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

Alexander Shlapunov, Alexander Polkovnikov, Victor Mironov

TL;DR

This work presents a unifying algebraic approach to generating and analyzing PDE systems via elliptic differential complexes. By constructing Maxwell's and Stokes' type operators on these complexes, both steady and time-dependent variants are developed, with explicit criteria for Petrovskii/Douglis–Nirenberg ellipticity and via parametrices built from generalized Laplacians $\Delta_{j,\boldsymbol{\mu}}$. The de Rham complex is used as a canonical example to connect the abstract framework with classical vector calculus and widely studied physical models, such as electromagnetism, hydrodynamics, and quantum-field contexts, while the Koszul and Dolbeault examples illustrate broader applicability in higher dimensions. The approach yields systematic pathways to solvability results, fundamental solutions, and time-dependent generalizations, offering a versatile toolkit for modeling and analyzing PDEs emerging in mathematical physics.

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 Lamé 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.

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

TL;DR

This work presents a unifying algebraic approach to generating and analyzing PDE systems via elliptic differential complexes. By constructing Maxwell's and Stokes' type operators on these complexes, both steady and time-dependent variants are developed, with explicit criteria for Petrovskii/Douglis–Nirenberg ellipticity and via parametrices built from generalized Laplacians . The de Rham complex is used as a canonical example to connect the abstract framework with classical vector calculus and widely studied physical models, such as electromagnetism, hydrodynamics, and quantum-field contexts, while the Koszul and Dolbeault examples illustrate broader applicability in higher dimensions. The approach yields systematic pathways to solvability results, fundamental solutions, and time-dependent generalizations, offering a versatile toolkit for modeling and analyzing PDEs emerging in mathematical physics.

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 Lamé 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.
Paper Structure (15 sections, 11 theorems, 182 equations)

This paper contains 15 sections, 11 theorems, 182 equations.

Table of Contents

  1. Differential complexes
  2. Differential operators
  3. Compatibility differential complexes
  4. Elliptic differential complexes
  5. The induced complexes and time dependent processes
  6. Maxwell's and Stokes' type operators for differential complexes
  7. Steady Maxwell's and Stokes' type operators for elliptic complexes
  8. Maxwell's and Stokes' type operators for induced elliptic complexes
  9. Parametrices for steady Maxwell' and Stokes' operators.
  10. Parametrices for evolutionary operators
  11. Natural perturbations.
  12. Maxwell's and Stokes' type operators for the de Rham complex
  13. The de Rham complex
  14. Maxwell's and Stokes' type systems in ${\mathbb R}^3$
  15. Some typical elliptic differential complexes

Key Result

Lemma 1.1

Complex eq.complex is elliptic if and only if the mappings $\delta_q: \pi^* E_q \to \pi^* E_q$ are bijective for all $(x,z) \in T^{\ast} X \setminus \{0\}$ and all $0\leq q \leq N$.

Theorems & Definitions (25)

  • Lemma 1.1
  • Lemma 1.3
  • Remark 1.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 15 more