Department of
Mathematics
University of North Carolina
Wednesdays,
4:00-4:50pm, Phillips 381
| Wed.,
9/23/09 |
Nathan
Pennington, UNC |
TITLE:
Solutions to LANS equations in integrable in time spaces ABSTRACT |
| Mon.,
9/28/09 (GMA Visions) |
Parul
Laul, UNC |
TITLE: Local Energy Estimates of the Wave Equation |
| Wed.,
9/30/09 |
Nathan Pennington, UNC | TITLE:
Solutions to LANS equations in integrable in time spaces (Continued) ABSTRACT |
| Wed.,
10/7/09 |
Michael
Taylor, UNC |
TITLE:
Flat 2D tori with sparse spectra ABSTRACT: Flat 2D tori have as the spectra of their Laplace operators the negatives of square-lengths of elements of lattices in 2D Euclidean space. One example is the 2D integer lattice. In such a case, all the square length are integers, but most positive integers are not square lengths. We have "sparse spectrum." The talk will deal with the construction of other flat 2D tori with sparse spectra. There will be a little Fourier analysis, a little algebra, and a computer program. Some time before the talk, Taylor's web page will have a pdf file on this topic, which can be downloaded. |
| Wed.,
10/14/09 |
Michael
Taylor, UNC |
TITLE:
Linear and nonlinear waves on hyperbolic space ABSTRACT: We give a sequence of two talks. We begin by producing formulas for solutions to the linear wave equation on 3D hyperbolic space. These formulas are then used together with some harmonic analysis to establish "dispersive estimates" and "Strichartz estimates" on solutions to wave equations. We then show how this leads to results on global existence for a class of nonlinear wave equations. This is a report on joint work with Jason Metcalfe. |
| Wed.,
10/21/09 |
Michael
Taylor, UNC |
TITLE:
Linear and nonlinear waves on hyperbolic space ABSTRACT: We give a sequence of two talks. We begin by producing formulas for solutions to the linear wave equation on 3D hyperbolic space. These formulas are then used together with some harmonic analysis to establish "dispersive estimates" and "Strichartz estimates" on solutions to wave equations. We then show how this leads to results on global existence for a class of nonlinear wave equations. This is a report on joint work with Jason Metcalfe. |
| Wed.,
11/4/09 |
Joseph
Cima, UNC |
TITLE:
Introduction to H^1 ABSTRACT: There will be a brief introduction to the analytic theory on the disc.Then we will define the "real" Fefferman and Stein Hardy space(s) on R^n. Much of this material is taken from Stein's book, Harmonic Analysis (Princeton Press- which I have checked out of the library!), Chapters 3,4. I will give a few of the shorter proofs and just quote other pertinent results. Mainly how can one tell if an L^1 function can be in the Hardy 1 space, definitions of atoms, definition of BMO ,the topological dual etc. This talk is suitable for graduate students. A second lecture will be given two weeks later. |
| Wed.,
11/11/09 |
Ian
Zwiers, University of Toronto |
TITLE:
Blowup of the Cubic Focusing Nonlinear Schrodinger Equation on a Ring ABSTRACT: We prove there exist solutions to the
three-dimensional
cubic focusing nonlinear Schrodinger equation that blowup on a circle,
in the sense of L2 concentration on a ring, bounded H1 norm outside any
surrounding toroid, and growth of the global H1 norm with the log-log
rate.
