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.
Skip Nav Destination
Article navigation
March 2014
Research Article|
March 24 2014
Conservation properties and potential systems of vorticity-type equations Available to Purchase
Alexei F. Cheviakov
Alexei F. Cheviakov
a)
Department of Mathematics and Statistics,
University of Saskatchewan
, Saskatoon, Saskatchewan, S7N 5E6, Canada
Search for other works by this author on:
Alexei F. Cheviakov
a)
Department of Mathematics and Statistics,
University of Saskatchewan
, Saskatoon, Saskatchewan, S7N 5E6, Canada
a)
Electronic mail: [email protected]
J. Math. Phys. 55, 033508 (2014)
Article history
Received:
April 17 2013
Accepted:
February 27 2014
Citation
Alexei F. Cheviakov; Conservation properties and potential systems of vorticity-type equations. J. Math. Phys. 1 March 2014; 55 (3): 033508. https://doi.org/10.1063/1.4868218
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity
Ramón G. Plaza, Delyan Zhelyazov
Connecting stochastic optimal control and reinforcement learning
J. Quer, Enric Ribera Borrell
Related Content
Multidimensional partial differential equation systems: Nonlocal symmetries, nonlocal conservation laws, exact solutions
J. Math. Phys. (October 2010)
Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems
J. Math. Phys. (October 2010)
On locally and nonlocally related potential systems
J. Math. Phys. (July 2010)
Nonlocal symmetries classifications and exact solution of Chaplygin gas equations
J. Math. Phys. (August 2018)
New similarity reductions and exact solutions for helically symmetric viscous flows
Physics of Fluids (May 2020)