Variational Dual Solutions for Incompressible Fluids

We consider a construction proposed in \cite{acharyaQAM} that builds on the notion of weak solutions for incompressible fluids to provide a scheme that generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them to recover standard solutions. The scheme provides a generalisation of a construction of Y$\mathrm{.}$Brenier for the Euler equations. We rigorously analyze the scheme, extending the work of Y$\mathrm{.}$Brenier for Euler, and also provide an extension of it to the case of the Navier-Stokes equations. Furthermore we obtain the inviscid limit of Navier-Stokes to Euler as a $\mathrm{\Gamma }$-limit.