On Well-Posedness for a Multi-particle Fluid Model

Abstract

In this paper, we study a one-dimensional fluid modelled by the Burgers equation influenced by an arbitrary but finite number of particles N(t) moving inside the fluid, each one acting as a point-wise drag force with a particle-related friction constant $\lambda$. For given particle paths $h_i(t)$, we only assume finite speed of particles, allowing for crossing, merging and splitting of particles. This model is an extension of existing models for fluid interactions with a single particle; compare (Andreianov et al., SIAM J Math Anal 46(2):1030–1052, 2014, [3], Lagoutière et al., J Differ Equ 245(11):3503–3544, 2008, [10]):

$$ \partial _t u(x,t) + \partial _x \left( \frac{u^2}{2}\right) = \sum _{i=1}^N \lambda (h_i'(t)-u(t,h_i(t))\delta (x-h_i(t))$$

Well-posedness for the Cauchy problem, as well as an $L^\infty$ bound, is proven under the weak assumption that particle paths are Lipschitz continuous. In this context, an entropy admissibility criteria are shown, using the theory of $L^1$-dissipative germs, compare (Andreianov et al., Arch Ration Mech Anal 201:26–86, 2011, [2]), to deal with the moving interfaces resulting from the point-wise particles and the shock waves from the fluid equation interacting with them.