A new class of elliptic quasi-variational-hemivariational inequalities for fluid flow with mixed boundary conditions. (English) Zbl 1524.49020
Summary: In this paper we study a class of quasi-variational-hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani-Ky Fan fixed point theorem. The applicability of the results is illustrated by the steady-state Oseen model of a generalized Newtonian incompressible fluid with mixed boundary conditions. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and a generalization of the threshold slip and leak condition of frictional type.
MSC:
| 49J40 | Variational inequalities |
| 35Q30 | Navier-Stokes equations |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 74M15 | Contact in solid mechanics |
Keywords:
quasi-variational-hemivariational inequality; obstacle; variational inequality; Clarke subgradient; mosco convergence; fixed pointReferences:
| [1] | Abergel, F.; Temam, R., On some control problems in fluid mechanics, Theor. Comput. Fluid Dyn., 1, 303-325 (1990) · Zbl 0708.76106 |
| [2] | An, R.; Li, Y.; Li, K., Solvability of Navier-Stokes equations with leak boundary conditions, Acta Math. Appl. Sin., 25, 225-234 (2009) · Zbl 1180.35404 |
| [3] | Baiocchi, C.; Capelo, A., Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems (1984), John Wiley: John Wiley Chichester · Zbl 0551.49007 |
| [4] | Barbu, V., Optimal Control of Variational Inequalities (1984), Pitman: Pitman Boston · Zbl 0574.49005 |
| [5] | Bensoussan, A.; Lions, J.-L., Controle impulsionnel et inéquations quasi-variationnelles d’évolutions, C. R. Acad. Sci. Paris Sér. A-B, 276, A1333-A1338 (1974) · Zbl 0264.49005 |
| [6] | Brézis, H., Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18, 115-175 (1968) · Zbl 0169.18602 |
| [7] | Browder, F. E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Am. Math. Soc., 71, 780-785 (1965) · Zbl 0138.39902 |
| [8] | Carl, S.; Le, V. K.; Motreanu, D., Nonsmooth Variational Problems and Their Inequalities (2007), Springer: Springer New York · Zbl 1109.35004 |
| [9] | Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley, Interscience: Wiley, Interscience New York · Zbl 0582.49001 |
| [10] | Denkowski, Z.; Migórski, S.; Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Theory (2003), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers Boston, Dordrecht, London, New York · Zbl 1040.46001 |
| [11] | Denkowski, Z.; Migórski, S.; Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Applications (2003), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers Boston, Dordrecht, London, New York · Zbl 1030.35106 |
| [12] | Dudek, S.; Kalita, P.; Migórski, S., Stationary flow of non-Newtonian fluid with frictional boundary conditions, Z. Angew. Math. Phys., 66, 2625-2646 (2015) · Zbl 1327.76053 |
| [13] | Dudek, S.; Kalita, P.; Migórski, S., Stationary Oberbeck-Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations, Appl. Anal., 96, 2192-2217 (2017) · Zbl 1386.76056 |
| [14] | Dudek, S.; Migórski, S., Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law, J. Math. Fluid Mech., 20, 1317-1333 (2018) · Zbl 06955609 |
| [15] | Duvaut, G.; Lions, J.-L., Inequalities in Mechanics and Physics (1976), Springer: Springer Berlin · Zbl 0331.35002 |
| [16] | Fujita, H., A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 888, 199-216 (1994) · Zbl 0939.76527 |
| [17] | Fujita, H., Non stationary Stokes flows under leak boundary conditions of friction type, J. Comput. Math., 19, 1-8 (2001) · Zbl 0993.76014 |
| [18] | Fujita, H., A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math., 149, 57-69 (2002) · Zbl 1058.76023 |
| [19] | (Gunzburger, M. D., Flow Control. Flow Control, The IMA Volumes in Mathematics and Its Applications (1995), Springer-Verlag: Springer-Verlag New York) |
| [20] | (Han, W.; Migórski, S.; Sofonea, M., Advances in Variational and Hemivariational Inequalities with Applications. Theory, Numerical Analysis, and Applications. Advances in Variational and Hemivariational Inequalities with Applications. Theory, Numerical Analysis, and Applications, Advances in Mechanics and Mathematics, vol. 33 (2015), Springer) · Zbl 1309.49002 |
| [21] | Kashiwabara, T., On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differ. Equ., 254, 756-778 (2013) · Zbl 1253.35102 |
| [22] | Kashiwabara, T., On a finite element approximation of the Stokes equations under a slip boundary condition of the friction type, Jpn. J. Ind. Appl. Math., 30, 227-261 (2013) · Zbl 1263.76043 |
| [23] | Khan, A. A.; Motreanu, D., Inverse problems for quasi-variational inequalities, J. Glob. Optim., 70, 401-411 (2018) · Zbl 1387.35620 |
| [24] | Kikuchi, N.; Oden, J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (1988), SIAM: SIAM Philadelphia · Zbl 0685.73002 |
| [25] | Kim, T.; Huang, F., The non-steady Navier-Stokes systems with mixed boundary conditions including friction conditions, Methods Appl. Anal., 25, 13-50 (2018) · Zbl 1406.35228 |
| [26] | Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, vol. 31 (2000), SIAM: SIAM Philadelphia · Zbl 0988.49003 |
| [27] | Kravchuk, A. S.; Neittaanmäki, P. J., Variational and Quasi-variational Inequalities in Mechanics (2007), Springer: Springer Dordrecht · Zbl 1131.49001 |
| [28] | Le Roux, C., Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Arch. Ration. Mech. Anal., 148, 309-356 (1999) · Zbl 0934.76005 |
| [29] | Le Roux, C.; Tani, A., Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30, 595-624 (2007) · Zbl 1251.76008 |
