Title : TBA
Abstract : TBA