Abstract
In this paper we examine a class of elliptic quasi-variational inequalities, which involve a constraint set and a set-valued map. First, we establish the existence of a solution and the compactness of the solution set. The approach is based on results for an elliptic variational inequality and the Kakutani-Ky Fan fixed point theorem. Next, we prove an existence and compactness result for a quasi-variational-hemivariational inequality. The latter involves a locally Lipschitz continuous functional and a convex potential. Finally, we present an application to the stationary incompressible Navier-Stokes equation with mixed boundary conditions which model a generalized Newtonian fluid of Bingham type.
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Acknowledgements
The authors are greatly indebted to the reviewers for their helpful comments concerning the preliminary version of the paper.
Funding
Project is supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019, and the National Science Centre of Poland under Project No. 2021/41/B/ST1/01636.
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S.D. and S.M. were involved in conceptualization, methodology, and validation. S.M. wrote the first draft of the manuscript, which was reviewed, edited and corrected by S.D. S.D. prepared subsequent versions of the manuscript, these versions were corrected by S.M. S.D. and S.M. were equally involved in writing. All authors read and approved the final manuscript.
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Migórski, S., Dudek, S. A Quasi-Variational-Hemivariational Inequality for Incompressible Navier-Stokes System with Bingham Fluid. Set-Valued Var. Anal 32, 14 (2024). https://doi.org/10.1007/s11228-024-00717-3
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DOI: https://doi.org/10.1007/s11228-024-00717-3