Seminars and Colloquia by Series

Approximation of p-ground states

Series
PDE Seminar
Time
Tuesday, October 6, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ryan HyndUniversity of Pennsylvania
The smallest eigenvalue of a symmetric matrix A can be expressed through Rayleigh's formula. Moreover, if the smallest eigenvalue is simple, it can be approximated by using the inverse iteration method or by studying the large time behavior of solutions of the ODE x'(t)=-Ax(t). We discuss surprising analogs of these facts for a nonlinear PDE eigenvalue problem involving the p-Laplacian.

Instability index, exponential trichotomy, and invariant manifolds for Hamiltonian PDEs: Part II

Series
PDE Seminar
Time
Tuesday, September 8, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chongchun ZengSchool of Mathematics, Georgia Tech
Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that R(L) is closed, L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded and with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^{tJL}. More specifically, we obtain some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^{tJL}. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases (particularly ground states), we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This work will be discussed in two talks. In the first talk, we will motivate the problem by several Hamiltonian PDEs, describe the main results, and demonstrate how they are applied. In the second talk, some ideas of the proof will be given.

Instability index, exponential trichotomy, and invariant manifolds for Hamiltonian PDEs: Part I

Series
PDE Seminar
Time
Tuesday, September 1, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zhiwu LinSchool of Mathematics, Georgia Tech
Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that R(L) is closed, L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded and with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^{tJL}. More specifically, we obtain some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^{tJL}. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases (particularly ground states), we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This work will be discussed in two talks. In the first talk, we will motivate the problem by several Hamiltonian PDEs, describe the main results, and demonstrate how they are applied. In the second talk, some ideas of the proof will be given.

Dynamics for the Fractional Nonlinear Schrodinger Equation

Series
PDE Seminar
Time
Tuesday, August 25, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shihui ZhuDepartment of Mathematics, Sichuan Normal University
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)

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 WangAMSS, 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 JiuCapital 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 JabinUniversity 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 WangThe 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 PausaderPrinceton 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.

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