A universal total anomalous dissipator

Abstract

For all $\alpha \in (0,1)$, we construct an explicit divergence-free vector field $V\in L_t^{\infty }{C}_x^{\alpha }\cap C_t^{\frac{\alpha }{1-\alpha }}{L}_x^{\infty }$ so that the solutions to the drift-diffusion equations $${\partial }_t{\theta }^{\kappa }-\kappa \Delta {\theta }^{\kappa }+V\cdot \nabla {\theta }^{\kappa }=0$$ exhibit asymptotic total dissipation for all mean-zero initial data: ${\lim }_{\kappa \to 0}\Vert {\theta }^{\kappa }(1,\cdot ){\Vert }_{L^2}=0$. Additionally, we give explicit rates in $\kappa$ and uniform dependence on initial data.

30 pages, 4 figures