AI 中文总结
该研究证明ℝᵈ上平方可积无散向量场的三个性质等价，还构造出一类特殊向量场，其对应连续性方程柯西问题的解时间正向唯一、反向不唯一。
AI 中文摘要
针对ℝᵈ上的平方可积无散向量场$\boldsymbol{v}$，我们证明以下性质是等价的：1）算子$A_0 \rho = \boldsymbol{v} \bullet \nabla \rho$（其中$\rho \boldsymbol{\u2208} C^\u221e_c(\u211d^d)$）在$L^2(\u211d^d)$上是本质斜自伴的；2）连续性方程的平方可积（关于空间变量）广义解是重正规化的；3）对应的连续性方程柯西问题的平方可积（关于空间变量）广义解在时间正向和反向均唯一。我们还构造了一个具有紧支集的有界无散向量场$\boldsymbol{v}\boldsymbol{\u223a} \u211d^3 \to \u211d^3$，对于该向量场，对应的连续性方程柯西问题的平方可积（关于空间变量）解在时间正向是唯一的，但在时间反向不唯一。
英文摘要
For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 ρ= \boldsymbol{v} \cdot \nabla ρ$ (where $ρ\in C^\infty_c(\mathbb{R}^d)$) is essentially skew-adjoint on $L^2(\mathbb{R}^d)$; 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field $\boldsymbol{v}\colon \mathbb{R}^3 \to \mathbb{R}^3$ for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.
Comments27 pages, 4 figures
