AI 中文总结
本文将 Ambrosio-DiPerna-Lions 理论推广至向量场满足 $L^p$ Osgood 条件的输运方程，通过构造加权能量而非对易子消失的方法，证明其有界分布解唯一且可重正化。
AI 中文摘要
本文将 Ambrosio-DiPerna-Lions 理论关于输运方程适定性的研究推广至向量场满足 $L^p$ Osgood 条件的情形，证明了输运方程的有界分布解是唯一的，因此是可重正化的。与经典证明不同，本文未依赖对易子消失，而是证明了由解的 Littlewood-Paley 分解构造的加权能量是守恒的。
英文摘要
In this note we extend the Ambrosio-DiPerna-Lions theory on the well-posedness of the transport equation to the case in which the vector field satisfies an $L^p$ Osgood condition. In particular we show that bounded distributional solutions to the transport equation are unique and thus renormalized. As opposed to the classical proof, we do not rely on the vanishing of a commutator, but we show that a weighted energy built out of the Littlewood-Paley decomposition of the solution is conserved.
Comments18 pages
