PDE Geometric Analysis seminar: Difference between revisions
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|Xuwen Chen (Brown University)[http://www.math.brown.edu/~chenxuwen/] | |Xuwen Chen (Brown University)[http://www.math.brown.edu/~chenxuwen/] | ||
|[[#Xuwen Chen| | |[[#Xuwen Chen| The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution]] | ||
|C.Kim | |C.Kim | ||
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A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang. | A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang. | ||
If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes. | If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes. | ||
===Xuwen Chen=== | |||
''The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution'' | |||
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant. | |||
===Kyudong Choi=== | ===Kyudong Choi=== | ||
Revision as of 13:32, 28 September 2014
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 Spring 2015
Seminar Schedule Fall 2014
| date | speaker | title | host(s) |
|---|---|---|---|
| September 15 | Greg Kuperberg (UC-Davis) | Cartan-Hadamard and the Little Prince | Viaclovsky |
| September 22 (joint with Analysis Seminar) | Steve Hofmann (U. of Missouri) | Quantitative Rectifiability and Elliptic Equations | Seeger |
| Oct 6th, | Xiangwen Zhang (Columbia University) | Alexandrov's Uniqueness Theorem for Convex Surfaces | B.Wang |
| October 13 | Xuwen Chen (Brown University)[1] | The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution | C.Kim |
| October 20 | Kyudong Choi (UW-Madison) |
Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system |
C.Kim |
| October 27 | Chanwoo Kim (UW-Madison) |
BV-Regularity of the Boltzmann Equation in Non-Convex Domains |
Local |
| November 3 | Myoungjean Bae (POSTECH) | TBA | M.Feldman |
| November 10 | Philip Isett (MIT) | TBA | C.Kim |
| November 17 | Lei Wu | Geometric Correction for Diffusive Expansion in Neutron Transport Equation | C.Kim |
| November 24 | Hongnian Huang (Univeristy of New Mexico) | TBA | B.Wang |
Fall Abstracts
Greg Kuperberg
Cartan-Hadamard and the Little Prince.
Steve Hofmann
Quantitative Rectifiability and Elliptic Equations
A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis. In this talk, we discuss recent developments in an ongoing program to find scale-invariant, higher dimensional versions of the F. and M. Riesz Theorem, as well as converses. In particular, we discuss substitute results that continue to hold in the absence of any connectivity hypothesis.
Xiangwen Zhang
Alexandrov's Uniqueness Theorem for Convex Surfaces
A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang. If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.
Xuwen Chen
The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.
Kyudong Choi
Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system
In connection with the recent proposal for possible singularity formation at the boundary for solutions of the 3d axi-symmetric incompressible Euler's equations / the 2D Boussinesq system (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.
Myoungjean Bae
TBA
Lei Wu
Geometric Correction for Diffusive Expansion in Neutron Transport Equation
We revisit the diffusive limit of a steady neutron transport equation in a 2-D unit disk with one-speed velocity. The traditional method is Hilbert expansions and boundary layer analysis. We will carefully study the classical theory of the construction of boundary layers, and discuss the necessity and specific method to add the geometric correction.