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A universal total anomalous dissipator (2501.18526v1)
Published 30 Jan 2025 in math.AP and math.PR
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: $\lim_{\kappa\rightarrow 0}|\theta\kappa(1,\cdot)|_{L2}=0$. Additionally, we give explicit rates in $\kappa$ and uniform dependence on initial data.