Uniqueness in the Cauchy problem for transport in R2 and R1+2

Suppose that X(x1,x2) is a planar vector field with bounded coefficients and bounded divergence. Suppose in addition that X preserves a volume with Lipschitz continuous density. We prove that there is uniqueness in the Cauchy problem for the transport operator ∂t+X and also uniqueness in the Cauchy problem for X at noncharacteristic hypersurfaces.