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Well-posedness for a one-dimensional fluid-particle interaction model. (English) Zbl 1302.35263
The fluid-particle interaction model introduced by the three last authors in [F. Lagoutière et al., J. Differ. Equations 245, No. 11, 3503–3544 (2008; Zbl 1151.76033)] is the object of this 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, K. H. Karlsen and N. H. Risebro in [Arch. Ration. Mech. Anal. 201, No. 1, 27–86 (2011; Zbl 1261.35088)]. In this paper, the authors prove well-posedness and justify an approximation strategy for the particle in Burgers system in the case of initial data of bounded variation. The existence results for \(L^\infty\) data is also given.

MSC:
35L81 Singular hyperbolic equations
35R06 PDEs with measure
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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