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Structure of shocks in Burgers turbulence with stable noise initial data


Bertoin, Jean (1999). Structure of shocks in Burgers turbulence with stable noise initial data. Communications in Mathematical Physics, 203(3):729-741.

Abstract

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval.

Abstract

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1999
Deposited On:25 Jul 2013 06:47
Last Modified:05 Apr 2016 16:52
Publisher:Springer
ISSN:0010-3616
Publisher DOI:https://doi.org/10.1007/s002200050633
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1700933
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0943.60055

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