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

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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 include the equations based on elliptic Laplace and Lamé 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 advantages of our approach are that (1) the proposed method of PDE generation covers a broad class of generated systems, especially in high dimensions, due to different underlying algebraic structures and (2) it enables a direct constructing of the fundamental solutions/parametrices for the generated systems.