A remark on duality solutions for some weakly nonlinear scalar conservation laws

François James (1) , Nicolas Vauchelet (2, 3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LJLL - Laboratoire Jacques-Louis Lions
3 BANG - Nonlinear Analysis for Biology and Geophysical flows

We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following the work of Bouchut and James (Comm. Partial Diff. Eq., 24, 1999), a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. Then we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.

Domaines: Equations aux dérivées partielles [math.AP]

François James, Nicolas Vauchelet. A remark on duality solutions for some weakly nonlinear scalar conservation laws. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2011, 349, pp.657-661. ⟨10.1016/j.crma.2011.05.004⟩. ⟨hal-00591119⟩