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arXiv · 2611.01130

Local and global conservation laws of partial differential equations on moving domains in three dimensions

Alexey F. Shevyakov·Stephen C. Anco
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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.

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Alexey F. Shevyakov, Stephen C. Anco. 2026-10-31. Local and global conservation laws of partial differential equations on moving domains in three dimensions. https://arxiv.org/abs/2611.01130

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