On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients

We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients $a(x)$. It is shown that any generalized solution satisfies the renormalization property if and only if the operator $a\cdot \mathrm{\nabla }u$, $u\in C_0^1({\mathbb{R}}^n)$ in the Hilbert space $L^2({\mathbb{R}}^n)$ is an essentially skew-adjoint operator, and this is equivalent to the uniqueness of generalized solutions. We also establish existence of a contractive semigroup, which provides generalized solutions, and give a criterion of its uniqueness.