Simple Systems with Anomalous Dissipation and Energy Cascade |
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Authors: | Jonathan C Mattingly Toufic Suidan Eric Vanden-Eijnden |
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Institution: | (1) Department of Mathematics and CNCS, Duke University, Durham, NC 27708, USA;(2) Mathematics Department, University of California, Santa Cruz, CA 95064, USA;(3) Courant Institute, New York University, New York, NY 10012, USA |
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Abstract: | We analyze a class of dynamical systems of the type where f
n
(t) is a forcing term with only for and the coupling coefficients c
n
satisfy a condition ensuring the formal conservation of energy . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined)
and, as a result, may admit unique (statistical) steady states when the forcing term f
n
(t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific
choices of the coupling coefficients c
n
. The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely scales as as n→∞. Here the exponents α depend on the coupling coefficients c
n
and denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions
of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence,
these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in
that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable. |
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Keywords: | |
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