论文标题
Sobolev速度场的对流扩散方程
Advection diffusion equations with Sobolev velocity field
论文作者
论文摘要
在本说明中，我们研究了与不可压缩的$ w^{1，p} $速度字段相关的，具有$ p> 2 $的速度。我们提出了有关耗能率的新估计值，并讨论了对上限的应用，以增强耗散速率，密度的$ l^2 $规范的下限以及定量消失的粘度估计。我们论点中采用的关键工具是对规律性结果的传播，来自运输方程的研究，以及将能量耗散率与运输方程的规律性估计联系起来的新结果。最终，我们提供了强调我们估计清晰度的示例。
In this note we study advection diffusion equations associated to incompressible $W^{1,p}$ velocity fields with $p>2$. We present new estimates on the energy dissipation rate and we discuss applications to the study of upper bounds on the enhancing dissipation rate, lower bounds on the $L^2$ norm of the density, and quantitative vanishing viscosity estimates. The key tools employed in our argument are a propagation of regularity result, coming from the study of transport equations, and a new result connecting the energy dissipation rate to regularity estimates for transport equations. Eventually we provide examples which underline the sharpness of our estimates.
