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We show the existence of a velocity field $v$, solution of (randomly) forcedNavier-Stokes equations, which produces total dissipation of kinetic energy infinite time when advecting a passive scalar $\rho$. The total dissipation holdstrue uniformly in the viscosity parameter and the initial conditions $\rho_0$,in particular the dissipation is anomalous. Dissipation induced by singlerealizations of $v$ is also discussed. Our results extend to the case when$\rho$ is replaced by a solution to the two or three dimensional(deterministic) Navier-Stokes equations advected by $v$.

Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations