On flows generated by square-integrable vector fields

Abstract

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

27 pages, 4 figures