Weakly dissipative solutions and weak–strong uniqueness for the Navier–Stokes–Smoluchowski system |
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| Institution: | 1. Department of Mechanical and Materials Engineering, Portland State University, Portland, OR 97207, United States;2. Institute for Energy Technology (IFE), Department of Process and Fluid Flow Technology, 2027 Kjeller, Norway;3. École Centrale de Lille, Université Lille Nord de France, 59655 Villeneuve d’Ascq, France;1. IAC–CNR “Mauro Picone” (sezione di Roma), Via dei Taurini, 19 – 00185 Rome, Italy;2. Sorbonne Universités, UPMC Univ Paris 06, Inria, Lab. J.L. Lions UMR 7598, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France;1. IPSO, INRIA, 263, avenue du Général-Leclerc, 35000 Rennes, France;2. IRMAR, Université de Rennes-1, campus de Beaulieu, 35000 Rennes, France;3. IRMAR, CNRS, 35000 Rennes, France |
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| Abstract: | This article deals with a fluid–particle interaction model for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier–Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually. The existence of weakly dissipative solutions is established under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, a weak–strong uniqueness result is established via the relative entropy method yielding that a weakly dissipative solution agrees with a classical solution with the same initial data when such a classical solution exists. |
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| Keywords: | Weak–strong uniqueness Relative entropy Navier–Stokes equations Smoluchowski equation |
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