PDEs Seminar (fall 2009)

Organizers: Zhiwu Lin

Meets Tuesdays at 3:05 pm in Skiles 255 unless otherwise indicated

08/25/09

Speaker: David Hoff, Indiana University, Bloomington

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

Abstract: 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.

09/01/09

Speaker: Lincoln Chayes, UCLA

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

Abstract: 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.

09/08/09

Speaker: Konstantina Trivisa, University of Maryland, College Park

Title: On asymptotics, structure and stability for multicomponent reactive flows

Abstract: 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.

09/15/09

Speaker: Zhang, Lei, University of Florida

Title: Convergence properties of solutions to several classes of PDEs

Abstract: 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.

09/22/09

Speaker: Shigeaki Koike, Saitama University, Japan

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

Abstract: 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.

09/29/09

Speaker: Stephen Pankavich, University of Texas at Arlington

Title: TBA

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10/06/09 No seminar due to fall recess.

10/13/09

Speaker: Xiaoming Wang, Florida State University

Title: Boundary layer associated with the Darcy-Brinkman-Boussinesq system

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

10/20/09

Speaker: Hongqiu Chen, University of Memphis

Title: TBA

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10/27/09

Speaker: Dongho Chae, Sungkyunkwan University, Korea

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11/03/09

Speaker: Qing Han, University of Notre Dame

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11/10/09

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11/17/09

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11/24/09

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12/01/09

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12/08/09 (finals week, possibly no talk)

Please contact me to volunteer to talk or recommend speakers.

zlin@math.gatech.edu