Transport equation and Cauchy problem for BV vector fields.

*(English)*Zbl 1075.35087The author extends the result by DiPerna and Lions about weak solutions \(w\) of the Cauchy problem for the transport equation \(B\cdot\nabla w=c\), to the case of \(B\) vector field of bounded variation only (DiPerna and Lions studied the case of \(B\) in a Sobolev space). In particular, he proves that every locally bounded weak solution is renormalizable whenever \(B\) has locally bounded variation, and its distributional divergence locally belongs to \(L^1\). The main idea starts from one of the previous improvements of DiPerna’s and Lions’ result, given by Colombini and Lerner. They proved a uniqueness result in the case that the derivative of the vector field is a measure only along one direction, and there is absolute continuity along the other ones.

Using Alberti’s theorem about the rank-one structure of the singular part of the derivative, the present author is able to prove that, in some sense, any vector field with bounded variation asymptotically behaves on a small scale as the vector fields considered by Colombini and Lerner. The renormalization property permits then to prove uniqueness and comparison results for the Cauchy problem. Then, the author uses his results to investigate, in a measure-theoretic framework, the concept of characteristics for the transport equation, and also to study the Lagrangian flow for bounded vector fields of locally bounded variation.

Finally, it is worth noticing that the results of the present paper are applied in other works by the author and collaborators, to give an existence result for systems of conservation laws, in dimensions more than 2, and with only bounded variation initial datum. Such a result considerably improves previous ones.

Using Alberti’s theorem about the rank-one structure of the singular part of the derivative, the present author is able to prove that, in some sense, any vector field with bounded variation asymptotically behaves on a small scale as the vector fields considered by Colombini and Lerner. The renormalization property permits then to prove uniqueness and comparison results for the Cauchy problem. Then, the author uses his results to investigate, in a measure-theoretic framework, the concept of characteristics for the transport equation, and also to study the Lagrangian flow for bounded vector fields of locally bounded variation.

Finally, it is worth noticing that the results of the present paper are applied in other works by the author and collaborators, to give an existence result for systems of conservation laws, in dimensions more than 2, and with only bounded variation initial datum. Such a result considerably improves previous ones.

Reviewer: Fabio Bagagiolo (Trento)

##### MSC:

35Q72 | Other PDE from mechanics (MSC2000) |

35K15 | Initial value problems for second-order parabolic equations |

35L65 | Hyperbolic conservation laws |

##### Keywords:

transport equation; bounded variation vector fields; Lagrangian flow; renormalization; uniqueness; comparison; characteristic
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##### References:

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