Simple systems with anomalous dissipation and energy cascade

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2007-11-01

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Mattingly, JC
Suidan, T
Vanden-Eijnden, E

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Abstract

We analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. 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 E |an|2 scales as n-α as n→∞. Here the exponents α depend on the coupling coefficients c n and E 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. © 2007 Springer-Verlag.

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Science & Technology, Physical Sciences, Physics, Mathematical, Physics, EULER, EQUATIONS

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10.1007/s00220-007-0333-0

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Mattingly, JC, T Suidan and E Vanden-Eijnden (2007). Simple systems with anomalous dissipation and energy cascade. Communications in Mathematical Physics, 276(1). pp. 189–220. 10.1007/s00220-007-0333-0 Retrieved from https://hdl.handle.net/10161/24756.

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Mattingly

Jonathan Christopher Mattingly

Kimberly J. Jenkins Distinguished University Professor of New Technologies

Jonathan Christopher  Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day.  He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a Professor of Mathematics and of Statistical Science.

His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.

Since 2013 he has also been working to understand and quantify gerrymandering and its interaction of a region's geopolitical landscape. This has lead him to testify in a number of court cases including in North Carolina, which led to the NC congressional and both NC legislative maps being deemed unconstitutional and replaced for the 2020 elections. 

He is the recipient of a Sloan Fellowship and a PECASE CAREER award.  He is also a fellow of the IMS and the AMS. He was awarded the Defender of Freedom award by  Common Cause for his work on Quantifying Gerrymandering.



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