In this talk we will described the bootstrapping scheme of
the
proof. Namely that, for sufficient decay outside a surrounding toroid,
the dynamic near the circle of concentration is essentially
two-dimensional. Then, for appropriate data, the robust two-dimensional
log-log blowup regime occurs. Merle & Raphael's precise description
of this singular behaviour allows us to prove an unusual persistance of
regularity away from the circle of concentration, which re-establishes
the sufficient decay outside the surrounding toroid.
|
| Wed.,
11/18/09 |
Joseph
Cima, UNC |
TITLE:
Introduction to H^1 ABSTRACT: A proof of a homogeneous "div-curl" lemma and again how can one get functions in the Hardy 1 space. (Paper of Meyer, Semmes, Coiffman, et.al.) See a chapter in M.Taylor's book on tools used in PDE. If time allows I will give a proof of a result of Jones , Zinsmeister, Iwaniec, Bonami and an interesting bounded operator mapping BMO into a space called the "Hardy-Orliz" space. |
| Wed.,
12/2/09 |
Jeff
Jauregui, Duke University |
|
| Wed., 1/28/09 |
Benjamin Dodson, UNC |
TITLE: Global existence for some
nonlinear Schrodinger equations ABSTRACT |
| Wed., 2/4/09 |
Benjamin Dodson, UNC |
TITLE: Global existence for some
nonlinear Schrodinger equations (Continued) |
| Fri.
1/30/09 through Sun. 2/1/09 |
Carolina
Meeting on Harmonic Analysis and PDE |
http://www.unc.edu/~metcalfe/conference09.html |
| Wed., 2/25/09 |
Monica Visan, University of
Chicago |
TITLE: Nonlinear Schrodinger
equations at critical regularity ABSTRACT: We introduce the nonlinear Schrodinger equation (NLS) and define criticality. We then survey the history of the two most studied NLS at critical regularity, namely, the mass- and energy-critical NLS. We review some of the techniques developed to solve these problems and describe recent progress on the energy-critical and energy-supercritical NLS. This includes joint work with Rowan Killip, Terence Tao, and Xiaoyi Zhang. |
| Wed., 3/4/09 |
John Anderson, Holy Cross |
TITLE: $H^1$-BMO duality and
Hardy-Orlicz spaces ABSTRACT: In 1971 Charles Fefferman identified the dual of the Hardy space $H^1$ with the space BMO of functions of bounded mean oscillation. The dual pairing is somewhat subtle, because the product of an $H^1$ function and a BMO function need not be integrable. In a recent paper Bonami, Iwaniec, Jones and Zinsmeister (BIJZ) identify the product of an $H^1$ function and an analytic BMO function (acting as a distribution) with a function in a certain Hardy-Orlicz space. I will give an exposition of this and other results in the BIJZ paper. |
| Wed.,
3/25/09 |
Benjamin
Dodson, UNC |
Dissertation
Defense TITLE: The Pinsky phenomenon and indefinite signature Schrodinger equation ABSTRACT: The thesis examines the focusing phenomenon and Gibbs phenomenon for the linear Schrodinger equation of indefinite signature. Local and global existence results for the nonlinear equation are also obtained. |
| Thurs.,
3/26/09 (Colloquium) |
Steve
Hofmann, University of Missouri |
TITLE:
Local $Tb$ Theorems and Applications in PDE ABSTRACT: Singular integral operators ("SIOs") arise often in complex analysis and in the theory of partial differential equations. For example, the prototypical SIOs are the Hilbert Transform, which relates the real and imaginary parts of the boundary values of an analytic function in the upper half plane and, in higher dimensions, the Riesz Transforms, which relate the normal and tangential components of the boundary values of the gradient of a harmonic function in the half space. The terminology "singular integral" refers to the fact that the Schwarz kernels of these integral operators possess a singularity which just fails to be integrable, and therefore the operators must be defined in some limiting or "principal value" ("p.v.") sense. The aforementioned prototypical SIOs are of convolution type, so their boundedness on L^2 (which is the funadmental desired property of such operators) may be verified via Plancherel's Theorem (that is, one may exploit the fact that convolution operators are "diagonalized" by the Fourier transform). On the other hand, there are many other important examples of SIOs, arising, e.g., in the theory of variable coefficient elliptic PDE, and in the theory of analytic functions in domains with non-smooth boundaries, which are not of convolution type, and for these, Plancherel's Theorem is not available as a tool to establish L^2 boundedness. In this talk, we shall present a survey of progress on the development of criteria to verify the L^2 boundedness of non-convolution SIOs, and we shall discuss some applications. |
