[ Mathematics | University of North Carolina, Chapel Hill

UNC PDE Graduate Mini-Schools 2013-2014

Where:
Phillips Hall
University of North Carolina, Chapel Hill
Department of Mathematics

When: Wednesday, Thursday, and Friday from 4-5 PM
on weeks to be determined as of yet.

For more info: uncpdeminischools "at" gmail "dot" com



  • For more information and regular announcements, check us out at:


    Speakers:


  • Maciej Zworski lecturing April 3rd and 4th, 2014 on dynamical zeta functions and an introduction to microlocal analysis.

    Current Schedule -
    Thursday, April 3rd:
    9:00-10:00 AM - Zworski Lecture I in Peabody 008.
    Talk Slides.
    10:00-11:00 AM - Coffee and Refreshments in 331 Phillips
    11:00 AM - 12:00 PM Zworski Lecture II in Wilson Library Pleasant's Room
    Friday, April 4th:
    1:00-2:00 PM - Zworski Lecture III in 332 Phillips
    2:00-3:00 PM - Grad Lecture I - Jeffrey Galkowski
    Talk Slides.
    3:00-4:00 PM - Coffee and Refreshments in 331 Phillips
    4:00-5:00 PM - Zworski Lecture IV in 381 Phillips



  • Chris Sogge lecturing in November 21-22, 2013 on a topic related to the spectrum of the Laplacian on manifolds.

    Current Schedule -
    Thursday, November 21st:
    3:00 PM - Coffee/Snacks in 330 Philips
    3:30-4:30 PM - Lecture 1 from Professor Sogge in 381 Phillips
    Talk I: Focal points and sup-norms of eigenfunctions.
    abstract: If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x_0 in M for the geodesic flow at which the associated Perron-Frobenius operator U: L^2 (S_{x_0}^* M ) -> L^2 (S_{x_0}^* M ) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase.
    Talk Slides.
    4:35-5:20 PM - 1st 30 minute talk by Hongtan Sun in 381 Phillips -
    Title: Strichartz type estimates for wave equations on hyperbolic trapped domain.
    Abstract: We establish Strichartz type estimate for the wave equation on Riemannian manifolds $(\Omega,g)$, for the case that $\Omega$ is the exterior of a smooth, normally hyperbolic trapped obstacle in $n$ dimensional Euclidean space, and $n$ is a positive odd integer. As for the normally hyperbolic trapped obstacles, we will get some loss of derivatives for data in the local energy decay estimate. Hence the Strichartz estimate has a derivative loss, and we need two different $L^p_tL^q_x$ norm of the forcing term to bound the solution of inhomogeneous equation.
    Talk Slides.

    Friday, November 22nd:
    3:00 PM - Coffee/Snacks in 330 Philips
    3:30-4:30 PM - Lecture 2 from Professor Sogge in 328 Phillips
    Talk II: Focal points and sup-norms of eigenfunctions.
    abstract: If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x_0 in M for the geodesic flow at which the associated Perron-Frobenius operator U: L^2 (S_{x_0}^* M ) -> L^2 (S_{x_0}^* M ) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase.
    Talk Slides.
    4:35-5:20 PM - 2nd 30 minute talk by Min Xue in 328 Phillips -
    Title and Abstract.
    Talk Slides.



  • Kevin Zumbrun lecturing October 2-4, 2013 on "Modulation and stability of periodic traveling waves, and the link through Whitham's modulation equations to hyperbolic conservation laws and shock fronts."

