Variational principle for the Navier-Stokes equations

A variational principle is presented for the Navier-Stokes equations in the case of a contained boundary-driven, homogeneous, incompressible, viscous fluid. Based upon making the fluid’s total viscous dissipation over a given time interval stationary subject to the constraint of the Navier-Stokes equations, the variational problem looks overconstrained and intractable. However, introducing a nonunique velocity decomposition, $\mathit{u}(\mathit{x},t)=\mathit{\phi }(\mathit{x},t)+\mathit{\nu }(\mathit{x},t),$ “opens up” the variational problem so that what is presumed a single allowable point over the velocity domain $\mathit{u}$ corresponding to the unique solution of the Navier-Stokes equations becomes a surface with a saddle point over the extended domain $(\mathit{\phi },\mathit{\nu }).\mathrm{}$ Complementary or dual variational problems can then be constructed to estimate this saddle point value strictly from above as part of a minimization process or below via a maximization procedure. One of these reduced variational principles is the natural and ultimate generalization of the upper bounding problem developed by Doering and Constantin. The other corresponds to the ultimate Busse problem which now acts to lower bound the true dissipation. Crucially, these reduced variational problems require only the solution of a series of linear problems to produce bounds even though their unique intersection is conjectured to correspond to a solution of the nonlinear Navier-Stokes equations.