On flows generated by square-integrable vector fields

Abstract

For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 \rho = \boldsymbol{v} \cdot \nabla \rho$ (where $\rho \in C^\infty_c(\mathbb{R}^d)$) 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.