An elementary approach to mixing and dissipation enhancement by transport noise

We investigate the mixing properties of solutions to the stochastic transport equation $du=\circ dW\cdot \mathrm{\nabla }u$, where the driving noise $W(t,x)$ is white in time, colored and divergence-free in space. Furthermore, we prove the dissipation enhancement in the presence of a small viscous term. Applying our results, we also derive the mixing properties for a regularized stochastic 2D Euler equation.