Chaos in partial differential equations

Li, Y. Charles

doi:10.1090/conm/301/05160

Cited by 20 publications

(26 citation statements)

References 185 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Local well-posedness is often enough. In fact, this is the case in my proof on the existence of chaos in partial differential equations [Li04].…”

Section: Global Well-posedness Of the Navier-stokes Equationsmentioning

confidence: 76%

“…We will see below that studies on chaos in PDEs indicate that turbulence can have Bernoulli shift dynamics which results in the wandering of a turbulent solution in a fat domain in the phase-space; thus, turbulence can not be averaged. The hope is that turbulence can be controlled [Li04]. The first demonstration of existence of an unstable recurrent pattern in a 3D turbulent hydrodynamic flow was performed in [KK01], using the full numerical simulation, a 15,422-dimensional discretization of the 3D Plane Couette turbulence at the Reynold's number Re = 400.…”

Section: Turbulencementioning

confidence: 99%

“…In both the perturbed sine-Gordon equation and the complex Ginzburg-Landau equation, one can incorporate as many unstable modes as one likes, the resulting Bernoulli shift dynamics is still the same. On a computer, the solution with more unstable modes may look rougher, but it is still chaos [Li04].…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

See 2 more Smart Citations

High-Dimensional Chaotic and Attractor Systems

Ivancevic¹

2007

View full text Add to dashboard Cite

“…Local well-posedness is often enough. In fact, this is the case in my proof on the existence of chaos in partial differential equations [Li04].…”

Section: Global Well-posedness Of the Navier-stokes Equationsmentioning

confidence: 76%

Section: Turbulencementioning

confidence: 99%

See 1 more Smart Citation

High-Dimensional Chaotic and Attractor Systems

Ivancevic¹

2007

View full text Add to dashboard Cite

“…The sensitive dependence on the initial and boundary conditions of the dynamical evolution of such systems, and the broadband and continuous power spectra are the indicators of deterministic chaos. A mathematical proof on the existence of chaotic behavior in Navier-Stokes equations and turbulence has been conducted by Li (2004). On other hand, Simonnet et al (2009) analyzed the presence of bifurcations in ocean, atmospheric and climate models for understanding the variability of oceanic and atmospheric flows as well as the climate system.…”

Section: Storm Surge Modelingmentioning

confidence: 99%

Nonlinear chaotic model for predicting storm surges

Siek

Solomatine

2010

Nonlin. Processes Geophys.

View full text Add to dashboard Cite

Abstract. This paper addresses the use of the methods of nonlinear dynamics and chaos theory for building a predictive chaotic model from time series. The chaotic model predictions are made by the adaptive local models based on the dynamical neighbors found in the reconstructed phase space of the observables. We implemented the univariate and multivariate chaotic models with direct and multi-steps prediction techniques and optimized these models using an exhaustive search method. The built models were tested for predicting storm surge dynamics for different stormy conditions in the North Sea, and are compared to neural network models. The results show that the chaotic models can generally provide reliable and accurate short-term storm surge predictions.

show abstract

“…In some cases, chaos (turbulence) can be rigorously proved to exist using e.g. shadowing techniques [15]. For the Navier-Stokes equations, the problem is much more difficult than a typical dynamical system.…”

Section: Introductionmentioning

confidence: 99%

A resolution of the turbulence paradox: Numerical implementation

Lan

2013

International Journal of Non-Linear Mechanics

View full text Add to dashboard Cite

Sommerfeld paradox (turbulence paradox) roughly says that mathematically the Couette linear shear flow is linearly stable for all values of the Reynolds number, but experimentally transition from the linear shear to turbulence occurs under perturbations of any size when the Reynolds number is large enough. In [18], we offered a resolution of this paradox. The aim of this paper is to provide a numerical implementation of the resolution. The main idea of the resolution is that even though the linear shear is linearly stable, slow orbits (also called quasi-steady states) in arbitrarily small neighborhoods of the linear shear can be linearly unstable. The key is that in infinite dimensions, smallness in one norm does not mean smallness in all norms. Our study here focuses upon a sequence of 2D oscillatory shears which are the Couette linear shear plus small amplitude and high frequency sinusoidal shear perturbations. In the fluid velocity variable, the sequence approaches the Couette linear shear (e.g. in L 2 norm of velocity), thus it can be viewed as Couette linear shear plus small noises; while in the fluid vorticity variable, the sequence does not approaches the Couette linear shear (e.g. in H 1 norm of velocity). Unlike the Couette linear shear, the sequence of oscillatory shears has inviscid linear instability; furthermore, with the sequence of oscillatory shears as potentials, the Orr-Sommerfeld operator has unstable eigenvalues when the Reynolds number is large enough, this should lead to transient nonlinear growth which manifests as transient turbulence as observed in experiments. The main result of this paper verifies this transient growth.

show abstract

Chaos in partial differential equations

Cited by 20 publications

References 185 publications

High-Dimensional Chaotic and Attractor Systems

High-Dimensional Chaotic and Attractor Systems

Nonlinear chaotic model for predicting storm surges

A resolution of the turbulence paradox: Numerical implementation

Contact Info

Product

Resources

About