| Wed., 4/15/09 |
Yuri Latushkin, University of Missouri | TITLE: An index theorem, the
spectral flow, and the spectral shift function for relatively trace
class perturbations ABSTRACT: This is a joint work with Fritz Gesztesy, Konstantin Makarov, Fedor Sukochev, and Yuri Tomilov. Under relatively trace class assumptions, we calculate the Fredholm index of the operator d/dt+A(t) via Krein's spectral shift function of the operators A(+infinity) and A(-infinity). The main technical result is that the difference of the corresponding Morse projections is of trace class; this is proved using double operator integrals. |
| Wed., 4/22/09 |
Walter Strauss,
Brown University |
TITLE: Stability Criteria in a
Collisionless Plasma ABSTRACT: Stability of a state in a physical system refers to the asymptotic behavior of the nearby states. For a collisionless plasma (solar wind, hot fusion,...) that is modeled by the relativistic Vlasov-Maxwell system, many equilibria are stable but many others are unstable. In this talk, presenting joint work with Zhiwu Lin, I will consider axisymmetric equilibria of the form f(e, p) that are decreasing in the particle energy e and also depend on the particle angular momentum p. Then a necessary and sufficient condition for linear stability is the positivity of a certain linear operator L^0. This operator L^0 is much less complicated than the generator of the full linearized system. It has an interesting non-local term that can definitely affect its positivity. There is a similar reduction in the simpler case of 1.5 dimensional symmetry. For the important example of a purely magnetic equilibrium, explicit conditions for the linear/nonlinear stability/instability are obtained. |
| Thurs.,
4/23/09 (Colloquium) |
Walter
Strauss, Brown University |
TITLE:
Steady Rotational Water Waves ABSTRACT: Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,...) with an arbitrary vorticity function. Consider such a wave traveling at a constant speed over a flat bed. Using local and global bifurcation theory and topological degree, one can prove that there exist many such waves of large amplitude. I will outline the existence proof, joint with Adrian Constantin, and also exhibit some recent computations, joint with Joy Ko, of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid. |
Fall 2008
| Wed., 9/3/08 |
Benjamin Dodson,
UNC |
TITLE: Nonlinear
Perturbations of the Minkowski Schrodinger equation ABSTRACT |
| Wed., 9/17/08 |
Becca Thomases, UC Davis |
TTILE: Mixing Transitions
and Oscillations in Low-Reynolds Number Viscoelastic Fluids ABSTRACT: In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations. Next we study this and related models numerically for low-Reynolds number flows in two dimensions. For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state. The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices. |
| Wed., 9/24/08 |
Nathan Pennington, UNC |
TITLE: Derivation of the
Lagrangian averaged Navier-Stokes equation and local existence with
small initial data ABSTRACT |
| Wed., 10/22/08 |
Mihai Tohaneanu, UC
Berkeley |
TITLE: Local energy decay
on Schwarzchild and Kerr backgrounds ABSTRACT: Understanding the decay of linear waves is crucial in dealing with the problem of stability of the Kerr space time. I will talk about one way to measure this decay, namely local energy estimates, from which one can deduce many other useful estimates (uniform energy bounds, pointwise bounds, Strichartz estimates etc). This is joint work with Jeremy Marzuola, Jason Metcalfe and Daniel Tataru (for Schwarzschild) and Daniel Tataru (for Kerr). |
| Wed., 10/29/08 |
