Transport equations with partially BV velocities

Abstract

In this article, we prove the uniqueness of weak solutions for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the vector field X = a1(x1) .?x1 + a2(x1,x2) .?x2 where a1(x1) is a BV function and a2(x1,x2) is only L1 with respect to x1 and BV with respect to x2, with a boundedness condition on the divergence of each vector field a1, a2. This model was studied in a recent paper by P.-L.Lions and C.Le Bris with a W1,1 regularity assumption replacing our BV hypothesis. This settles partly a question raised in a forthcoming paper by L.Ambrosio. We examine the details of the argument of that article and we combine some consequences of the Alberti rank-one structure theorem for BV vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a BV function.