We first show that these flows are nonlinearly stable: smooth solutions never cross into the elliptic domain. We conjecture and show some numerical evidence that this is also true for more layers. Then, we consider shocks at the interface of miscible fluids - a problem of geophysical importance in the atmosphere and ocean. We require an additional postulate that would yield the mixing rate at the shock: one can imagine that the energy dissipated at the shock can now flow both into small scale turbulence and into mixing the fluid, but the partition between these sinks of macroscopic energy is unknown. We discuss two possibilities for deriving the additional constraint on the problem: kinematics and entropy maximization. We show that these are in fact equivalent and yield upper bounds on the mixing rates.
In this lecture, we will focus on the KdV equation. It will be shown that when the initial data is analytic, then the convergence rate is actually exponential. This concurs with numerical experiments which also exhibit exponentially-fast convergence.
We have designed a wavelet-based method [1, 2, 3] which decomposes each flow realization into two orthogonal components: the coherent vortices resulting from the nonlinear term of Navier-Stokes equation, whose statistical behaviour is non-Gaussian and long-range correlated, and the incoherent background flow corresponding to the linear dissipation term, whose statistical behaviour is quasi-Gaussian and decorrelated. In this talk we will show that these components correspond in wavelet space to two simply connected domains. We will visualize the time evolution of the interface separating them in order to study the transfers associated to the turbulent flow dynamics. We will apply this method to experimental and numerical datasets of 2D and 3D turbulent flows (see as example Figures 1, 2, 3).
The existence of such an interface in wavelet space is the basis of the CVS (Coherent Vortex Simulation) method [1, 2, 3], which computes the time evolution of turbulent flows with a reduced number of wavelet modes corresponding to the coherent vortices only. CVS combines a nonlinear filtering of the solution at time step t, which gives the interface in wavelet space, with the addition of the wavelet coe cients adjacent to this interface, which corresponds to dealiasing and defines the security zone, in order to compute the flow at time step (t+ 1). The principle of the CVS method is to retain the nonlinear activity at all active scales, and model the linear activity which is also multiscale, being noise-like, but presents a different statistical behaviour than the coherent vortices.
References
[1] M. Farge, K. Schneider and N. Kevlahan, 1999.
Phys. Fluids, 11(8), 2187 2201.
[2] M. Farge and K. Schneider, 2001.
Flow, Turbulence and Combustion, 66(4), 393 426.
[3] M. Farge, K. Schneider, G. Pellegrino, A. Wray
and B. Rogallo, 2003. Phys. Fluids, 15(10), 2886 2896.
This talk is based on the paper with Bernd Sturmfels and Seth Sullivant which is posted at http://front.math.ucdavis.edu/math.ST/0509390.
Joint work with: Nilima Nigam, Department of Mathematics and Statistics, McGill University.
Using the theory of Lyapunov exponents, we give a definition of the Evans function for quite general first order nonatonomous matrix differential equations on the line in the context of perturbation theory. The central result of the talk is a formula relating the Evans function and the modified Fredholm determinant of the so-called ``sandwiched resolvent". This result is obtained under the assumption that the unperturbed equation has exponential dichotomy on the line, and that the perturbation has certain exponential decay at infinities controlled by the general Lyapunov exponents of the unperturbed system.
This talk is based on a joint paper with Fritz Gesztesy and Konstantin A. Makarov.
They were students togther in 1946 at Moscow State University and both were strongly influenced by Petrovsky and Gelfand to study PDE. Oleinik remained in Moscow and ultimately became the Head of the "department of PDE ". Ladyzhenskaya moved to Leningrad and became the leader at the Steklov institute of a famous group in PDE and mathematical physics. Both proved fundamental theorems in PDE and both were challenged and inspired by problems in fluid dynamics.
As a tool we use a new characterization of the Sacker-Sell spectrum of a dynamical system in terms of Mane sequences.
This lecture will survey the mathematical theory of Babenko's equation and discuss in particular the possibility of solutions that are not very regular and correspond to Stokes waves with stagnation points on the free boundary, or possibly not corresponding to Stokes waves at all.. Open questions, such as "can there can be uncountably many stagnation points" arise naturally in the theory? The theory raises important questions about the extent to which weak solutions of the new version of Zakharov's equations give rise to true water waves with the constant pressure condition on the free surface.
In this presentation, we review the derivation of a hydrodynamic model, which is familiar in the kinetic theory of gases. We describe closure assumptions and relaxation approximations commonly used today in charge transport. Following this well-known material, we will review some recent mathematical results. The use of such hydrodynamic models in charge transport dates to 1986 in semiconductors and to 1995 in ion channels. The talk will review this chronology, and will also sketch the role of the Kato semigroup theory to obtain local smooth solutions for the Cauchy problem for the system. A few simulation results will be presented.