Maxwell's and Stokes' operators associated with elliptic differential complexes

TL;DR

The paper introduces a new method for generating systems of PDEs from elliptic complexes, linking algebraic structures to models in physics like electromagnetism, fluid dynamics, and quantum mechanics.

Contribution

It presents a novel technique to produce PDE systems from elliptic complexes, expanding the modeling toolkit for natural sciences and mathematical physics.

Findings

Generates PDE systems related to physical models from elliptic complexes.

Includes a wide range of equations like Laplace, Navier-Stokes, Maxwell, and Klein-Gordon.

Applicable to higher dimensions and various algebraic structures.

Abstract

We propose a new technique to generate reasonable systems of partial differential equations (PDE) that could be potential candidates for depicting models in natural sciences related to quasi-linear equations. Such systems appear within typical constructions of the Homological Algebra as complexes of differential operators describing compatibility conditions for overdetermined systems of PDE's. The related models can be both steady and evolutionary. Additional assumptions on the ellipticity of the differential complex provide a wide class of elliptic, parabolic and hyperbolic operators that could be generated in this way. In particular, it appears that an essentially large amount of equations related to the modern Mathematical Physics is generated by the de Rham complex of differentials on the exterior differential forms. These includes the elliptic Laplace and Lam\'e type operators; the parabolic heat transfer equation; the Euler type and Navier-Stokes type equations in Hydrodynamics; the hyperbolic wave equation and the Maxwell equations in Electrodynamics; the Klein-Gordon equation in Relativistic Quantum Mechanics; and so on. Our model generation method covers a broad class of generating systems, especially in higher spatial dimensions, due to different basic algebraic structures at play.