Renormalization for Autonomous Nearly Incompressible Bv Vector Fields in 2d

Given a bounded autonomous vector field b : R2 → R2, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation ∂tu + b · ∇u = 0. Assuming that b is of class BV and it is nearly incompressible, 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 [10] (where the steady nearly incompressible 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].