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 (Formula Presented). 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 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 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].
On uniqueness of weak solutions to transport equation with non-smooth velocity field / Bonicatto, P.. - 236:(2018), pp. 191-203. (Intervento presentato al convegno 16th International Conference on Hyperbolic Problems: Theory, Numerics and Applications, 2016 tenutosi a deu nel 2016) [10.1007/978-3-319-91545-6_15].
On uniqueness of weak solutions to transport equation with non-smooth velocity field
Bonicatto P.
2018-01-01
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 (Formula Presented). 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 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 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].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione
