Spontaneously stochastic solutions in one-dimensional inviscid systems. (28th June 2016)

• Record Type: Journal Article
• Authors: Mailybaev, Alexei A
• Is Part Of: Nonlinearity. Volume 29:Number 8(2016:Aug.)
• Page Start: 2238
• Page End: 2252
• Publication Date: 2016-06-28
• Subjects: spontaneous stochasticity -- inviscid solutions -- hydrodynamic model -- blowup -- turbulence -- probability measure -- chaos; 76F20 -- 76F55 -- 35L67; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/8/2238
• Languages: English
• ISSNs: 0951-7715

Abstract

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.