2008-2009 Dispersive Seminar

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This page contains information about an informal seminar on Dispersive PDEs during Fall 2008. The seminar is organized by J. Colliander. We will meet on Thursdays, 11:00am--12:00pm in BA2139, which is on the 2nd floor of the Bahen Centre.

We plan to survey aspects of the following topics:

We will read through the classical paper of Grillakis and the more recent work of Kenig-Merle.

We will survey the theory developed by Merle-Raphaël and their collaborators.

We will discuss weak turbulence theory of Zakharov in the context of the Majda-McLaughlin-Tabak model. We will also link these questions with works by Bourgain and W.-M. Wang and CKSTT concerning the behavior of high Sobolev norms during Schrödinger evolutions.

We mentioned works by de Bouard et. al., and recent works of Abou-Salem and Sulem, Oh et. al. J. Quastel is also interested in this topic.

B. Pigott and M. Zoghi are studying aspects of completely integrable systems. Some lectures in this direction may be planned for later in the semester.


The PDE/AppliedMath/Analysis seminar is now posted in the form of a google calendar.


Contents

4 December, 2008, 11:00am--12:00pm

Speaker: Tadahiro Oh
Title: Invariant Gibbs measure, white noise, a.s. GWP and all that. (Part 3)
Abstract: I will talk about abstract Wiener spaces, Fernique's Theorem, and its application to Schrödinger-Benjamin-Ono system.

27 November, 2008, 2:00pm--3:00pm; BA3000 (Different Room)

Speaker: I. Michael Sigal
Title: On collapse dynamics in Yang-Mills and wave map equations
Abstract: In this talk I discuss Yang-Mills and wave map equations coming from elementary particle physics (related to the gauge fields and sigma-model, respectively) and which are interesting from the geometric viewpoint. Solutions of these equations in 4+1 and 2+1 dimensions are expected to exhibit a 'critical' collapse for certain open set of initial conditions. This is supported by a considerable body of numerical simulations produced by the numerical gravity community studying the critical black hole formation and some recent rigorous results. We present some recent analytical and numerical results about the collapse dynamics for these equations.
References: This talk is partly based on this article. Some related work by Rodnianski and Sterbenz.

Image:2008 11 27 Sigal YM WM DispersiveSeminar.pdf

27 November, 2008, 11:00am--12:00pm

Speaker: Tadahiro Oh
Title: Invariant Gibbs measure, white noise, a.s. GWP and all that. (Part 2)

20 November, 2008, 11:00am--12:00pm

Speaker: Tadahiro Oh
Title: Invariant Gibbs measure, white noise, a.s. GWP and all that. (Part 1)
Abstract (for Part 1, 2, and 3): In this talk, I will talk about invariant (Gibbs) measures to Hamiltonian PDEs. Liouville's theorem states that a Hamiltonian ODE is volume preserving. Thus, given a Hamiltonian ODE with the Hamiltonian H(p,q), we see that the measure \scriptstyle d \mu_n = Z^{-1}e^{-\beta H(p, q)} \prod_{j = 1}^n d p_j d q_j is invariant under the flow. Now, we'd like to consider the same issue for a Hamiltonian PDE with the Hamiltonian H(u). i.e. Is the Gibbs measure " \scriptstyle d \mu = Z^{-1} e^{-\beta H(u)} \prod_{x \in \mathbb{T}} d u(x) " invariant under the flow? First, we will briefly go over the theory of Gaussian measures on Hilbert spaces. In particular, we discuss for which \scriptstyle s \in \mathbb{R} the Gaussian measure \scriptstyle d \rho = Z^{-1} e^{-\int_{x \in \mathbb{T}} |D_x^\alpha u|^2 dx } \prod_{x \in \mathbb{T}} d u(x) is countably additive and supported on Hs. Then, we'll discuss the precise meaning of "μ" (for α = 1 with an L2 cutoff in the focusing case) by following the work by Lebowitz-Rose-Speer (1988) and Bourgain(1994). If the PDE is globally well-posed on the support of such μ, then the invariance of μ follows via the invariance of μn, the invariant Gibbs measure of the finite dimensional flow to the PDE. If only the local well-posedness is known, then we use the finite dimensional flow and its invariant Gibbs measure μn to show the global well-posedness of the PDE almost surely on the statistical ensemble (i.e. on the support of μ) and the invariance of μ.
Next, we consider the case where the PDE barely fails to be locally well-posed in Hs containing the support of μ. In this case, using the abstract Wiener space theory, we present several function spaces (other than the usual Sobolev spaces) containing the support of μ. Then, we show how one may show LWP of a PDE in these new spaces by taking specific examples, and hence can obtain a.s. GWP as well as the invariance of the measure.
Lastly, we consider the issue of the (invariant) measures supported below L2. In this case, we can not follow the previous argument by Bourgain since the L2 cutoff does not make sense. For Schrödinger-Benjamin-Ono system, one can use a simple probablistic argument to handle this issue. Next, we introduce Wick ordering and discuss a construction of the Gibbs measure of Benjamin-Ono by Tzvetkov. Then, we use Wick ordering to show a formal invariance of the white noise to mKdV (or cubic NLS) by establishing a weak convergence of the grand canonical ensemble to the white noise.
Lecture Notes
References:
  • A. Ayache, N. Tzvetkov, Lp properties for Gaussian random series, arXiv:math/0610139.
  • J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1--26.
  • J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421--445.
  • J. Bourgain, Invariant measures for NLS in infinite volume, Comm. Math. Phys. 210 (2000), no. 3, 605--620.
  • J. Lebowitz, H. Rose, E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50 (1988), no. 3-4, 657--687.
  • H. Kuo, Gaussian Measures in Banach Spaces, Lec. Notes in Math. 463 Springer-Verlag (2006).
  • T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: invariant Gibbs measure and a.s. global well-posedness, www.math.toronto.edu/oh.
  • T. Oh, Invariance of the Gibbs measure for the Sch\"odinger-Benjamin-Ono system, www.math.toronto.edu/oh.
  • T. Oh, Invariance of the white noise for the KdV, www.math.toronto.edu/oh.
  • T. Oh, J. Quastel, B. Valko, work in progress.
  • J. Quastel, B. Valko, KdV Preserves White Noise, arXiv:math/0611152.
  • B. Roynette, Mouvement brownien et espaces de Besov, Stochastics Stochastics Rep., 43 (1993), pp. 221–260.
  • N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, arXiv:math/0610626.
  • N. Tzvetkov, Invariant measures for the Nonlinear Schrodinger equation on the disc, arXiv:math/0603112.
  • P. Zhidkov, Korteweg-de Vries and Nonlinear Schrodinger Equations: Qualitative Theory, Lec. Notes in Math. 1756 Springer-Verlag (2001).

