Difference between revisions of "PDE Geometric Analysis seminar"
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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. | 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. | ||
+ | |||
+ | ===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. |
Revision as of 20:56, 12 March 2018
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 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) | TBD | 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) | TBD | Tran |
April 21-22 (Saturday-Sunday) | Midwest PDE seminar | Angenent, Feldman, Kim, Tran. | |
April 25 (Wednesday) | Hitoshi Ishii (Wasow lecture) | TBD | 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.
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.