Abstract
In this paper, we study existence, dependence, and optimal control results concerning solutions to a class of hemivariational inequalities for stationary Navier–Stokes equations but without making use of the theory of pseudo-monotone operators. To do so, we consider a classical assumption, due to Rauch, which constrains us to make a slight change on the definition of a solution. The Rauch assumption, although it insures the existence of a solution, does not allow the conclusion that the non-convex functional is locally Lipschitz. Moreover, two dependence results are proved, one with respect to changes of the boundary condition and the other with respect to the density of external forces. The later one will be used to prove the existence of an optimal control to the distributed parameter optimal control problem where the control is represented by the external forces.
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Al-Homidan, S., Ansari, Q.H., Chadli, O.: Noncoercive stationary Navier–Stokes equations of heat-conducting fluids modeled by hemivariational inequalities: an equilibrium problem approach. Results Math. 74(132), 1–31 (2019)
Alekseev, G.V., Smishliaev, A.B.: Solvability of the boundary-value problems for the Boussinesq equations with inhomogeneous boundary conditions. J. Math. Fluid Mech. 3(1), 18–39 (2001)
Aubin, J.P.: Applied Functional Analysis. Wiley, New York (1979)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Berlin (1990)
Ben Aadi, S., Chadli, O., Koukkous, A.: Evolution hemivariational inequalities for non-stationary Navier–Stokes equations: existence of periodic solutions by an equilibrium problem approach. Minimax Theory Appl. 3(1), 107–130 (2018)
Bykhovskii, É.B., Smirnov, N.V.E.: Orthogonal decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis. Trudy Matematicheskogo Instituta imeni VA Steklova 59, 5–36 (1960)
Chang, K.C.: Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80(1), 102–129 (1981)
Chebotarev, A.Y.: Subdifferential boundary value problems for stationary Navier–Stokes equations. Differentsial’nye Uravneniya 28(8), 1443–1450 (1992)
Chebotarev, A.Y.: Stationary variational inequalities in the model of inhomogeneous incompressible fluids. Sibirsk. Math. Zh. (Siberian Math. J.) 38, 1184–1193 (1997)
Chebotarev, A.Y.: Variational Inequalities for Navier–Stokes type operators and one-sided problems for equations of viscous heat-conducting fluids. Math. Notes 70(1–2), 264–274 (2001)
Chebotarev, A.Y.: Modeling of steady flows in a channel by Navier–Stokes variational inequalities. J. Appl. Mech. Tech. Phys. 44, 852–857 (2003)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
Clarke, F.H.: Generalized gradient and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)
Dudek, S., Kalita, P., Migórski, S.: Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law. Comput. Math. Appl. 74(8), 1813–1825 (2017)
Dudek, S., Kalita, P., Migórski, S.: Stationary flow of non-Newtonian fluid with nonmonotone frictional boundary conditions. Z. Angew. Math. Phys. 66(5), 2625–2646 (2015)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Studies in Mathematics and its Applications. North-Holland, Amsterdam (1976)
Fang, C., Czuprynski, K., Han, W., Cheng, X., Dai, X.: Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition. IMA J. Numer. Anal. 00, 1–21 (2019)
Fang, C., Han, W., Migórski, S.: A class of hemivariational inequalities for nonstationary Navier–Stokes equations. Nonlinear Anal. Ser. B Real World Appl. 31, 257–276 (2016)
Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Science and Business Media, New York (2012)
Kalita, P., Łukaszewicz, G.: On Large Time Asymptotics for Two Classes of Contact Problems. In Advances in Variational and Hemivariational Inequalities, pp. 299–332. Springer, Cham (2015)
Kalita, P., Łukaszewicz, G.: Attractors for Navier–Stokes flows with multivalued and nonmonotone subdifferential boundary conditions. Nonlinear Anal. Real World Appl. 19, 75–88 (2014)
Konovalova, D.S.: Subdifferential boundary value problems for evolution Navier–Stokes equations. Differ. Uravn. (Differential Equations) 36, 792–798 (2000)
Lions, J.L.: Quelques méthodes de résolution des problemes aux limites non linéaires. Dunod, Paris (1969)
Migórski, S., Dudek, S.: Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone Friction law. J. Math. Fluid Mech. 20(3), 1317–1333 (2018)
