Mixing by Statistically Self-similar Gaussian Random Fields

We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on \(\mathbb {R}^d\) ${\mathbb{R}}^d$ . If the velocity field u is homogeneous, isotropic, and statistically self-similar, we derive an exact formula which captures non-diffusive mixing. For zero diffusivity, the formula takes the shape of \(\mathbb {E}\ \Vert \theta _t \Vert _{\dot{H}^{-s}}^2 = \textrm{e}^{-\lambda _{d,s} t} \Vert \theta _0 \Vert _{\dot{H}^{-s}}^2\) $\mathbb{E}\Vert {\theta }_t{\Vert }_{{\dot{H}}^{-s}}^2={\text{e}}^{-{\lambda }_{d,s}t}{\Vert {\theta }_0\Vert }_{{\dot{H}}^{-s}}^2$ with any \(s\in (0,d/2)\) $s\in (0,d/2)$ and \(\frac{\lambda _{d,s}}{D_1}:= s(\frac{\lambda _{1}}{D_1}-2s)\) $\frac{{\lambda }_{d,s}}{D_1}:=s(\frac{{\lambda }_1}{D_1}-2s)$ where \(\lambda _1/D_1 = d\) ${\lambda }_1/D_1=d$ is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and \( D_1\) $D_1$ is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity.