On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field

Abstract

Given a bounded, autonomous vector field \(\mathbf b :\mathbb {R}^d \rightarrow \mathbb {R}^d\), we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation $$\begin{aligned} \partial _t u + \mathbf b \cdot \nabla u = 0. \end{aligned}$$ (1) This problem is related to a conjecture made by A. Bressan, raised while studying the well-posedness of a class of hyperbolic conservation laws. Furthermore, from the Lagrangian point of view, this gives insights on the structure of the flow of non-smooth vector fields. In this work, we will discuss the two-dimensional case and we prove that, if \(d=2\), uniqueness of weak solutions for (1) holds under the assumptions that \(\mathbf b\) is of class \(\mathrm {BV}\) and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa in J Eur Math Soc (JEMS) 16(2):201–234, 2014, [2]) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem. This is joint work with S. Bianchini and N.A. Gusev (SIAM J Math Anal 48(1):1–33, 2016), [6].