Well-posedness for a one-dimensional fluid-particle interaction model

Boris Andreianov (1) , Frédéric Lagoutière (2) , Nicolas Seguin (3, 4) , Takéo Takahashi (5, 6)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LJLL - Laboratoire Jacques-Louis Lions
4 ANGE - Numerical Analysis, Geophysics and Ecology
5 CORIDA - Robust control of infinite dimensional systems and applications
6 IECL - Institut Élie Cartan de Lorraine

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.

Mots clés: Burgers equation; Fluid-particle interaction; Splitting; Non-conservative coupling; Wave-front tracking; Well-posedness; BV estimates; Fixed-Point

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Well-posedness for a one-dimensional fluid-particle interaction model. SIAM Journal on Mathematical Analysis, 2014, 46 (2). ⟨hal-00789315⟩