Seminars and Colloquia by Series

Stability of wave patterns to the bi-polar Vlasov-Poisson-Boltzmann system

Series
PDE Seminar
Time
Tuesday, August 18, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yi Wang – AMSS, Chinese Academy of Sciences
We investigate the nonlinear stability of elementary wave patterns (such as shock, rarefaction wave and contact discontinuity, etc) for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the elementary wave patterns to the system. Then, the time-asymptotic stability of the planar rarefaction wave, viscous shock waves and viscous contact wave (viscous version of contact discontinuity) are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. These results imply that these basic wave patterns are still stable in the transportation of charged particles under the binary collision, mutual interaction, and the effect of the electrostatic potential force. The talk is based on the joint works with Hailiang Li (CNU, China), Tong Yang (CityU, Hong Kong) and Mingying Zhong (GXU, China).

Global Classical Solution to the Two-dimensional Compressible Navier-Stokes Equations with Density-dependent Viscosity

Series
PDE Seminar
Time
Tuesday, April 28, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Quansen Jiu – Capital Normal University, China
In this talk, we will present some results on global classical solution to the two-dimensional compressible Navier-Stokes equations with density-dependent of viscosity, which is the shear viscosity is a positive constant and the bulk viscosity is of the type $\r^\b$ with $\b>\frac43$. This model was first studied by Kazhikhov and Vaigant who proved the global well-posedness of the classical solution in periodic case with $\b> 3$ and the initial data is away from vacuum. Here we consider the Cauchy problem and the initial data may be large and vacuum is permmited. Weighted stimates are applied to prove the main results.

Mean field limits for many-agents models

Series
PDE Seminar
Time
Tuesday, April 14, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre-Emmanuel Jabin – University of Maryland, College Park
We consider some recent models from stochastic or optimal control involving a very large number of agents. The goal is to derive mean field limits when the number of agents increases to infinity. This presents some new unique difficulties; the corresponding master equation is a non linear Hamilton-Jacobi equation for instance instead of the linear transport equations that are more typical in the usual mean field limits. We can nevertheless pass to the limit by looking at the problem from an optimization point of view and by using an appropriate kinetic formulation. This is a joint work with S. Mischler, E. Sere, D. Talay.

Compactness on Multidimensional Steady Euler Equations

Series
PDE Seminar
Time
Thursday, April 9, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skile 005
Speaker
Tian-Yi Wang – The Chinese University of Hong Kong
This is a special PDE seminar in Skiles 005. In this talk, we will introduce the compactness framework for approximate solutions to sonic-subsonic flows governed by the irrotational steady compressible Euler equations in arbitrary dimension. After that, similar results will be presented for the isentropic case. As a direct application, we establish several existence theorems for multidimensional sonic-subsonic Euler flows. Also, we will show the recent progress on the incompressible limits.

The Euler-Maxwell system in 2D

Series
PDE Seminar
Time
Tuesday, April 7, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benoit Pausader – Princeton University
The Euler-Maxwell system describes the interaction between a compressible fluid of electrons over a background of fixed ions and the self-consistent electromagnetic field created by the motion.We show that small irrotational perturbations of a constant equilibrium lead to solutions which remain globally smooth and return to equilibrium. This is in sharp contrast with the case of neutral fluids where shock creation happens even for very nice initial data.Mathematically, this is a quasilinear dispersive system and we show a small data-global solution result. The main challenge comes from the low dimension which leads to slow decay and from the fact that the nonlinearity has some badly resonant interactions which force a correction to the linear decay. This is joint work with Yu Deng and Alex Ionescu.

Stability of periodic waves for 1D NLS

Series
PDE Seminar
Time
Tuesday, March 31, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stephen Gustafson – UBC
Cubic focusing and defocusing Nonlinear Schroedinger Equations admit spatially (and temporally) periodic standing wave solutions given explicitly by elliptic functions. A natural question to ask is: are they stable in some sense (spectrally/linearly, orbitally, asymptotically,...), against some class of perturbations (same-period, multiple-period, general...)? Recent efforts have slightly enlarged our understanding of such issues. I'll give a short survey, and describe an elementary proof of the linear stability of some of these waves. Partly joint work in progress with S. Le Coz and T.-P. Tsai.

Global well-posedness for some cubic dispersive equations

Series
PDE Seminar
Time
Tuesday, March 24, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin Dodson – Johns Hopkins University
In this talk we examine the cubic nonlinear wave and Schrodinger equations. In three dimensions, each of these equations is H^{1/2} critical. It has been showed that such equations are well-posed and scattering when the H^{1/2} norm is bounded, however, there is no known quantity that controls the H^{1/2} norm. In this talk we use the I-method to prove global well posedness for data in H^{s}, s > 1/2.

Onsager's Conjecture

Series
PDE Seminar
Time
Tuesday, March 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tristan Buckmaster – Courant Institute, NYU
In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Hölder Continuous Euler Flows

Series
PDE Seminar
Time
Tuesday, March 3, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Phillip Isett – MIT
Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations, which are designed to reduce the error term in solving the equation. In this talk, I will discuss an improved convex integration framework, which yields solutions with Holder regularity 1/5- as well as other recent results.

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