PDE Geometric Analysis seminar
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm  4:30pm, unless indicated otherwise.
Contents
Previous PDE/GA seminars
Tentative schedule for Fall 2018
PDE GA Seminar Schedule Fall 2018Spring 2019
date  speaker  title  host(s)


September 10,  Hiroyoshi Mitake (University of Tokyo)  TBA  Tran 
September 17,  Changyou Wang (Purdue)  TBA  Tran 
September 24/26,  Gunther Uhlmann (UWash)  TBA  Li 
October 1,  Matthew Schrecker (UW)  TBA  Kim and Tran 
October 8,  Anna Mazzucato (PSU)  TBA  Li and Kim 
October 15,  Lei Wu (Lehigh)  TBA  Kim 
October 22,  Annalaura Stingo (UCD)  TBA  Mihaela Ifrim 
October 29,  Jessica Lin (McGill University)  TBA  Tran 
November 5,  Albert Ai (University of Berkeley)  TBA  Mihaela Ifrim 
March 4 2019  Vladimir Sverak (Minnesota)  TBA(Wasow lecture)  Kim}
AbstractsDan KnopfTitle: NonKähler Ricci flow singularities that converge to KählerRicci solitons Abstract: We describe Riemannian (nonKähler) Ricci flow solutions that develop finitetime TypeI 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 nonKähler solutions of Ricci flow that become asymptotically Kähler, in suitable spacetime 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 SeegerTitle: 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 seminorm 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 KrupaTitle: 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 HamiltonJacobi 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 HamiltonJacobi. 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 TaskovicTitle: Exponential tails for the noncutoff 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 nonintegrable 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 nonintegrable 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 noncutoff 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 MittagLeffler 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 PandeyTitle: Instabilities in shallow water wave models Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop nontrivial 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 NguyenTitle: 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 BurgersPoisson and BurgersHilbert equations. Some open questions will be discussed. Hongwei GaoTitle: Stochastic homogenization of certain nonconvex HamiltonJacobi equations Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex HamiltonJacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (levelset) convex Hamiltonians and a sequence of (levelset) concave Hamiltonians through the minmax 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 NguyenTitle : 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 NavierStokes 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 solutiondependent potential: the active potential. The method of proof uses the BreschDesjardins entropy and the analysis of the evolution of the active potential. InJee JeongTitle: Singularity formation for the 3D axisymmetric Euler equations Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilonz)^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 wellposedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local wellposedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of HouLuo, KiselevSverak, 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 CalderTitle: 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 graphbased semisupervised 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 HamiltonJacobi 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 graphbased learning problems have PDE continuum limits corresponding to weighted pLaplace 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 IshiiTitle: Asymptotic problems for HamiltonJacobi equations and weak KAM theory Abstract: In the lecture, I discuss two asymptotic problems related to HamiltonJacobi equations. One concerns the longtime behavior of solutions of time evolutionary HamiltonJacobi equations and the other is the socalled vanishing discount problem for stationary HamiltonJacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of HamiltonJacobi 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. 