Geometric properties of the Maxwell set and a vortex filament structure for Burgers equation
type: Item
instance of: scholarly article
title: Geometric properties of the Maxwell set and a vortex filament structure for Burgers equation
zbMATH Open document ID: 1180.35594
author: Andrew Neate
author: Aubrey Truman
published in: Letters in Mathematical Physics
publication date: 2007-05-31T00:00:00Z
review text: The inviscid limit (\(\mu\to 0\)) of the viscous stochastic Burgers equation \[ \frac{\partial v^\mu}{\partial t}+\left(v^\mu\cdot \nabla\right)v^\mu =\frac{\mu^2}{2}\Delta v^\mu -\nabla V(x)-\epsilon\nabla k_t(x)\dot W_t \] is discussed in terms of the level surfaces of the minimizing Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The geometry of the Maxwell set is examined in terms of the behaviour of the pre-Maxwell set, the pre-caustic and the pre-level surfaces. In particular it is proved that even if initially the fluid flow is irrotational, in the inviscid limit, associated with the advent of the Maxwell set a non-zero vorticity vector forms in the fluid with vortex lines on the Maxwell set. This suggests that in quite general circumstances for small viscosity there is a vortex filament structure near the Maxwell set for both deterministic and stochastic Burgers equations.
reviewed by: Nikita Ratanov
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 76M35
zbMATH DE Number: 5159810
zbMATH Keywords: stochastic Burgers equation
zbMATH Keywords: Maxwell set
zbMATH Keywords: caustic
zbMATH Keywords: vorticity
MaRDI profile type: Publication
full work available at URL: s11005-007-0145-3
OpenAlex ID: W2002856980
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DOI: 10.1007/S11005-007-0145-3
description: scientific article; zbMATH DE number 5159810