| [30] | Li, Y.; Li, K., Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions, J. Math. Anal. Appl., 381, 1-9 (2011) · Zbl 1221.35282 |
| [31] | Lions, J.-L., Quelques méthodes de resolution des problémes aux limites non linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603 |
| [32] | Lions, J.-L.; Stampacchia, G., Variational inequalities, Commun. Pure Appl. Math., 20, 493-519 (1967) · Zbl 0152.34601 |
| [33] | Liu, Z. H.; Migórski, S.; Zeng, S. D., Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differ. Equ., 263, 3989-4006 (2017) · Zbl 1372.35008 |
| [34] | Málek, J.; Nečas, J.; Rokyta, M.; Ružička, M., Weak and Measure-Valued Solutions to Evolutionary PDEs (1996), Springer · Zbl 0851.35002 |
| [35] | Málek, J.; Rajagopal, K. R., Mathematical issues concerning the Navier-Stokes equations and some of their generalizations, (Dafermos, C.; Feireisl, E., Handbook of Evolutionary Equations, vol. II (2005), Elsevier) · Zbl 1095.35027 |
| [36] | Migórski, S.; Dudek, S., A new class of variational-hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions, Z. Angew. Math. Mech., 100, Article e201900112 pp. (2020) · Zbl 07794850 |
| [37] | Migórski, S.; Khan, A. A.; Zeng, S. D., Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of p-Laplacian type, Inverse Probl., 35, Article 035004 pp. (2019), 14 p · Zbl 1516.35530 |
| [38] | Migórski, S.; Khan, A. A.; Zeng, S. D., Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems, Inverse Probl., 36 (2020) · Zbl 1433.35462 |
| [39] | Migórski, S.; Ochal, A., Hemivariational inequalities for stationary Navier-Stokes equations, J. Math. Anal. Appl., 306, 197-217 (2005) · Zbl 1109.35089 |
| [40] | Migórski, S.; Ochal, A.; Sofonea, M., Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26 (2013), Springer: Springer New York · Zbl 1262.49001 |
| [41] | Migórski, S.; Ochal, A.; Sofonea, M., A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elast., 127, 151-178 (2017) · Zbl 1368.47045 |
| [42] | Migórski, S.; Paczka, D., On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces, Nonlinear Anal. Ser. B, Real World Appl., 39, 337-361 (2018) · Zbl 1448.76010 |
| [43] | Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3, 510-585 (1969) · Zbl 0192.49101 |
| [44] | Motreanu, D.; Sofonea, M., Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials, Abstr. Appl. Anal., 4, 255-279 (1999) · Zbl 0974.58019 |
| [45] | Naniewicz, Z.; Panagiotopoulos, P. D., Mathematical Theory of Hemivariational Inequalities and Applications (1995), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel, Hong Kong · Zbl 0968.49008 |
| [46] | Navier, C. L.M. H., Memoire sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. A, 6, 389-440 (1823) |
| [47] | Panagiotopoulos, P. D., Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech., 65, 29-36 (1985) · Zbl 0574.73015 |
| [48] | Panagiotopoulos, P. D., Inequality Problems in Mechanics and Applications (1985), Birkhäuser: Birkhäuser Boston · Zbl 0579.73014 |
| [49] | Panagiotopoulos, P. D., Hemivariational Inequalities, Applications in Mechanics and Engineering (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0826.73002 |
| [50] | Saito, N., On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. RIMS, Kyoto Univ., 40, 345-383 (2004) · Zbl 1050.35029 |
| [51] | Saito, N.; Fujita, H., Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition, Lect. Notes Pure Appl. Math., 223, 73-86 (2001) · Zbl 0995.35048 |
| [52] | Saito, N.; Sugitani, Y.; Zhou, G., Unilateral problem for the Stokes equations: the well-posedness and finite element approximation, Appl. Numer. Math., 105, 124-147 (2016) · Zbl 1381.76201 |
| [53] | Sofonea, M., Optimal control of a class of variational-hemivariational inequalities in reflexive Banach spaces, Appl. Math. Optim. (2017) |
| [54] | Sofonea, M.; Matei, A., Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note, vol. 398 (2012), Cambridge University Press · Zbl 1255.49002 |
| [55] | Sofonea, M.; Migórski, S., Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics (2018), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 1384.49002 |
| [56] | (Sritharan, S. S., Optimal Control of Viscous Flow. Optimal Control of Viscous Flow, SIAM Frontiers in Applied Mathematics (1998), Philadelphia) · Zbl 0920.76004 |
| [57] | Temam, R., Navier-Stokes Equations (1979), North-Holland: North-Holland Amsterdam · Zbl 0454.35073 |
| [58] | Xiao, Y. B.; Sofonea, M., Generalized penalty method for elliptic variational-hemivariational inequalities, Appl. Math. Optim. (2019) |
| [59] | Zeng, S. D.; Liu, Z. H.; Migórski, S.; Yao, J.-C., Convergence of a generalized penalty method for variational-hemivariational inequalities, Commun. Nonlinear Sci. Numer. Simul., 92, Article 105476 pp. (2021) · Zbl 07274884 |
| [60] | Zeng, S. D.; Migórski, S.; Khan, A. A., Nonlinear quasi-hemivariational inequalities: existence and optimal control, SIAM J. Control Optim. (2021) · Zbl 07332073 |
| [61] | Zhang, S.; Yang, Z.; Li, X., A projection method based on self-adaptive rules for Stokes equations with nonlinear slip boundary conditions, J. Math. Anal. Appl., 491, Article 124306 pp. (2020) · Zbl 07244666 |
| [62] | Zhou, G.; Saito, N., The Navier-Stokes equations under a unilateral boundary condition of Signorini’s type, J. Math. Fluid Mech., 18, 481-510 (2016) · Zbl 1346.35119 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
