On flows generated by square-integrable vector fields

Abstract: For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}<sup>d$ we prove that the following properties are equivalent: 1) the operator $A_0 ρ= \boldsymbol{v} \cdot \nabla ρ$ (where $ρ\in C<sup>\infty_c(\mathbb{R}<sup>d)$) is essentially skew-adjoint on $L<sup>2(\mathbb{R}<sup>d)$; 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 $$\boldsymbol{v}\colon \mathbb{R}³ \to \mathbb{R}<sup>3$$ 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.