Chaos in Miles' equations

Li, Y.Charles

doi:10.1016/j.chaos.2004.03.018

Cited by 10 publications

(6 citation statements)

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“…subject to the same boundary conditon as (3.3). For the first time, one can prove the existence of homoclinic orbits for a water wave equation (3.4) [13]. The Bernoulli shift dynamics was also established under generic assumptions [13].…”

Section: Chaos In Partial Differential Equationsmentioning

confidence: 95%

“…For the first time, one can prove the existence of homoclinic orbits for a water wave equation (3.4) [13]. The Bernoulli shift dynamics was also established under generic assumptions [13]. That is, for the first time, one can prove the existence of chaos in water waves under generic assumptions.…”

Section: Chaos In Partial Differential Equationsmentioning

confidence: 96%

“…But the Bernoulli shift dynamics was established under generic assumptions [12]. A real fluid example is the amplitude equation of Faraday water wave, which is also a complex Ginzburg-Landau equation [13],…”

Section: Chaos In Partial Differential Equationsmentioning

confidence: 99%

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On the True nature of turbulence

2007

The Mathematical Intelligencer

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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 [10].Fluid dynamicists believe that Navier-Stokes equations accurately describe turbulence. A mathematical proof on the global regularity of the solutions to the Navier-Stokes equations is a very challenging problem. Such a proof or disproof does not solve the problem of turbulence. It may help understanding turbulence. Turbulence is more of a dynamical system problem. Studies on chaos in partial differential equations 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.

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Section: Chaos In Partial Differential Equationsmentioning

confidence: 95%

Section: Chaos In Partial Differential Equationsmentioning

confidence: 96%

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On the True nature of turbulence

2007

The Mathematical Intelligencer

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“…For the first time, one can prove the existence of homoclinic orbits for a water wave equation (8.83). The Bernoulli shift dynamics was also established under generic assumptions [Li04d]. That is, one can prove the existence of chaos in water waves under generic assumptions.…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

“…But the Bernoulli shift dynamics was established under generic assumptions [Li04b]. A real fluid example is the amplitude equation of Faraday water wave, which is also a complex Ginzburg-Landau equation [Li04d], (8.83) subject to the same boundary condition as (8.82). For the first time, one can prove the existence of homoclinic orbits for a water wave equation (8.83).…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%