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

Abstract

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 evolution $ν_t$ and its linear evolution $μ_t$ are mutually singular for almost every time. Moreover, the same holds for the nonlinear evolution of an arbitrary initial measure $ν_0$. This is in sharp contrast to the available results for Hamiltonian PDEs, where Gaussian measures are commonly quasi-invariant. We achieve this by showing that there exists a set $\mathscr G_t$ such that $μ_t(\mathscr G_t) = 1$, but for any smooth solution of Navier-Stokes, the set of times $t$ such that $u(t) \in \mathscr G_t$ is null, which we deduce from a quantitative convexity bound for frequency-localised versions of the (typically infinite) quantity $\| \mathrm{e}^{-t Δ} u(t) \|_{L^2}^2$.