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Conservation properties and potential systems of vorticity-type equations

### Abstract

Partial differential equations of the form , involving two vector functions in depending on t, x, y, z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in . The corresponding potential system has a clear physical meaning. For the Maxwell's equations, it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.

© 2014 AIP Publishing LLC

Received 17 April 2013
Accepted 27 February 2014
Published online 24 March 2014

Acknowledgments:
The author is grateful to Martin Oberlack for discussions, and to NSERC of Canada and Alexander von Humboldt Foundation of Germany for research support.

Article outline:

I. INTRODUCTION
II. CONSERVATION LAWS AND POTENTIAL SYSTEMS
A. Divergence-type conservation laws
B. Direct construction of divergence-type conservation laws
C. Potential systems following from divergence-type conservation laws
III. A FAMILY OF DIVERGENCE-TYPE CONSERVATION LAWS OF THE SYSTEM (1.2)
A. The conservation laws
B. Examples
1. Vorticity conservation laws in viscous and inviscid fluid dynamics
2. Magnetic conservation laws in general magnetohydrodynamics
3. Conservation laws of general and vacuum Maxwell's equations
IV. THE DEGREE TWO CONSERVATION LAW STRUCTURE OF EQs. (1.2)
A. General and lower-degree conservation laws
B. Potential equations following from lower-degree conservation laws
1. Three independent variables
2. Four independent variables
C. The system (1.2) as a conservation law of degree two. Resulting potential equations
D. Physical applications
1. Equations of incompressible fluid dynamics
2. Equations of isotropic magnetohydrodynamics. Generalization of the Galas-Bogoyavlenskij potential
3. General and vacuum Maxwell's equations
V. APPLICATIONS TO SYSTEMS IN LOWER DIMENSIONS
A. Potential system for vacuum Maxwell's equations in 2 + 1 dimensions
B. Potential equations and nonlocal conservation laws of MHD equations in three and 2 + 1 dimensions
VI. SUMMARY AND DISCUSSION

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2014-03-24

2016-09-29

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