On flows generated by square-integrable vector fields

For square-integrable divergence-free vector field $\mathit{v}$ on ${\mathbb{R}}^d$ we prove that the following properties are equivalent: 1) the operator $A_0\rho =\mathit{v}\cdot \mathrm{\nabla }\rho$ (where $\rho \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$) 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 $\mathit{v}\text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ 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.