    Current Schedule -
    Wednesday, October 2nd:
    3:30 PM - Coffee/Snacks in 330 Philips
    4-5 PM - Lecture 1 from Professor Zumbrun in 381 Phillips
    Talk I: Periodic patterns with conservation laws: the Whitham equations and rigorous long-time asymptotics.
    abstract: Periodic patterns and traveling waves arise quite generally in optics, biology, chemistry, and many other applications. A great success story over the past couple decades for the dynamical systems approach to PDE has been the rigorous treatment of modulation of periodic patterns in reaction diffusion systems. However, the techniques used were designed for modulations with a single degree of freedom. For systems possessing one or more conservation laws, hence one or more additional degrees of freedom, these methods do not apply. Here, motivated by applications to thin film flow, we present an approach applying also to this more general situation, emphasizing connections to the hyperbolic-parabolic ``Whitham equations'' formally governing slow modulations, and through them an analogy to viscous shock wave theory.
    Talk Slides.
    reading list:
    arXiv:1105.5040 - Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability by Mathew Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin Zumbrun;
    arXiv:1105.5044 - Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation by Mathew Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin Zumbrun;
    arXiv:1211.2156 - Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations by Mathew A. Johnson, Pascal Noble, L.Miguel Rodrigues, Kevin Zumbrun;
    arXiv:1307.6957 - Nonlinear stability of source defects in the complex Ginzburg-Landau equation by Margaret Beck, Toan T. Nguyen, Bjorn Sandstede, Kevin Zumbrun;
    PDF File - Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model by U. Frisch, Z.S. She, O. Thual.

    5:00-5:30 PM - 1st 30 minute talk by Blake Barker in 381 Phillips -
    "Numerical stability analysis for thin film flow: toward rigorous verification"
    Talk Slides.

    Thursday, October 3rd:
    3:30 PM - Coffee/Snacks in 330 Philips
    4-5 PM - Lecture 2 from Professor Zumbrun in 332 Phillips
    Talk II: Spectral stability analysis I: numerical methods and phenomena in thin film flow.
    abstract: Having reduced the study of behavior to the study of Floquet spectra, we discuss in various numerical methods for approximating the spectra of general, possibly large-amplitude waves, and give an overview of numerically observed phenomena in thin film flow. These include``viscoelastic behavior'' in cellular Kuramoto-Sivashinsky behavior, ``dispersion-enhanced stability,'' and the ``homoclinic paradox'' in inclined thin-film flow, the latter concerning the puzzling phenomenon that asymptotic behavior appears to consist of solitary waves, despite that solitary waves are readily seen to be exponentially unstable.
    Talk Slides.
    reading list:
    arXiv:1203.3795 - Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation by Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin Zumbrun;
    arXiv:1011.5695 - $2$-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardner by Kevin Zumbrun;
    arXiv:1008.4729 - Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law by Blake Barker, Mathew A. Johnson, Pascal Noble, L.Miguel Rodrigues, Kevin Zumbrun;
    arXiv:1007.5262 - Metastability of solitary roll wave solutions of the St. Venant equations with viscosity by Blake Barker, Mathew A. Johnson, L. Miguel Rodrigues, Kevin Zumbrun.

    5:00-5:30 PM - 2nd 30 minute talk by Soyeun Jung in 332 Phillips -
    "Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves"


    Friday, October 4th:
    2:30-3:30 PM - Lecture 3 from Professor Zumbrun in 224 Phillips
    Talk III: Spectral stability analysis II: rigorous existence and stability in the weakly unstable regime.
    abstract: Finally, we discuss rigorous existence and spectral stability theory in the ``weakly unstable'' regime at the threshold of instability of constant (e.g., laminar) solutions, comparing the classical Turing scenario involving Ginzburg Landau equations and the familiar Eckhaus stability boundaries to the much more complicated scenario arising in inclined thin film flow, involving a singularly perturbed KdV equation and physical wavelengths approaching infinity.
    Talk Slides.
    reading list:
    arXiv:0006002 - Pattern formation with a conservation law by P. C. Matthews, S. M. Cox;
    arXiv:1202.6402 - Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit by Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin Zumbrun;
    PDF File - Stability of periodic waves govemed by the modified Kawahara equation by D.E. Bar, A.A. Nepomnyashchy.

    3:30-4:00 PM - 3rd 30 minute talk by Fang Yu in 224 Phillips -
    "Stability of supersonic contact discontinuities for three dimensional compressible steady Euler flows"

    4:00-4:30 PM - Coffee/Snacks in 330 Philips

    4:30-5:30 PM - Tom Beale speaks in the Applied Math Seminar in 332 Phillips -
    "Numerical methods for nearly singular integrals and moving interfaces in fluids"


    Links

  • Practical Information regarding getting to UNC.

  • Some graduate funding support will be available for young researchers within the United States to attend each lecture series. To inquire about funding, please e-mail us at our Mini-School E-mail Address.

  • Funding generously provided by the NSF as a no cost extension of our conference grant from July, 2012.