The paper is concerned with the existence and uniqueness of solutions to the Cauchy problem for transport equations on \(\mathbb{R}^+\times\mathbb{R}^n,\) \[ \begin{cases} \partial_{t}u(t,x) +b(t,x) \cdot\nabla_{x}u(t,x) =0,\\ u( 0,x) =\bar{u}(x) , \end{cases} \] where \(b:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}^n\) is a given smooth vector field, \(\bar{u}\) is a given smooth initial condition and \(u\) is an unknown function. If \(b\) is a Lipschitz continuous function, the answer to the question is given by the classical Cauchy-Lipschitz theorem. However, many physical phenomena lead to problems with discontinuous coefficients (one may think, for instance, of jump discontinuities in coefficients in the problems arising in the theory of shock waves).
The purpose of the paper is to illustrate the most important ideas of the theory of generalized solutions to transport equations and ordinary differential equations with Sobolev coefficients introduced by R. J. DiPerna and P. L. Lions [Invent. Math. 98, No. 3, 511–547 (1989; Zbl 0696.34049)] and its extensions to BV functions (summable functions whose derivatives are Radon measures) obtained by L. Ambrosio [Invent. Math. 158, No. 2, 227–260 (2004; Zbl 1075.35087)]. Further developments of the theory, conjectures and open problems in three different research directions are also discussed.
For the entire collection see [Zbl 1151.00015].