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Hemivariational Inequality for Navier–Stokes Equations: Existence, Dependence, and Optimal Control

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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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References

  1. 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)

    MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aubin, J.P.: Applied Functional Analysis. Wiley, New York (1979)

    MATH  Google Scholar 

  4. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Berlin (1990)

    MATH  Google Scholar 

  5. 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)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chebotarev, A.Y.: Subdifferential boundary value problems for stationary Navier–Stokes equations. Differentsial’nye Uravneniya 28(8), 1443–1450 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Chebotarev, A.Y.: Stationary variational inequalities in the model of inhomogeneous incompressible fluids. Sibirsk. Math. Zh. (Siberian Math. J.) 38, 1184–1193 (1997)

    MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chebotarev, A.Y.: Modeling of steady flows in a channel by Navier–Stokes variational inequalities. J. Appl. Mech. Tech. Phys. 44, 852–857 (2003)

    Article  MathSciNet  Google Scholar 

  12. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

    MATH  Google Scholar 

  13. Clarke, F.H.: Generalized gradient and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Studies in Mathematics and its Applications. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  17. 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)

    MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Science and Business Media, New York (2012)

    MATH  Google Scholar 

  20. 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)

    MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Konovalova, D.S.: Subdifferential boundary value problems for evolution Navier–Stokes equations. Differ. Uravn. (Differential Equations) 36, 792–798 (2000)

    Google Scholar 

  23. Lions, J.L.: Quelques méthodes de résolution des problemes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Migórski, S., Ochal, A.: Navier–Stokes problems modelled by evolution hemivariational inequalities. Discrete Contin. Dyn. Syst. Supplement, 731–740 (2007)

  30. Migórski, S.: A note on optimal control problem for a hemivariational inequality modeling fluid flow. Dyn. Syst., 545–554 (2013)

  31. Migórski, S., Szafraniec, P.: Nonmonotone slip problem for miscible liquids. J. Math. Anal. Appl. 471(1–2), 342–357 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5(2), 217–228 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Moreau, J.J., Panagiotopoulos, P.D.: Nonsmooth Mechanics and Applications. CISM Courses and Lectures, vol. 302. Springer, Wien, pp. 81–176 (1988)

  34. Naniewicz, Z.: Hemivariational inequalities with functions fulfilling directional growth condition. Appl. Anal. 55(3–4), 259–285 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Naniewicz, Z.: Hemivariational inequalities with functionals which are not locally Lipschitz. Nonlinear Anal. Theory Methods Appl. 25(12), 1307–1320 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. Theory Methods Appl. 16, 209–231 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Panagiotopoulos, P.D.: Variational-hemivariational inequalities in nonlinear elasticity. Aplikace Matematiky 33, 249–268 (1988)

    MathSciNet  MATH  Google Scholar 

  39. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Panagiotopoulos, P.D., Koltsakis, E.: The nonmonotone skin effect in plane elasticity. Problems obeying to subdifferential materials laws. ZAMM 70, 13–21 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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)

    Article  Google Scholar 

  42. Panagiotopoulos, P.D.: Nonconvex superpotentials in the sense of F. H. Clarke and applications. Mech. Res. Commun. 8, 335–340 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Panagiotopoulos, P.D.: Nonconvex energy functions. Hemivariational inequalities and substationary principles. Acta Mech. 42, 160–183 (1983)

    Google Scholar 

  44. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birhhauser, Basel (1985). MIH Publisher, Moscow, Russian translation (1989)

  45. 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)

    MATH  Google Scholar 

  46. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications: Convex and nonconvex energy functions. Springer Science and Business Media, New York (2012)

    Google Scholar 

  47. Rauch, J.: Discontinuous semilinear differential equations and multiple valued maps. Proc. Am. Math. Soc. 64(2), 277–282 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rockafellar, R.T.: Generalized derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  49. Szafraniec, P.: Evolutionary Boussinesq model with nonmonotone friction and heat flux boundary conditions. Nonlinear Anal. Real World Appl. 34, 403–415 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. Temam, R.: Navier–Stokes Equations. Studies in Mathematics and its Applications. Mathematical Surveys and Monographs. North-Holland Publishing Co., Amsterdam (1979)

    Google Scholar 

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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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Correspondence to Sultana Ben Aadi.

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Communicated by Sohrab Effati.

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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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