Related papers: Weak stablity and closure in turbulence
Considerable effort has been expended over the last 2 centuries into explaining the behavior of fluid flow after the onset of turbulence. While perturbations in the velocity field have been shown to explain turbulent transitions, a physical…
This contribution covers the topics presented by the authors at the {\it ``Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961"} meeting that took place in Marseille in 2011. It focuses on some of the…
We survey results of recent activity towards studying controllability and accessibility issues for equations of dynamics of incompressible fluids controlled by low-dimensional or, degenerate, forcing. New results concerning controllability…
We study stationary solutions to the continuity equation for weakly compressible flows. These describe non-equilibrium steady states of weakly dissipative dynamical systems. Compressibility is a singular perturbation that changes the steady…
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given…
The Navier-Stokes equation describes the deterministic evolution of incompressible fluids. The effects of random initial conditions on solutions of this equation are studied. It is shown that there is an infrared stable fixed point…
We consider the stability of a system of equations which are a singular perturbation of the incompressible rigid-plastic flow equations used to model granular flow. A linear stability analysis shows that solutions of these equations are…
The turbulent energy flow of the onedimensional Majda-McLaughlin-Tabak equation is studied numerically. The system exhibits weak turbulence for weak driving forces, while weak turbulence coexists with strongly nonlinear intermittent…
This review paper is devoted to a presentation of recent progress in wave turbulence. I first present the context and state of the art of this field of research both experimentally and theoretically. I then focus on the case of wave…
In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an…
To date it has not been possible to prove whether or not the three-dimensional incompressible Euler equations develop singular behaviour in finite time. Some possible singular scenarios, as for instance shock-waves, are very important from…
We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long time numerical simulations makes this system extremely valuable for wave turbulence studies.…
Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of \emph{incompressible} fluids is nowadays available not only for Navier--Stokes fluids but also for various…
We will discuss various aspects of the incompressible Euler equation. We will discuss, in particular, problems related to the least action principle, the existence of special solutions, the problem of solvability, singularity formation, and…
A calculational approach in fluid turbulence is presented. Use is made of the attracting nature of the fluid-dynamic dynamical system. An approximate approach is offerred that effectively propagates the statistics in time. Loss of…
In this article, I would like to express some of my views on the nature of turbulence. These views are mainly drawn from the author's recent results on chaos in partial differential equations \cite{Li04}. Fluid dynamicists believe that…
The dynamics of fluids is a long standing challenge that remained as an unsolved problem for centuries. Understanding its main features, chaos and turbulence, is likely to provide an understanding of the principles and non-linear dynamics…
In this course we review the theory of incompressible homogeneous turbulence at an elementary level, and discuss the similarities and differences expected in the compressible case, relevant to the interstellar medium and molecular clouds.…
Recent numerical explorations of extremely intense circulation fluctuations at high Reynolds number flows have brought to light novel aspects of turbulent intermittency. Vortex gas modeling ideas, introduced alongside such developments,…
Hydrodynamic equations for ideal incompressible fluid are written in terms of generalized stream function. Two-dimensional version of these equations is transformed to the form of one dynamic equation for the stream function. This equation…