Migórski, S., Paczka, D.: Frictional contact problems for steady flow of incompressible fluids in Orlicz spaces. In: Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham, pp. 1–53 (2019)
Migórski, S., Paczka, D.: On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces. Nonlinear Anal. Real World Appl. 39, 337–361 (2018)
Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)
Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)
Migórski, S., Ochal, A.: Navier–Stokes problems modelled by evolution hemivariational inequalities. Discrete Contin. Dyn. Syst. Supplement, 731–740 (2007)
Migórski, S.: A note on optimal control problem for a hemivariational inequality modeling fluid flow. Dyn. Syst., 545–554 (2013)
Migórski, S., Szafraniec, P.: Nonmonotone slip problem for miscible liquids. J. Math. Anal. Appl. 471(1–2), 342–357 (2019)
Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5(2), 217–228 (2004)
Moreau, J.J., Panagiotopoulos, P.D.: Nonsmooth Mechanics and Applications. CISM Courses and Lectures, vol. 302. Springer, Wien, pp. 81–176 (1988)
Naniewicz, Z.: Hemivariational inequalities with functions fulfilling directional growth condition. Appl. Anal. 55(3–4), 259–285 (1994)
Naniewicz, Z.: Hemivariational inequalities with functionals which are not locally Lipschitz. Nonlinear Anal. Theory Methods Appl. 25(12), 1307–1320 (1995)
Ovcharova, N., Gwinner, J.: A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities. J. Optim. Theory Appl. 162(3), 754–778 (2014)
Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. Theory Methods Appl. 16, 209–231 (1991)
Panagiotopoulos, P.D.: Variational-hemivariational inequalities in nonlinear elasticity. Aplikace Matematiky 33, 249–268 (1988)
Panagiotopoulos, P.D., Stavroulakis, G.E.: A variational-hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions. Q. Appl. Math. 46, 409–430 (1988)
Panagiotopoulos, P.D., Koltsakis, E.: The nonmonotone skin effect in plane elasticity. Problems obeying to subdifferential materials laws. ZAMM 70, 13–21 (1990)
Panagiotopoulos, P.D., Baniotopoulos, C.C.: A hemivariational inequality and substationary approach to the interface problem: theory and prospects of applications. Eng. Anal. 1, 20–31 (1984)
Panagiotopoulos, P.D.: Nonconvex superpotentials in the sense of F. H. Clarke and applications. Mech. Res. Commun. 8, 335–340 (1981)
Panagiotopoulos, P.D.: Nonconvex energy functions. Hemivariational inequalities and substationary principles. Acta Mech. 42, 160–183 (1983)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birhhauser, Basel (1985). MIH Publisher, Moscow, Russian translation (1989)
Panagiotopoulos, P.D.: Hemivariational inequalities and their applications. In: Moreau, J.J., Panagiotopoulos, P.D., Strang, G. (eds.) Topics in Nonsmooth Mechanics. Birkhauser, Basel (1988)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications: Convex and nonconvex energy functions. Springer Science and Business Media, New York (2012)
Rauch, J.: Discontinuous semilinear differential equations and multiple valued maps. Proc. Am. Math. Soc. 64(2), 277–282 (1977)
Rockafellar, R.T.: Generalized derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)
Szafraniec, P.: Evolutionary Boussinesq model with nonmonotone friction and heat flux boundary conditions. Nonlinear Anal. Real World Appl. 34, 403–415 (2017)
Temam, R.: Navier–Stokes Equations. Studies in Mathematics and its Applications. Mathematical Surveys and Monographs. North-Holland Publishing Co., Amsterdam (1979)
Acknowledgements
We would like to express our gratitude to the Editor for taking time to handle the manuscript and to anonymous referees whose constructive comments are very helpful for improving the quality of our paper. We would like also to thank Prof. S. Migórski for pointing out that the Rauch and the growth conditions are completely independent.
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Mahdioui, H., Ben Aadi, S. & Akhlil, K. Hemivariational Inequality for Navier–Stokes Equations: Existence, Dependence, and Optimal Control. Bull. Iran. Math. Soc. 47, 1751–1774 (2021). https://doi.org/10.1007/s41980-020-00470-x
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DOI: https://doi.org/10.1007/s41980-020-00470-x
Keywords
- Navier–Stokes equations
- Hemivariational inequalities
- Galerkin method
- Optimal control
- Non-convex Optimization
- Subdifferential