Title | Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions |

Publication Type | Journal Article |

Year of Publication | 2016 |

Authors | Bianchini, S, Bonicatto, P, Gusev, NA |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 48 |

Pagination | 1-33 |

Abstract | Given a bounded autonomous vector field $b \colon \mathbb{R}^d \to \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, 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 [7] (where the steady 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]. |

URL | https://doi.org/10.1137/15M1007380 |

DOI | 10.1137/15M1007380 |

## Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions

Research Group: