Georgia Institute of Technology

Margaret Beck

Tue, 01/26/2010 - 3:00pm, Skiles 255

TBA, Boston University        Organizer: Maria Westdickenberg

Thin domains with a highly oscillating boundary

Tue, 12/01/2009 - 3:01pm, Skiles 255

Jose Arrieta, Universidad Complutense de Madrid; visiting faculty at GT        Organizer: Maria Westdickenberg

 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.

A Nonlinear Degenerate Free-Boundary Problem and Subsonic-sonic flows

Thu, 11/19/2009 - 3:00pm, 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.

Kinetic-Fluid Boundary Layers and Applications to Hydrodynamic Limits of Boltzmann Equation (canceled)

Tue, 11/17/2009 - 3:05pm, 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.

Classical Solutions of Two Dimensional Inviscid Rotating Shallow Water System

Tue, 11/10/2009 - 3:05pm, 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.

The Linearized System for Isometric Embeddings and Its Characteristic Variety

Tue, 11/03/2009 - 3:05pm, Skiles 255

Qing Han, University of Notre Dame        Organizer: Zhiwu Lin

We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Notes on the blow-up problem of the Euler equations

Tue, 10/27/2009 - 3:05pm, Skiles 255

Dongho Chae, Sungkyunkwan University, Korea and Universty of Chicago        Organizer: Zhiwu Lin

We first discuss blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. In the second part of the talk we discuss some observations on the Euler equations with symmetries, which shows that the point-wise behavior of the pressure along the flows is closely related to the blow-up of of solutions.

Boundary Value Problems for Nonlinear Dispersive Wave Equations

Tue, 10/20/2009 - 3:05pm, Skiles 255

Hongqiu Chen, University of Memphis        Organizer: Zhiwu Lin

Under the classical small-amplitude, long wave-length assumptions in which the Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects, the KdV-equation u_t+ u_x + uu_x + u_xxx = 0 (1) and the regularized long wave equation, or BBM-equation u_t + u_x + uu_x-u_xxt = 0 (2) are formal reductions of the full, two-dimensional Euler equations for free surface flow. This talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave motion is specified at both ends of a finite stretch of length L of the media of propagation. After ascertaining natural boundary specifications that constitute well posed problems, it is shown that the solution of the two-point boundary value problem, posed on the interval [0;L], say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the quarter plane problem for both the KdV-equation and the BBM-equation.

Boundary layer associated with the Darcy-Brinkman-Boussinesq system

Tue, 10/13/2009 - 3:05pm, Skiles 255

Xiaoming Wang, Florida State University        Organizer: Zhiwu Lin

We study the asymptotic behavior of the infinite Darcy-Prandtl number Darcy-Brinkman-Boussinesq model for convection in porous media at small Brinkman-Darcy number. This is a singular limit involving a boundary layer with thickness proportional to the square root of the Brinkman-Darcynumber . This is a joint work with Jim Kelliher and Roger Temam.

The Vlasov-Poisson System with Steady Spatial Asymptotics

Tue, 09/29/2009 - 3:05pm, Skiles 255

Stephen Pankavich, University of Texas, Arlington        Organizer: Zhiwu Lin

We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.

Comparison principle for unbounded viscosity solutions of elliptic PDEs with superlinear terms in $Du$

Tue, 09/22/2009 - 3:05pm, Skiles 255

Shigeaki Koike, Saitama University, Japan        Organizer: Zhiwu Lin

We discuss comparison principle for viscosity solutions of fully nonlinear elliptic PDEs in $\R^n$ which may have superlinear growth in $Du$ with variable coefficients. As an example, we keep the following PDE in mind:$$-\tr (A(x)D^2u)+\langle B(x)Du,Du\rangle +\l u=f(x)\quad \mbox{in }\R^n,$$where $A:\R^n\to S^n$ is nonnegative, $B:\R^n\to S^n$ positive, and $\l >0$. Here $S^n$ is the set of $n\ti n$ symmetric matrices. The comparison principle for viscosity solutions has been one of main issues in viscosity solution theory. However, we notice that we do not know if the comparison principle holds unless $B$ is a constant matrix. Moreover, it is not clear which kind of assumptions for viscosity solutions at $\infty$ is suitable. There seem two choices: (1) one sided boundedness ($i.e.$ bounded from below), (2) growth condition.In this talk, assuming (2), we obtain the comparison principle for viscosity solutions. This is a work in progress jointly with O. Ley.

Convergence properties of solutions to several classes of PDEs

Tue, 09/15/2009 - 3:05pm, Skiles 255

Zhang, Lei, University of Florida, email        Organizer: Zhiwu Lin

Many problems in Geometry, Physics and Biology are described by nonlinear partial differential equations of second order or four order. In this talk I shall mainly address the blow-up phenomenon in a class of fourth order equations from conformal geometry and some Liouville systems from Physics and Ecology. There are some challenging open problems related to these equations and I will report the recent progress on these problems in my joint works with Gilbert Weinstein and Chang-shou Lin.

On asymptotics, structure and stability for multicomponent reactive flows

Tue, 09/08/2009 - 3:05pm, Skiles 255

Konstantina Trivisa, University of Maryland, College Park        Organizer: Michael Westdickenberg

Multicomponent reactive flows arise in many practical applicationssuch as combustion, atmospheric modelling, astrophysics, chemicalreactions, mathematical biology etc. The objective of this work isto develop a rigorous mathematical theory based on the principles ofcontinuum mechanics. Results on existence, stability, asymptotics as wellas singular limits will be discussed.

Global Existence of a Free Boundary Problem with Non--Standard Sources

Tue, 09/01/2009 - 3:00pm, Skiles 255

Lincoln Chayes, UCLA        Organizer: Zhiwu Lin

This seminar concerns the analysis of a PDE, invented by J.M. Lasry and P.L. Lions whose applications need not concern us. Notwithstanding, the focus of the application is the behavior of a free boundary in a diffusion equation which has dynamically evolving, non--standard sources.  Global existence and uniqueness are established for this system.  The work to be discussed represents a collaborative effort with Maria del Mar Gonzalez, Maria Pia Gualdani and Inwon Kim.

Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow

Tue, 08/25/2009 - 3:05pm, Skiles 255

David Hoff, Indiana University, Bloomington        Organizer: Michael Westdickenberg

We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories. I'm going to integrate into the talk something like a "pretalk" in an attempt to motivate the more technical material and to make things accessible to a general analysis audience.