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PDEs & Applications Seminar

Friday, February 14, 2025

Time: 10:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Alexey Shevyakov (University of Saskatchewan)

Title: Conservation laws of differential equations: computation, connections, and applications

Abstract: Local conservation laws of a system of differential equations (DE) are given by one or several divergence expressions that hold on solutions of that system. For ordinary differential equations, conservation laws lead to first integrals and the reduction of order. For partial differential equations (PDE), they provide globally conserved quantities, such as energy, momentum, as well as more exotic ones. Conservation laws used for analysis of global solution behaviour, are related to multiple other analytical properties of PDEs, and play an important role in the numerical treatment of PDEs.

In this talk, we will review the general theory, including trivial and equivalent conservation laws, their characteristic form, relationships with integrability, symmetries of DEs, Hamiltonians, variational systems, Lagrangians, and the first and second Noether's theorems. A systematic procedure to seek conservation laws will be discussed, applicable to virtually any PDE system; it will be compared to the Noether's theorem approach to seek conservation laws of variational models. A symbolic implementation of the direct method of conservation law computation in Maple will be presented. Examples of conservation laws and conserved quantities for classical PDEs and some nonlinear models arising in contemporary work will be discussed.

Time permitting, we will consider a common framework for different types of conservation laws of PDE systems in three space dimensions, including their global and local formulations in static and moving domains given by volumes, surfaces, and curves.