Abstract
This paper proves existence of a global weak solution to the inhomogeneous (i.e., non-constant density) incompressible Navier–Stokes system with mass diffusion. The system is well-known as the Kazhikhov–Smagulov model. The major novelty of the paper is to deal with the Kazhikhov–Smagulov model possessing the non-constant viscosity without any simplification of higher order nonlinearity. Every global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of Aubin–Lions–Simon type.
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Acknowledgements
This work was written during Kohei Soga’s one-year research stay in Fachbereich Mathematik, Technische Universität Darmstadt, Germany, with the grant Fukuzawa Fund (Keio Gijuku Fukuzawa Memorial Fund for the Advancement of Education and Research). He would like to express special thanks to Professor Dieter Bothe for his kind hosting in TU-Darmstadt. This work is also supported by JSPS Grant-in-aid for Young Scientists #18K13443 and JSPS Grants-in-Aid for Scientific Research (C) #22K03391.
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Kacedan, E., Soga, K. Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion. Z. Angew. Math. Phys. 75, 65 (2024). https://doi.org/10.1007/s00033-024-02209-8
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DOI: https://doi.org/10.1007/s00033-024-02209-8