## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, March 2, 2010 - 15:05 , Location: Skiles 255 , Marius Paicu , Université Paris-Sud , Organizer: Zhiwu Lin
We consider the three dimensional Navier-Stokes equations with a large initial data and we prove the existence of a global smooth solution. The main feature of the initial data is that it varies slowly in the vertical direction and has a norm which blows up as the small parameter goes to zero. Using the language of geometrical optics, this type of initial data can be seen as the ill prepared" case. Using analytical-type estimates and the special structure of the nonlinear term of the equation we obtain the existence of a global smooth solution generated by this large initial data. This talk is based on a work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z. Zhang.
Series: PDE Seminar
Tuesday, February 16, 2010 - 15:05 , Location: Skiles 255 , Wilfrid Gangbo , Georgia Tech , Organizer: Zhiwu Lin
Let $\mathbb{H}$ be a Hilbert space and $h: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}$ be such that $h(x, \cdot)$ is uniformly convex and grows superlinearly at infinity, uniformy in $x$. Suppose $U: \mathbb{H} \rightarrow \mathbb{R}$ is strictly convex and grows superlinearly at infinity. We assume that both $H$ and $U$ are smooth. If $\mathbb{H}$ is of infinite dimension, the initial value problem $\dot x= -\nabla_p h(x, -\nabla U(x)), \; x(0)=\bar x$ is not known to admit a solution. We study a class of parabolic equations on $\mathbb{R}^d$ (and so of infinite dimensional nature), analogous to the previous initial value problem and establish existence of solutions. First, we extend De Giorgi's interpolation method to parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. These interpolation reveal to be powerful tool for proving convergence of a time discrete algorithm. (This talk is based on a joint work with A. Figalli and T. Yolcu).
Series: PDE Seminar
Tuesday, February 9, 2010 - 15:00 , Location: Skiles 255 , Ronghua Pan , Georgia Tech , , Organizer: Ronghua Pan
Darcy's law was observed in the motion of porous medium flows. This talk aims at the mathematical justification on Darcy's law as long time limit from compressible Euler equations with damping. In particularly, we shall showthat any physical solution with finite total mass shall converges in L^1 distance toward the Barenblatt's solution of the same mass for the Porous Medium Equation. The approach will explore the dissipation of the entropy inequality motivated by the second law of thermodynamics. This is a joint work with Feimin Huang and Zhen Wang.
Series: PDE Seminar
Tuesday, February 2, 2010 - 15:10 , Location: Skiles 255 , Zhiwu Lin , Georgia Tech , Organizer: Zhiwu Lin
Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^s (s<3/2) norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^s (s<3/2) norm. We also showed that in (vorticity) H^s (s>3/2) neighborhood of Couette flows, the only steady structures (including travelling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^s (s>3/2) space might be simpler. Similar results will also be discussed for the problem of nonlinear Landau damping in 1D electrostatic plasmas.
Series: PDE Seminar
Tuesday, January 26, 2010 - 15:00 , Location: Skiles 255 , Margaret Beck , Boston University , Organizer:
The large-time behavior of solutions to Burgers equation with small viscosity isdescribed using invariant manifolds. In particular, a geometric explanation is provided for aphenomenon known as metastability, which in the present context means that solutions spend avery long time near the family of solutions known as diffusive N-waves before finallyconverging to a stable self-similar diffusion wave. More precisely, it is shown that in termsof similarity, or scaling, variables in an algebraically weighted L^2 space, theself-similar diffusion waves correspond to a one-dimensional global center manifold ofstationary solutions. Through each of these fixed points there exists a one-dimensional,global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus,metastability corresponds to a fast transient in which solutions approach this metastable"manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally,convergence to the self-similar diffusion wave. This is joint work with C. Eugene Wayne.
Series: PDE Seminar
Tuesday, January 19, 2010 - 15:05 , Location: Skiles 255 , Michael I. Weinstein , Columbia University , Organizer: Zhiwu Lin
I will discuss the intermediate and long time dynamics of solutions of the nonlinear Schroedinger - Gross Pitaevskii equation, governing nonlinear dispersive waves in a spatially non-homogeneous background. In particular, we present results (with B. Ilan) on solitons with frequencies near a spectral band edge associated with periodic potential, and results (with Z. Gang) on large time energy distribution in systems with multiple bound states. Finally, we discuss how such results can inform strategies for control of soliton-like states in optical and quantum systems.
Series: PDE Seminar
Tuesday, December 1, 2009 - 15:01 , Location: Skiles 255 , Jose Arrieta , Universidad Complutense de Madrid; visiting faculty at GT , Organizer:
In this talk we will present several results concerning the behavior of the Laplace operator with Neumann boundary conditions in a thin domain where its boundary presents a  highly oscillatory behavior.  Using homogenization and domain perturbation techniques, we obtain the asymptotic limit as the thickness of the domain goes to zero even for the case where the oscillations are not necessarily periodic.  We will also indicate how this result can be applied to analyze the asymptotic dynamics of reaction diffusion equations in these domains.
Series: PDE Seminar
Thursday, November 19, 2009 - 15:00 , Location: Skiles 255 , Zhouping Xin , The Chinese University of Hong Kong , Organizer: Zhiwu Lin
One of the challenges in the study of transonic flows is the understanding of the flow behavior near the sonic state due to the severe degeneracy of the governing equations. In this talk, I will discuss the well-posedness theory of a degenerate free boundary problem for a quasilinear second elliptic equation arising from studying steady subsonic-sonic irrotational compressible flows in a convergent nozzle. The flow speed is sonic at the free boundary where the potential flow equation becomes degenerate. Both existence and uniqueness will be shown and optimal regularity will be obtained. Smooth transonic flows in deLaval nozzles will also be discussed. This is a joint work with Chunpeng Wang.
Series: PDE Seminar
Tuesday, November 17, 2009 - 15:05 , Location: Skiles 255 , Ning Jiang , Courant Institute, New York University , Organizer: Zhiwu Lin
In a bounded domain with smooth boundary (which can be considered as a smooth sub-manifold of R3), we consider the Boltzmann equation with general Maxwell boundary condition---linear combination of specular reflection and diffusive absorption. We analyze the kinetic (Knudsen layer) and fluid (viscous layer) coupled boundary layers in both acoustic and incompressible regimes, in which the boundary layers behave significantly different. The existence and damping properties of these kinetic-fluid layers depends on the relative size of accommodation number and Kundsen number, and the differential geometric property of the boundary (the second fundamental form.) As applications, first we justify the incompressible Navier-Stokes-Fourier limit of the Boltzmann equation with Dirichlet, Navier, and diffusive boundary conditions respectively, depending on the relative size of accommodation number and Kundsen number. Using the damping property of the boundary layer in acoustic regime, we proved the convergence is strong. The second application is that we derive and justified the higher order acoustic approximation of the Boltzmann equation. This is a joint work with Nader Masmoudi.
Series: PDE Seminar
Tuesday, November 10, 2009 - 15:05 , Location: Skiles 255 , Chunjing Xie , University of Michigan, Ann Arbor , Organizer: Zhiwu Lin
In this talk, we will discuss the global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. This is joint work with Bin Cheng.