Alexandros Alexakis École Normale Supérieure One of the ways to quantify mixing efficiency of a flow is by looking at the spatial variation of the mixed scalar described by the dissipation wavenumber k_d. We derive a set of upper bounds on k_d for a scalar field stirred by spatially smooth, divergence-free flows and maintained by a steady source-sink distribution. The bounds yield different regimes for the scaling behavior of k_d. The transition between these regimes is controlled by the value of the Peclet number and the ratio L_u/L_s, where L_u and L_s are respectively, the characteristic lengthscales of the velocity and source fields. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their implications to realistic flows.