Monotonicity and vanishing-minimum-energy conjecture for discrete cavity flow

Fix the boundary velocity $v$ and the discretization $N$. Let $F$ be the kinetic-energy objective, let $u_0(v,N)$ denote the solution at $R=0$, and let $E_{\min}(R,v,N)$ be the minimum kinetic energy among solutions of the discrete cavity flow problem at Reynolds number $R$.

Minimum-energy conjecture. For every $R\geq 0$,

$$F(u_0(v,N))=E_{\min}(0,v,N)\geq E_{\min}(R,v,N)\geq 0,$$

and

$$E_{\min}(R,v,N)\longrightarrow 0\quad\text{as }R\longrightarrow\infty.$$

The conjecture is motivated by the numerical observation that the minimum energy decreases with $R$. The supplied text says that the vanishing of the minimum kinetic energy for increasing $R$ is left for future research; no resolution is given.