A universal total anomalous dissipator

TL;DR

This paper constructs a divergence-free vector field demonstrating total dissipation in solutions to drift-diffusion equations for all mean-zero initial data, with explicit rates and uniform dependence on initial conditions.

Contribution

It provides an explicit example of a divergence-free vector field causing total dissipation in drift-diffusion equations across all relevant parameters.

Findings

Solutions exhibit asymptotic total dissipation as diffusivity vanishes.

Explicit rates of dissipation in terms of diffusivity parameter.

Uniform dependence of dissipation on initial data.

Abstract

For all $\alpha\in(0,1)$, we construct an explicit divergence-free vector field $V\in L^\infty_tC^\alpha_x \cap C^{\frac{\alpha}{1-\alpha}}_t L^\infty_x$ 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: $\lim_{\kappa\rightarrow 0}\|\theta^\kappa(1,\cdot)\|_{L^2}=0$. Additionally, we give explicit rates in $\kappa$ and uniform dependence on initial data.