Carleton University

Purpose

In the current project, a new cost functional for the variational formulation is proposed, wherein all the intrinsic properties of the most complete version of the Navier-Stokes equation including the non-linear terms are incorporated. To achieve this, one needs to simultaneously consider the Navier-Stokes system and its dual by means of non-convex optimization theory. Due to the presence of a convex function and its Fenchel dual, this new and symmetric energy functional is constructed and used as follows: first, a general framework, in which solutions of the Navier-Stokes system – which is not of the Euler-Lagrange type – are identified as the minima of an appropriately devised non-negative self-dual energy functional. As the second step, such self-dual features are used to develop a systematic approach for a variational resolution and stability of the solutions of the developed equations.