# CAMP/Nonlinear PDE's Seminar

Wednesdays at 4 PM in Eckhart 202.

## Winter 2000

January 5
Charles Doering, University of Michigan
Laminar and turbulent energy dissipation in a shear layer with suction

The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. For sufficiently large values of the suction rate, a steady laminar flow is absolutely stable at all shear rates. For sufficiently small but nonzero suction, however, the laminar flow is linearly unstable at high Reynolds numbers. We find that the rigorous upper bound on the energy dissipation rate---valid even for turbulent (and weak) solutions of the Navier-Stokes equations---scale precisely the same as the dissipation in the laminar solution in the zero viscosity limit. Both the laminar and any possible turbulent flows display a finite nonvanishing residual dissipation as the viscosity is decreased. This is a manifestation of so-called Kolmogorov-type scaling in which the energy dissipation rate becomes independent of the viscosity at high Reynolds numbers. This result establishes the sharpness of the upper bound's scaling in the vanishing viscosity limit---for these boundary conditions. It also provides a mathematical illustration of the delicacy of corrections to high Reynolds number scaling (such as the logarithmic terms as appearing in the Prandtl-von Karman "law of the wall") to perturbations in the boundary conditions. This is joint work with Edward A. Spiegel (Columbia University) and Rodney A. Worthing (University of Michigan).

January 12
Philip Morrison, University of Texas at Austin
An Integral Transform for the Continuous Spectrum in Fluid and Plasma Dynamics

Fluid and plasma linear dynamical systems, obtained by expansion about equilibria, generally possess a continuous spectrum. An integral transform, which is a generalization of the Hilbert transform, will be described and shown to make the solution of such linear problems trivial. The transform will also be shown to amount to a coordinate change to action-angle variables for the infinite-dimensional Hamiltonian description of fluid and plasma dynamics. This affords a means for attaching Krein signature to the continuous spectrum, a quantity of importance for untangling bifurcations. Examples will include Vlasov-Poisson and shear flow dynamics.

January 19
George Papanicolaou, Stanford University
Super-Resolution in time reversal acoustics

Time reversal in acoustics or other wave propagation phenomena is an efficient way to locate sources and irregularities that cause scattering. What happens if small random inhomogeneities are also present in the medium? In the last few years a number of experiments were carried out showing that random inhomogeneities have a beneficial effect in that diffraction limits are reduced! I will explain theoretically how this is possible and present detailed numerical computations that show how prominent this effect is and how well the theory explains it. This is joint work with P. Blomgren and H. Zhao.

January 26
Cancelled
Xiao-Ping Wang, , University of Science and Technology, Hong Kong
An iterative grid redistribution method for singular problems in multiple dimensions

We introduce an iterative grid redistribution method based on the variational approach. The iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations. The method is particularly effective for solving PDE's with singular solutions (e.g. blow up solutions). Our method requires little prior information of the singular solutions and can handle multiple singularities. We will also show some numerical examples.

February 2
Jinchao Xu, Pennsylvania State University
Multigrid methods and applications

The speaker will first give a brief introduction to multigrid methods for solving partial differential equations and then talk about some recent developments and applications.

February 9
Norman Lebovitz, University of Chicago
Nonlinear Dynamical Systems for the Elliptical Instability

The elliptical instability is a widely occurring hydrodynamic instability of planar flows with noncircular streamlines, usually taken to be elliptical for mathematical simplicity. It has been invoked as a mechanism responsible for the onset of turbulence in shear flows, perhaps via successive bifurcations to periodic solutions, as outlined by Ruelle and Takens. To explore this it is necessary to understand the nonlinear development of the flow in parameter domains in which linear theory predicts instability. In this lecture I will describe an approach giving finite-dimensional dynamical systems providing an exact description of amplitude equations of the Euler equations of inviscid fluid dynamics up to a given order, and describe the kinds of bifurcations that have thus far been isolated.

February 16
Leonid Koralov, IAS, Princeton
Moment and Almost Sure Lyapunov Exponents for the Solution of the Parabolic Anderson Problem

We consider the Stochastic Partial Differential Equation $u_t = \kappa \Delta u + \xi(t,x)u, ~~~ t \geq 0, ~~~ x \in \mbox {\bbc Z}^d~.$ The potential is assumed to be Gaussian white noise in time, stationary in space. We obtain the asymptotics of the almost sure Lyapunov exponent for the solution as $\kappa \rightarrow 0$. We also study the dependence of the moment Lyapunov exponents upon the diffusion constant $\kappa$.

February 18, Friday, 3pm
Ronnie Sircar, University of Michigan
Mean-Reverting Stochastic Volatility

In modern financial markets, investors and trading institutions are faced with an environment of uncertain and changing volatility which must be modeled when pricing or managing the risks from derivative securities. However the situation is complicated because volatility is not directly observable. We describe an asymptotic and statistical analysis of the problem that exploits the tendency of volatility to come in bursts, or cluster. A singular perturbation approach provides a correction to the Black-Scholes pricing and hedging theory that adjusts for random volatility in a robust way, in that it does not depend on specific modeling of the volatility process. We illustrate how the modeling assumptions fit S&P 500 index data, and how the corrected theory identifies stable market quantities that are easily estimated from the observed implied volatility surface. Joint work with Jean-Pierre Fouque, George Papanicolaou and Knut Solna.

February 23
Jack Xin, University of Texas at Austin
Modeling Light Bullets with the Two-Dimensional Sine-Gordon Equation

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The Sine-Gordon (SG) equation comes as an asymptotic reduction of the two level dissipationless Maxwell-Bloch system. We derive a new and complete nonlinear Schr\"odinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusing-defocusing cycles. Numerical results are in qualitative agreement with asymptotics and reveal the SG light bullets, similar to the Maxwell light bullets. We achieve the understanding that the light bullets are manifestations of the persistence and robustness of the complete NLS asymptotics.

March 1
Kenneth McLaughlin, University of Arizona
On the semiclassical limit of the focusing nonlinear Schroedinger equation

While the focusing nonlinear Schroedinger equation is well posed (for reasonable initial data), in the semi-classical limit the underlying model equations are elliptic, and hence ill-posed as an initial value problem. I will describe some new analytical developments arising through a Riemann-Hilbert formulation of the semi-classical limit. The work is joint with Spyros Kamvissis and Peter Miller.

March 8
Peter Constantin, University of Chicago
An Eulerian-Lagrangian Approach to the Navier-Stokes Equations

I will discuss the Navier-Stokes equations governing the flow of incompressible fluids in a formulation that highlights the role played by the nontrivial commutator $$[\nabla _E, \nabla _L]$$ of Eulerian and Lagrangian gradients.

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