Paul Kessenich, University of
Michigan |
TITLE: Global Existence
for a 3D Incompressible Isotropic Viscoelastic Material ABSTRACT: Incompressible viscoelastic materials can be studied using Oldroyd-B model which views the material as an elastic polymer immersed in a Newtonian fluid. Previous global existence results for this model have used parabolic methods in which the size of the initial disturbance is inevitably dependent on the Newtonian viscosity in the Oldroyd-B equations. This talk will focus on the use of hyperbolic methods to prove global existence for a slightly more general set of equations so that the smallness of the initial data is not restricted by the viscosity parameter. As the viscosity goes to zero, a previously proven global existence result for incompressible isotropic elastodynamics can be recovered. |
| Wed., 11/5/08 |
Yaguang Wang, Northwestern |
TITLE: Zero Viscosity
Limit for Incompressible Navier-Stokes Equations with Navier Boundary
Conditions ABSTRACT: In this talk, we shall study the zero viscosity limit and behavior of boundary layers in a viscous incompressible flow with the Navier-friction boundary condition. For different relation of the slip lengths to the viscosity, we obtain several criteria on the convergence from the velocity of viscous fluids to that of in-viscous fluids in energy space when the viscosity tends to zero. The asymptotic behavior of boundary layers is derived by using multi-scale analysis. This is a joint work with Zhouping Xin. |
| Wed., 11/12/08 |
Tadeusz Iwaniec, Syracuse University | TITLE: Estimates of Jacobian
determinates - Hardy & Littlewood meet with Fefferman & Stein
in maximal inequalities ABSTRACT: This lecture features new types of maximal functions and commutators of singular integrals, linear and nonlinear. The topics are motivated by the study of nonlinear differential expresssions which arise naturally in geometric function theory, nonconvex calculus of variations and nonlinear elasticity. In particular, estimates of the Jacobian determinant in the Hardy space by means of subdeterminants involves an interpolation between maximal averages over the balls (Hardy maximal function) and spherical maximal averages (Fefferman-Stein maximal operator). |
| Thurs.,
11/13/08 (Colloquium), 4:00, Phillips 332 |
Tadeusz Iwaniec, Syracuse University | TITLE:
New Prospects of Quaiconformal Geometry, an Invitation to n-Harmonic
Hyperelasticity ABSTRACT: Quasiconformal geometry and nonlinear elasticity theory share common problems of compelling mathematical interest. Both theories are governed by specific energy integrals together with their associated Lagrange-Euler equations. The mappings of particular interest (elastic deformations) are the ones with smallest energy. In this talk I will present our latest advances and the results concerning existence, global injectivity (the principle of non-penetration of matter) and boundary behavior of deformations of smallest n-harmonic energy, a prototype of the planer Dirichlet problem. This brings us to a polyconvex variational integral for the inverse deformations. Distinctly, the inverse deformations are the mappings that minimize the L1 - average of the distortion function, among all homeomorphisms between two domains (very much reminiscent of the Teichmuller extremal problems). Then the total energy of a mapping and its inverse becomes of sufficient interest to call for closer examination. The great virtue of all our energy integrals is that they are invariant under conformal change of variables. For this reason we call this theory Quasiconformal Hyperelasticity. Our approach offers significantly larger class of mappings than the quasiconformal geometry. Interplay between nonlinear analysis and topology is critical in our approach. In particular, the underlying integration of rather special nonlinear differential expresssions (these are the free Lagrangians defined on homeomorphisms in a given homotopy class) becomes truly a work of art. Report is joint with Jani Onninen. This talk will be accessible to graduate students in analysis. |
| Wed., 11/19/08 |