13 November 2008, 11:00am--12:00pm

Speaker: B. Pigott
Title: Local well-posedness for modified KdV on the circle.
Abstract: I will review the paper of Takaoka and Tsutsumi concerning the local well-posedness for mKdV on the circle. The main new idea is to consider a modified evolution operator which (approximately) eliminates the worst terms in the nonlinearity. This modified evolution is then used to demonstrate continuity of the data-to-solution map.
References:
  • Takaoka, H., and Tsutsumi, Y., Well-Posedness of the Cauchy Problem for the Modified KdV Equation with Periodic Boundary Condition, IMRN 56 (2004), 3009--3040.

6 November 2008, 11:00am--12:00pm

Speaker: J. Colliander
Title: Global well-posedness for cubic defocusing NLS on {\mathbb{R}}^2 in H^s,~s>1/3.
Abstract: I will describe joint work with Tristan Roy of UCLA. We incorporate the improved almost conservation law of C-Keel-Staffilani-Takaoka-Tao in a (slightly improved) Morawetz-based bootstrap following C-Grillakis-Tzirakis.

30 October 2008, 11:00am--12:00pm

Speaker: G. Simpson
Title: Dispersive Wave Propagation through Heterogeneous Media
Abstract: I will discuss how heterogeneity in a medium can induce dispersive behavior in hyperbolic wave equations. This is largely motivated by applications described by systems of conservation laws, including acoustics and E&M. In the absence of heterogeneity and nonlinearity, one dimensional waves are non dispersive. Adding nonlinearity can lead to shock formation. Adding heterogeneity can lead to dispersive behavior. The combination of nonlinearity and heterogeneity can produce coherent structures. These structures, seen computationally in LeVeque & Yong '03, seem fundamentally different than the familiar nonlinear waves of equations with explicit linear dispersive operators.
References:
  • Thoo, J.B. and Hunter, J.K., 2003, Nonlinear hyperbolic wave propagation in a one-dimensional random medium, Wave Motion 37, 381--405.
  • LeVeque, R.J. and Yong, D.H., 2003, Solitary Waves in Layered Nonlinear Media, SIAM J. Appl. Math. 63, 1539--1560.
  • Rosales, R.R., Tabak, E.G., and Turner, C.V., 2002, Resonant Triads Involving a Nondispersive Wave, Stud. Appl. Math. 108, 105--122.
  • Goodman, R.H., Weinstein, M.I., and Holmes, P.J., 2001, Nonlinear Propagation of Light in One-Dimensional Periodic Structures, J. Nonlinear Sci. 11, 123--168.
  • Majda, A.J., Rosales, R.R., Tabak, E.G., and Turner, C.V., 1999, Interaction of Large-Scale Equatorial Waves and Dispersion of Kelvin Waves through Topographic Resonances, J. Atmos. Sci. 56, 4118--4133.
  • Hunter, J.K., Majda, A., and Rosales, R., 1986, Resonantly Interacting Weakly Nonlinear Hyperbolic Waves II: Several Space Variables, Stud. Appl. Math., 75, 187--226.
  • Majda, A. and Rosales, R., 1984, Resonantly Interacting Weakly Nonlinear Hyperbolic Waves I: A Single Space Variable, Stud. Appl. Math., 71, 149--179.
  • Hunter, J.K. and Keller, J.B., 1983, Weakly Nonlinear High-Frequency Waves, CPAM, 36, 547--569.

