A conditional three-dimensional computation with no uniqueness claim

The field $u_0$ is smooth and compactly supported in $D$, so it belongs to $H_0^1(D{)}^3$. Its divergence is ${\mathrm{\partial }}_x{\mathrm{\partial }}_y\psi -{\mathrm{\partial }}_y{\mathrm{\partial }}_x\psi ={\chi }^{\mathrm{\prime }}(x){\chi }^{\mathrm{\prime }}(y)\chi (z)-{\chi }^{\mathrm{\prime }}(x){\chi }^{\mathrm{\prime }}(y)\chi (z)=0,$ by direct product differentiation. Thus $u_0\in V_{\sigma }$ and has zero boundary trace.

For any mesh and finite partition, [F1] supplies a discrete trajectory; with $f=0$, its kinetic energy, time jumps and accumulated viscous dissipation obey the budget in Zero-forcing discrete energy budget. If additionally the Axiom of Choice holds and the continuum data admit a strong branch satisfying [F2], the comparison theorem makes that branch unique among energy weak solutions on its interval. Without that additional strong-solution hypothesis, neither the scheme's existence nor its energy budget selects a unique three-dimensional weak trajectory or asserts that the computed flow is turbulent.