Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions

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 ∂ₜ u + 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 present work extends the result obtained in [S. Bianchini and N. A. Gusev, Steady nearly incompressible vector fields in $2$D: Chain rule and Renormalization, preprint, 2014], 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 [G. Alberti, S. Bianchini, and G. Crippa, J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 201--234], using the results on the structure of level sets of Lipschitz maps obtained in [G. Alberti, S. Bianchini, and G. Crippa, Ann. Sc. Norm. Super. Pisa Cl. Sci.(s), 12 (2013), pp. 863--902]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [L. Ambrosio, Invent. Math., 158 (2004), pp. 227--260].