23 October 2008, 11:00am--12:00pm

Speaker: J. Colliander
Title: Remarks on the Hmidi-Keraani proof of mass concentration for L2-critical NLS
Abstract: I outline the proof of mass concentration for critical NLS following the argument of T. Hmidi and S. Keraani. The presentation will emphasize when explicit properties of the potential energy are used.

16 October 2008, 11:00am--12:00pm

Speakers: C. Sulem
Title: On the elliptic-elliptic Davey-Stewartson blowup
Abstract: The Davey - Stewartson system arises as envelope equations obtained for example from a modulation analysis of the full 3d water wave problem with surface tension. It has the form of a 2d NLS equation for the amplitude coupled with another equation for a mean field which is of elliptic or hyperbolic type. The Elliptic Elliptic DS system refers to the case where the dispersive term in the NLS equation is a Laplacian and the 2nd equation is of a Poisson equation. It can be rewritten as an cubic NLS equation with a nonlocal nonlinear term.
I will review the conserved quantities, well-posed results, possible occurrence of blowup, existence of solitary waves. An heuristic analysis of blowup solutions can be performed in the spirit of 2d crtical NLS and predicts a rate of blow up with a loglog law. Numerical simulations have confirmed this regime. A notable difference with critical NLS is the nonisotropy of solitary waves solutions.
A few references:
  • Ablowitz, M.J. and Segur, H., 1979, On the evolution of packets of water waves, J. Fluid Mech. 92, 691--715.
  • Cipolatti, R., 1992 On the existence of standing waves for the Davey--Stewartson system, Comm. Part. Diff. Eq. 17, 967--988.
  • Cipolatti, R., 1993 On the instability of ground states for a Davey--Stewartson system, Ann. Inst. H. Poincar\'e, Phys. Th\'eor. 58, 85--104.
  • Craig, W., Schanz, U., and Sulem, C., 1997 The modulational regime of three-dimensional water wave and the Davey--Stewartson system, Ann. Inst. H. Poincar\'e, Analyse non lin\'eaire, 14, 615--667.
  • Ghidaglia, J.M. and Saut, J.C., 1990 On the initial value problem for the Davey--Stewartson systems, Nonlinearity 3, 475--506.
  • Papanicolaou, G.C., Sulem, C., Sulem, P.L., and Wang, X.P., 1994 The focusing singularity of the Davey--Stewartson equations for gravity--capillary surface waves, Physica D 72, 61--86.
  • Sulem, C. , Sulem ,P.L. , 1999 NLS equation: Self-focusing and wave collapse, Springer, chapter 12.

9 October 2008, 11:00am--12:00pm

Speakers: G. Richards, I. Zwiers
Title: Log Log Blowup Regime

2 October 2008, 11:00am--12:00pm

Speakers: G. Richards, I. Zwiers
Title: Log Log Blowup Regime
Abstract:
References: See Log Log Blowup Regime
Image:LogLogLecture1 2October.pdf

25 September 2008, 11:00am--12:00pm

Speaker: M. Czubak
Title: Local well-posedness for the space-time Monopole Equation.
Abstract: The 2+1 Monopole Equation can be derived by a dimensional reduction from the Anti-Self-Dual Yang Mills equations on {\mathbb{R}}^{2+2}. It can be also viewed as the space-time analog of Bogomolny Equations. We show the equation is locally well-posed in the Coulomb gauge for initial data sufficiently small in Hs for s > 1 / 4, which is sharp by iteration methods for the formulation we use.
References: Preprint
My thesis with all the details omitted in the paper.
Slides from AMS Talk

18 September 2008 CANCELLED due to Morawetz Conference at Fields

11 September 2008, 11:00am--12:00pm

Title: Organizational Meeting
Abstract: Please come with ideas to discuss in the seminar this semester.
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