Calculus of Variations and Geometric Measure Theory

S. Bianchini - P. Bonicatto - E. Marconi

A Lagrangian approach for scalar multidimensional conservation laws

created by bonicatto on 12 Oct 2017



Inserted: 12 oct 2017
Last Updated: 12 oct 2017

Year: 2017


We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation laws in several space dimensions \[ \begin{cases} \partial_t u + \text{div}_x (\mathbf{f}(u)) = 0 & (t, x) \in (0, +\infty) \times \mathbb R^d,\\ u(0, \cdot) = u_0(\cdot) & t = 0. \end{cases} \] The construction is based on the transport collapse method introduced by Brenier. As a first application we show that if the solution $u$ is continuous, then its hypograph is given by the set \[ \{ (t, x, h) : h \le u_0 (x - f (h)t) \}, \] i.e. it is the translation of each level set of $u_0$ by its characteristic speed.