Local and global conservation laws of partial differential equations on moving domains in three dimensions
Abstract
For PDE systems in three spatial dimensions, firstly, a review of the various types of global conservation laws --- volumetric, surface-flux, circulatory --- is given using vector calculus, and their corresponding local forms --- divergence, curl, gradient --- are explained. Secondly, an extension of each type of conservation law to moving domains is presented, where the domain --- volume, surface, curve --- is transported by a prescribed velocity field. This provides a generalization of Reynolds' transport theorem known for material domains in which the velocity field is the fluid velocity. The main results are illustrated by numerous physical examples taken from gas dynamics, fluid flow, nonlinear acoustics, chemotaxis, plasma dynamics, and also Maxwell's equations, which do not involve any physically specified velocity field. As an additional main result, an extension of the multiplier method for finding local conservation laws of divergence-type is presented for local curl-type and gradient-type conservation laws on both static and moving domains.
Related
- Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions
- Mean field singular stochastic partial differential equations
- A new family of Darboux integrable partial differential equations
- A semigroup approach for rough partial differential equations with jumps
- The Neumann problem for stochastic partial differential equations
BibTeX
@misc{shevyakov2026local,
title = {{Local and global conservation laws of partial differential equations on moving domains in three dimensions}},
author = {Alexey F. Shevyakov and Stephen C. Anco},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01130},
eprint = {2611.01130},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01130},
}
RIS
TY - UNPB
TI - Local and global conservation laws of partial differential equations on moving domains in three dimensions
AU - Alexey F. Shevyakov
AU - Stephen C. Anco
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01130
UR - https://arxiv.org/abs/2611.01130
AB - For PDE systems in three spatial dimensions, firstly, a review of the various types of global conservation laws --- volumetric, surface-flux, circulatory --- is given using vector calculus, and their corresponding local forms --- divergence, curl, gradient --- are explained. Secondly, an extension of each type of conservation law to moving domains is presented, where the domain --- volume, surface, curve --- is transported by a prescribed velocity field. This provides a generalization of Reynolds' transport theorem known for material domains in which the velocity field is the fluid velocity. The main results are illustrated by numerous physical examples taken from gas dynamics, fluid flow, nonlinear acoustics, chemotaxis, plasma dynamics, and also Maxwell's equations, which do not involve any physically specified velocity field. As an additional main result, an extension of the multiplier method for finding local conservation laws of divergence-type is presented for local curl-type and gradient-type conservation laws on both static and moving domains.
ER -