Publications [#139685] of Jonathan C. Mattingly

Papers Published

1. Mattingly, Jonathan C. and Suidan, Toufic M. and Vanden-Eijnden, Eric, Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators, Journal of Statistical Physics, vol. 128 no. 5 (2007), pp. 1145--1152, ISSN 0022-4715 [MR2348788], [math-ph/0607047]
   (last updated on 2007/12/16)

   Abstract:
   We analyze a class of linear shell models subject to stochastic forcing in finitely many degrees of freedom. The unforced systems considered formally conserve 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 is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients. The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes ($a_n$) with higher $n$; this is responsible for solutions with interesting energy spectra, namely $\EE |a_n|^2$ scales as $n^{-\alpha}$ as $n\to\infty$. Here the exponents $\alpha$ depend on the coupling coefficients $c_n$ and $\EE$ 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.