Gaussian measures under the Navier-Stokes and heat flows are mutually singular

We consider the incompressible Navier-Stokes equations on the torus $\mathbb{T}^d$, $d \ge 2$, with translation invariant Gaussian random initial data. We show that, for the Gaussian measure formally given by $μ_0 \sim \exp(-\frac12 \| u \|_{\dot H^α}^2)$ on divergence-free fields, its nonlinear evo