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October  2018, 38(10): 4979-4996. doi: 10.3934/dcds.2018217

## Existence of weak solutions for particle-laden flow with surface tension

 1 Institute of Applied Mathematics and Mechanics of the NASU, 1, Dobrovol'skogo Str., 84100, Sloviansk, Ukraine 2 Department of Mathematics, University of California, Los Angeles, California 90095-1555, USA 3 Vasyl' Stus Donetsk National University, 21, 600-richya Str., 21021, Vinnytsia, Ukraine 4 Department of Mathematics, Duke University, Durham, North Carolina, 27708-0320, USA

* Corresponding author: Roman M. Taranets

Received  October 2017 Revised  May 2018 Published  July 2018

Fund Project: This work was supported in part by NSF grant DMS-1312543, and by a grant from Ministry of Education and Science of Ukraine (0118U003138 to Roman Taranets).

We prove the existence of solutions for a coupled system modeling the flow of a suspension of fluid and negatively buoyant non-colloidal particles in the thin film limit. The equations take the form of a fourth-order non-linear degenerate parabolic equation for the film height $h$ coupled to a second-order degenerate parabolic equation for the particle density $ψ$. We prove the existence of physically relevant solutions, which satisfy the uniform bounds $0 ≤ ψ/h ≤ 1$ and $h ≥ 0$.

Citation: Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
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