Minimax principle on energy dissipation of incompressible shear flow

The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin’s variational principle characterizes the upper bounds (maximum) of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical point of view was adopted to recast Doering-Constantin’s formulation into a minimax principle for the energy dissipation of an incompressible shear flow. Then, the Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin’s variational principle and Howard-Busse’s statistical turbulence theory.