Renormalization for autonomous nearly incompressible BV vector fields in two dimensions

Given a bounded autonomous vector field b : R d → R d , we study the uniqueness of bounded solutions to the initial value problem for the related transport equation ∂ t u + b · ∇u = 0. We are interested in the case where b is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension d = 2, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio’s superposition principle [4].