Structure of shocks in Burgers turbulence with stable noise initial data
type: Item
instance of: scholarly article
title: Structure of shocks in Burgers turbulence with stable noise initial data
zbMATH Open document ID: 0943.60055
published in: Communications in Mathematical Physics
publication date: 1999-12-16T00:00:00Z
review text: The author considers the Hopf-Cole solution \(u=u_0\) of the one-dimensional inviscid equation with the initial velocity given by a stable noise (the derivative in the Schwartz sense of a stable Lévy process) with index \(\alpha\in(1/2,2]\). (The solution is defined as the limit \(u_0=\lim_{\varepsilon\downarrow 0}u_\varepsilon u\), where \(u_\varepsilon\) is a unique solution of the Burgers equation \(\partial_t u+\partial(u^2/2)=\varepsilon\partial^2_{xx}u\).) It is proved that Lagrangian regular points exist (i.e., there are fluid particles that have not participated in shocks in the time interval \([0,t)\)) if and only if the noise is not completely asymmetric; otherwise, the shock structure is discrete. Moreover, in the Cauchy case \(\alpha=1\), it is shown that there are no rarefaction intervals, i.e., at any time \(t\), the locations of the fluid particles form an everywhere dense set a.s.
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1379945
zbMATH Keywords: Burgers equation
zbMATH Keywords: stable process
zbMATH Keywords: Hopf-Cole solution
author: Jean Bertoin
reviewed by: Vigirdas Mackevičius
MaRDI profile type: Publication
full work available at URL: s002200050633
OpenAlex ID: W2044596548
DOI: 10.1007/S002200050633
description: scientific article; zbMATH DE number 1379945