On the tangential set of a nearly incompressible vector field

In the theory of transport equation arises a problem of representation of a distribution div(β(u)b) where u is a scalar field, b is a vector field with bounded variation and β is a smooth function. Such representation has been obtained by L. Ambrosio, C. De Lellis and J. Maly, up to a singular measure with particular properties. The authors have also obtained a sufficient condition when this measure is zero. In connection with the Bressan's compactness conjecture they have posed a question of existence of a nearly incompressible vector field for which this sufficient condition is not satisfied. An example of such vector field, which has been recently constructed, will be presented.