Spontaneously stochastic solutions in one-dimensional inviscid systems

TL;DR

This paper demonstrates that solutions to the Sabra shell model of turbulence transition from deterministic to stochastic after finite-time blowup, supported by theoretical analysis and numerical evidence, highlighting universal stochastic behavior post-blowup.

Contribution

It introduces a novel theoretical framework showing the transition from deterministic to stochastic solutions in the inviscid limit of the Sabra model, supported by numerical confirmation.

Findings

Solutions become stochastic after blowup in the Sabra model.

Theoretical and numerical evidence supports the transition to stochasticity.

Universal onset of stochasticity immediately after blowup.

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 $\tau = \log(t-t_b)$, 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.