Nick Costanzino, Penn State University | TITLE: Front propagation
is discontinuous heterogeneous media ABSTRACT: We investigate the behaviour of multidimensional pulsating fronts in discontinuous heterogeneous media. In addition to extending the existence result of Berestycki-Hamel-Roque to the case of discontinuous coefficients, we show the dependence of the front speed on both bulk and local properties of the medium. This is joint work with A. Novikov (Penn State) and L. Heltai (SISSA). |
Spring 2008
| Wed., 1/16/08 |
Marius Mitrea,
University of Missouri |
TITLE: Optimal estimates
for the inhomogeneous problem for the bi-Laplacian in three-dimensional
Lipschitz domains ABSTRACT: We establish the well-posedness of the inhomogeneous Dirichlet problem for the bi-Laplacian in arbitrary three-dimensional Lipschitz domains, with data from Besov-Triebel-Lizorkin spaces, for the optimal range of indices (smoothness and integrability). The main novel contribution is to allow for certain non-locally convex spaces to be considered, and to establish integralrepresentations for the solution. |
| Thurs.,
1/17/08 (Colloquium) 4:00pm, Phillips 332 |
Marius
Mitrea, University of Missouri |
TITLE:
Singular
Integral Operators and Boundary Value Problems under Sharp Geometric
Measure Theoretic Assumptions ABSTRACT: It has long been recognized that there are subtle connections between the boundedness of singular integral operators (SIO) and the geometric measure-theoretic properties of sets. A fundamental result in this direction is the boundedness of SIO with reasonable kernels on uniformly rectifiable surfaces. These are Ahlfors regular surfaces (i.e., behave like a (n-1)-dimensional object at all scales), and contain ``big pieces of Lipschitz images of (n-1)-dimensional sets'' in a uniform fashion. This earlier work has interfaced tightly with geometric measure theory, but until now it has not been systematically applied to problems in PDE. The aim of this talk is to explore the role that SIO may play in the treatment of boundary value problems under sharp geometric measure theoretic assumptions on the underlying domain. In particular, I will describe some recent joint work with S. Hofmann and M. Taylor in which we forge new links between the analysis of SIO on uniformly rectifiable surfaces and problems in PDE, most notably boundary problems for the Laplace operator and other second order elliptic operators, including systems. |
| Wed., 1/23/08 |
Benjamin Dodson, UNC |
TTILE: Pinsky Phenomena
and the Minkowski Schrodinger Equation ABSTRACT: In $\mathbf{R}^{n}$, the linear Schrodinger equation $\frac{\partial u}{\partial t} - i \Delta u = 0$, $u(0,x) = \chi_{B(0;1)}$ exhibits some interesting behavior. But now change the metric on $\mathbf{R}^{n}$ to a more general metric of signature (p,q). The equation $\frac{\partial u}{\partial t} - i \Delta_{p,q} u = 0$ has significantly altered convergence phenomena. In particular there is a generalization of the Pinsky phenomenon. |
| Wed., 2/13/08 |
Nathan Pennington, UNC |
TITLE: The Lagrangian averaged
Navier-Stokes equation with small initial data ABSTRACT: The Lagrangian averaged Navier Stokes equation is a PDE that models the averaged motion of an ideal incompressible fluid filtering over spatial scales smaller than $\alpha$. The equation is used to reproduce the large-scale averaged motion of the Navier-Stokes equations without the use of artificial viscosity or dissipation. We will briefly discuss the derivation of the PDE and will show the existence of a short time solution $u$ to the Lagrangian averaged Navier Stokes equation $\partial_t u + (u\,\cdot\,\nabla)u+\text{div}\,\tau^\alpha = -\text{grad}\,p+\nu\Delta u$ with initial data $u(0)=u_0$ in the homogeneous Lebesque space $\dot{L}_{s,p}(\mathbb{R}^m)$ for $p\ge m$ and for $s>0$. |
| Wed., 2/20/08 |
CANCELED |
|
| Thurs.,
2/28/08 (Special day),
Time: 2-3 |
Sarah Raynor, Wake Forest
University |
TITLE: Neumann Fixed Boundary
