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On uniqueness of weak solutions to transport equation with non-smooth velocity field
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bonicatto, Paolo |
| Copyright Year | 2017 |
| Abstract | Given a bounded, autonomous vector field b : Rd → Rd , we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation ∂tu+b ·∇u = 0. (1) This problem is related to a conjecture made by A. Bressan, raised 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 nonsmooth 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 b is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa in [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 [5]. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://cvgmt.sns.it/media/doc/paper/3618/Hyp16-Proceedings_PB.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Analog Kinetic data structure Solutions Universal quantification Velocity (software development) Well-posed problem Whole Earth 'Lectronic Link |
| Content Type | Text |
| Resource Type | Article |
