DISSIPATION ENHANCEMENT BY SHEAR FLOWS FOR GENERALIZED DIFFUSION OPERATORS IN HIGH DIMENSION

We establish conditions for shear flows on the d-dimensional torus T^d, d≥2, that give enhanced dissipation for the associated linear advection-diffusion equation for well-prepared data. The diffusion operator can be of fractional or high order and does not need to have constant coefficients. We then construct flows that satisfy these assumptions and obtain a quantitative estimate on the dissipation enhancement. Our examples generalize known examples in two space dimensions to the high-dimensional setting, which is relevant in applications to sampling a distribution and in optimization.