Dissipation enhancement by mixing for evolution $p$--Laplacian advection equations

In this paper, we quantitatively consider the enhance-dissipation effect of the advection term to the $p$-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the $p$-Laplacian in the sense of $L^2$ decay, that is, the $L^2$ decay can be arbitrarily fast. The main ingredient of our argument is to understand the underlying iteration structure inherited from the evolution $p$-Laplacian advection equations. This extends the well-known results of the dissipation enhancement result of the linear Laplacian by Constantin, Kiselev, Ryzhik, and Zlatov{s} into a non-linear setting.