Regularity for an Elliptic Free Boundary Problem ABSTRACT: We examine the regularity properties of solutions to an elliptic free boundary problem near a Neumann fixed boundary. Consider a nonnegative function u, defined variationally, which is harmonic where it is positive and satisfies a gradient jump condition weakly along the free boundary (the boundary of the set where u is positive). Our main result is that u is Lipschitz continuous. Additionally, we prove various basic properties of such a minimizer near a portion of the fixed boundary on which Neumann conditions hold weakly. Our results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains, which have independent interest. |
| Wed., 3/19/08 |
Yaguang Wang, Shanghai Jiaotong University and Northwestern University | TITLE: Existence and
Stability of Compressible Current-Vortex Sheets in 3-D MHD |
| Thurs.,
3/20/08 (Colloquium) 4:00pm, Phillips 332 |
Christopher
Sogge, Johns Hopkins University |
TITLE:
How Focusing and Dispersion Influence the Solution of Certain Linear
and Nonllinear Equations ABSTRACT: In this survey, I shall present several estimates whose strength depends on whether certain types of dispersion is present and certain types of focusing is absent. These include size estimates for eigen functions, and Strichartz estimates both in manifolds without boundary and manifolds with boundary. I shall also go over some applications to the theory of nonlinear hyperbolic equations. |
| Wed., 3/26/08 |
CANCELLED |
|
| Thurs.,
3/27/08 (Colloquium) 4:00pm, Phillips 332 |
Victor
Guillemin, MIT |
TITLE: Classical and quantum Birkhoff canonical forms in one dimension |
| Wed., 4/2/08 |
Ramona Anton, Université Paris Sud, XI and Johns Hopkins University | TITLE: Non-linear
Schr\"odinger equations on domains with boundary ABSTRACT: We are interested in proving global existence results in the energy space for the semi-linear Schr\"odinger equation on domains of dimension 2 or 3. The main ingredients are generalized Strichartz inequalities adapted to the domains, which have some loss of derivatives. We present the results and the strategy for three types of domains. |
| Wed., 4/16/08 |
Michael Goldberg,
Johns Hopkins University |
TITLE: The Schr\"odinger
Equation with a Non-Smooth Magnetic Potential ABSTRACT: We prove Strichartz estimates for the absolutely continuous evolution of a Schr\"odinger operator $H = (i\nabla + A)^2 + V$ in ${\bf R}^n$, $n \ge 3$. Both the magnetic and electric potentials are time-independent and have polynomial pointwise decay. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This condition improves upon previous results that require half a derivative of smoothness or more. |
| Wed., 4/23/08 |
Gigliola Staffilani, MIT | TITLE: Weak turbulence for periodic 2D NLS |
| Thurs.,
4/24/08 (Colloquium) 4:00pm, Phillips 332 |
Gigliola
Staffilani, MIT |
TITLE:
"The nonlinear Schrodinger Equations: the old and the new" |
| 4/28
- 4/30/08 Brauer Lectures |
Charles
Fefferman, Princeton University |
TITLE:
Interpolation and extension of functions More info |
Fall 2007
| Wed., 10/10/07 |
Michael Shearer, NCSU |
TITLE: Particle-size
segregation in granular flow: a conservation law in two space dimensions. ABSTRACT: Kinetic sieving is the process by which large particles rise in granular avalanches, while smaller particles fall. Recent models of this effect reduce to a scalar conservation law in two dimensions and time, but with non-constant coefficients, reflecting the shear needed to induce segregation. Various topics of significance to applications are considered using the theory and constructions of scalar hyperbolic equations: steady solutions in which the direction of flow is time-like, leading to a sharp estimate of how long a chute should be to guarantee full segregation; breaking of interfaces, forming an evolving lens-shaped mixture zone; and the connection to recent experiments of Daniels on shear flow, for which the model is adjusted to account for nonuniform shear, with the consequent loss of constant solutions. |
| Wed., 10/24/07 |
Xiao-Biao Lin, NCSU |
TITLE: Gearhart-Pr¨uss
