## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, August 25, 2015 - 15:05 , Location: Skiles 006 , Shihui Zhu , Department of Mathematics, Sichuan Normal University , Organizer: Wilfrid Gangbo
In this talk, we consider the dynamical properties of solutions to the fractional nonlinear Schrodinger equation (FNLS, for short) arising from pseudorelativistic Boson stars. First, by establishing the profile decomposition of bounded sequences in H^s, we find the best constant of a Gagliardo-Nirenberg type inequality. Then, we obtain the stability and instability of standing waves for (FNLS) by the profile decomposition. Finally, we investigate the dynamical properties of blow-up solutions for (FNLS), including sharp threshold mass, concentration and limiting profile. (Joint joint with Jian Zhang)
Series: PDE Seminar
Tuesday, August 18, 2015 - 15:05 , Location: Skiles 006 , Yi Wang , AMSS, Chinese Academy of Sciences , Organizer: Wilfrid Gangbo
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).
Series: PDE Seminar
Tuesday, April 28, 2015 - 15:05 , Location: Skiles 006 , Quansen Jiu , Capital Normal University, China , Organizer:
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.
Series: PDE Seminar
Thursday, April 23, 2015 - 15:05 , Location: Skiles 005 , El Houcein Elabdelaoui , Universite de Rouen, France , Organizer:
Series: PDE Seminar
Tuesday, April 14, 2015 - 15:05 , Location: Skiles 006 , Pierre-Emmanuel Jabin , University of Maryland, College Park , Organizer:
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.
Series: PDE Seminar
Thursday, April 9, 2015 - 15:05 , Location: Skile 005 , Tian-Yi Wang , The Chinese University of Hong Kong , Organizer:
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.
Series: PDE Seminar
Tuesday, April 7, 2015 - 15:05 , Location: Skiles 006 , Benoit Pausader , Princeton University , Organizer:
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.
Series: PDE Seminar
Tuesday, March 31, 2015 - 15:05 , Location: Skiles 006 , Stephen Gustafson , UBC , Organizer:
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.
Series: PDE Seminar
Tuesday, March 24, 2015 - 15:05 , Location: Skiles 006 , Benjamin Dodson , Johns Hopkins University , Organizer:
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.
Series: PDE Seminar
Tuesday, March 10, 2015 - 15:05 , Location: Skiles 006 , Tristan Buckmaster , Courant Institute, NYU , Organizer:
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.