Abstract
We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lame system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
Highlights
The problem of describing the dynamics of incompressible viscous fluid is of great importance in applications
In 2006 the Clay Mathematics Institute announced it as the sixth prize millennium problem, see [5]
The dynamics is described by the Navier-Stokes equations and the problem consists in finding a classical solution to the equations
Summary
The problem of describing the dynamics of incompressible viscous fluid is of great importance in applications. The impulse equation of dynamics of (compressible) viscous fluid was formulated in differential form independently by Claude Navier (1827) and George Stokes (1845). This is ρ(u′t + u′xu) = −μ∆u + (λ + μ) ∇ div u − ∇p + f,. In contrast to [5], we believe that the main problem concerning the Navier-Stokes equations consists in removal of this gap, i.e., in specifying adequate function spaces in which both existence and uniqueness theorems are valid. This paper is aimed in elaborating another insight into the Navier-Stokes equations It consists in specifying this problem within the framework of global analysis of elliptic complexes on manifolds. Any differential operator A extends to a continuous mapping of Hs(X , F ) into Hs−m(X , G) for each s ∈ R, and the order m of A is completely determined by this action
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