Theorem and linear stability for Riemann solutions of conservation laws ABSTRACT: We first review the Hille-Yosida Theory, Paley-Wiener Theorem and Gearhart-Pr¨uss Theorem on the asymptotic behavior of semigroups. We then consider the spectral and linear stability of of the Riemann solutions with multiple Lax shocks for systems of conservation laws u_T + f(u)_X= 0. Using the self-similar change of variables x = X/T , t = ln(T), Riemann solutions become stationary to the system u_t +(Df(u)-xI)u_x = 0. In the space of O((1 + |x|)^(-m)) functions, we show that if the real part of \lambda is greater than -m, then \lambda is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, there are resonance values where the determinant can be arbitrarily small but nonzero. A C_0 semigroup is constructed and using the Gearhart-Pr¨uss Theorem, we show that the solutions are of O(e^{\gamma t}) if \gamma is greater than the largest real parts of the eigenvalues and the resonance values. We present examples where Riemann solutions have two or three Lax-shocks. We will discuss how the linear stability can be used to determine the nonlinear stability of Riemann solution with shocks. |
| Mon.,
10/29/07 (GMA Visions) 4:00pm, Phillips 367 |
Jason
Metcalfe, UNC |
TITLE:
Local energy estimates for wave equations on Schwarzschild black
hole backgrounds ABSTRACT: One measure of dispersion for wave equations which is known to be fairly robust is the localized energy estimates. We will briefly discuss these estimates in Minkowski space-time (i.e. flat space). We shall then discuss the wave equation on Schwarzschild black hole backgrounds and some recent attempts at proving analogous estimates in this setting. At the end, some related open problems will be outlined. |
| Wed., 10/31/07 |
Mark Williams, UNC |
TITLE: ODEs with fast
transitions and spherical viscous shocks. ABSTRACT: We discuss a new technique for constructing solutions to two-point ODEs that exhibit fast transitions in the interior of the domain. The technique is applied to the construction of spherical viscous shocks. |
| Wed., 11/7/07 |
Michael Lacey,
Georgia Tech. |
TITLE: Irregularities of
Distribution and Related Questions ABSTRACT: The subject of Irregularities of Distribution concerns identification of optimal rates of convergence to uniform distribution. This classical topic is mostly understood in average case analysis. Certain endpoint estimates remain stubbornly resistant, despite decades of research on the topic. These questions are in turn related to arising in Harmonic Analysis, Probability Theory, and Approximation Theory. Given $ N$ points $ P_N$ in the unit cube, define the Discrepancy Function by $$ D_N (x) = |P_N \cap [0,x)| - N |[0,x)| $$ where $x$ is the rectangle with antipodal verticies at the origin and at $x$ in the unit cube. The importance of this function is highlighted by the classical Koksma-Hlawka inequality. We describe the average case analysis of $ D_N$, namely a universal lower bound on the $ L^2 $ norm of $ D_N$ due to Klaus Roth. The endpoint lower bounds for $ D_N$ remain a mystery in dimensions three and higher. We describe a new result, joint with Dmitriy Bilyk and Armen Vagharshakyan, on the $ L ^{\infty}$ of the Discrepancy function, in all dimensions three and higher. |
| Wed., 11/28/07 |
Jeremy Marzuola,
Columbia University |
TITLE: Wave packet
parametrices for evolution equations governed by PDO's with rough
symbols ABSTRACT: We prove existence of solutions to a generic class of dispersive equations under certain integrability conditions along the Hamilton flow of the leading order pseudodifferential operator governing the evolution of the solution. |
| Wed., 12/5/07 |
Jason Metcalfe, UNC |
TITLE: On 4D nonlinear wave
equations in exterior domains ABSTRACT: This talk is on a recent joint work with Y. Du, C. Sogge, and Y. Zhou. Here, we discuss two results concerning 4D wave equations in exterior domains. The first is a proof of an exterior domain analog of the 4D Strauss conjecture. The second is an exterior domain analog of a result of Hormander concerning almost global existence for quasilinear wave equations with nonlinear dependence on the solution not just its derivatives. The key to the proof is a combination of certain localized energy estimates with a certain Hardy-type inequality. |