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arXiv · 1512.04465

Spontaneously stochastic solutions in one-dimensional inviscid systems

Alexei A. Mailybaev
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Abstract

In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finite-time blowup, $t < t_b$, must be continued as a stochastic process after the blowup, $t > t_b$, representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for the logarithmic time $τ= \log(t-t_b)$, which features a stable traveling wave solution for the inviscid Burgers equation, but a stochastic traveling wave for the Sabra model. The latter describes a universal onset of stochasticity immediately after the blowup.

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Alexei A. Mailybaev. 2016-06-28. Spontaneously stochastic solutions in one-dimensional inviscid systems. https://doi.org/10.1088/0951-7715%2F29%2F8%2F2238

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