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

Abstract: We propose a regular method for generating consistent systems of partial differential equations (PDEs) that describe a wide class of models in natural sciences. Such systems appear within typical constructions of the Homological Algebra as complexes of differential operators describing compatibility conditions for overdetermined PDEs. Additional assumptions on the ellipticity/parameter-dependent ellipticity of the differential complexes provide a wide range of elliptic, parabolic and hyperbolic operators. In particular, most equations related to modern Mathematical Physics are generated by the de Rham complex of differentials on exterior differential forms. These includes the equations based on elliptic Laplace and Lam\'e type operators; the parabolic heat and mass transfer equations; 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. The advantage of our approach is that this generation method covers a broad class of generating systems, especially in high dimensions, due to different underlying algebraic structures than the conventional ones.

This is joint work with V. L. Mironov and A. N. Polkovnikov.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods