Difference between revisions of "PDE Geometric Analysis seminar"
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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: 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 | ||
+ | \[ | ||
+ | u_t+ {\left(u^2\over 2\right)_x}~=~{\bf K}*u\qquad\mathrm{and}\qquad u(0,\cdot)~=~u_0 | ||
+ | \] | ||
+ | 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 $\left({\bf K}=-{d\over dx} {1\over 2}e^{-{|x|}}\right)$ and Burgers-Hilbert equations $({\bf K}={1\over \pi x})$. Some open questions will be discussed. |
Revision as of 19:57, 13 February 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) | TBD | 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) | TBD | Tran |
March 19 | Huy Nguyen (Princeton) | TBD | Lee |
March 26 | Spring recess (Mar 24-Apr 1, 2018) | ||
April 2 | TBD | ||
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.
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 \[ u_t+ {\left(u^2\over 2\right)_x}~=~{\bf K}*u\qquad\mathrm{and}\qquad u(0,\cdot)~=~u_0 \] 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 $\left({\bf K}=-{d\over dx} {1\over 2}e^{-{|x|}}\right)$ and Burgers-Hilbert equations $({\bf K}={1\over \pi x})$. Some open questions will be discussed.