TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID∗

doi:10.1007/s10473-023-0413-z

Abstract

Abstract: In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued, nonmonotone friction law. First, a variational formulation of the model is obtained; that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field. Then, an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated. Under mild assumptions, the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven. Furthermore, a uniqueness theorem for the abstract inequality is established by using a monotonicity argument. Finally, we employ the theoretical results to examine the nonstationary Oseen model.

Key words: nonstationary Oseen model, Newtonian incompressible fluid, hemivariational inequality, periodic solution, generalized subgradient

Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG. TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗[J].Acta mathematica scientia,Series B, 2023, 43(4): 1645-1667.

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References

[1] Aubin J P, Cellina A.Differential Inclusions: Set-Valued Maps and Viability Theory. Berlin: Springer-Verlag, 1984
[2] Barboteu M, Bartosz K, Han W, Janiczko T. Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM J Numer Anal, 2015, 53: 527-550
[3] Berkovits J, Mustonen V. Monotonicity methods for nonlinear evolution equations. Nonlinear Anal, 1966, 27: 1397-1405
[4] Brézis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011
[5] Carl S, Le V K, Motreanu D.Nonsmooth Variational Problems and Their Inequalities. New York: Springer, 2007
[6] Clarke F H.Optimization and Nonsmooth Analysis. New York: Wiley, 1983
[7] Denkowski Z, Migórski S, Papageorgiou N S. An Introduction to Nonlinear Analysis: Theory. Boston: Kluwer, 2003
[8] Denkowski Z, Migórski S, Papageorgiou N S. An Introduction to Nonlinear Analysis: Applications. Boston: Kluwer, 2003
[9] Djoko J K, Lubuma J M. Analysis of a time implicit scheme for the Oseen model driven by nonlinear slip boundary conditions. J Math Fluid Mech, 2018, 18: 717-730
[10] Dudek S, Kalita P, Migórski S. Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law. Comput Math Appl,2017, 74: 1813-1825
[11] Fang C J, Han W, Migórski S, Sofonea M. A class of hemivariational inequalities for nonstationary Navier-Stokes equations. Nonlinear Anal,2016, 31: 257-276
[12] Fujita H. A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku, 1994, 888: 199-216
[13] Fujita H. Non stationary Stokes flows under leak boundary conditions of friction type. J Comput Math, 2001, 19: 1-8
[14] Han Y, Huang N J, Lu J, Xiao Y B. Existence and stability of solutions to inverse variational inequality problems. Appl Math Mech, 2017, 38: 749-764
[15] Han W, Sofonea M, Barboteu M. Numerical analysis of elliptic hemivariational inequalities. SIAM J Numer Anal, 2017, 55: 640-663
[16] Han W, Migórski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal,2014, 46: 3891-3912
[17] Kashiwabara T. On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type. J Differential Equations, 2013, 254: 756-778
[18] Le Roux C, Tani A. Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions. Math Methods Appl Sci, 2007, 30: 595-624
[19] Gasi̤ski L, Winkert P. Existence and uniqueness results for double phase problems with convection term. J Differential Equations,2020, 268: 4183-4193
[20] Li Y, Li K. Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions. J Math Anal Appl, 2011, 381: 1-9
[21] Lions J L.Quelques Méthodes de Resolution des Problémes aux Limites non Linéaires. Paris: Dunod, 1969
[22] Liu Y J, Migórski S, Nguyen V T, Zeng S D. Existence and convergence results for an elastic frictional contact problem with nonmonotone subdifferential boundary conditions. Acta Math Sci,2021, 41B(4): 1151-1168
[23] Liu Z H, Motreanu D, Zeng S D. Generalized penalty and regularization method for differential variational-hemivariational inequalities. SIAM J Optim, 2021, 31: 1158-1183
[24] Malek 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. Amsterdam: Elsevier, 2005
[25] Migórski S, Dudek S. Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law. J Math Fluid Mech,2018, 20: 1317-1333
[26] Migórski S, Dudek S. A new class of variational-hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions. Zeitschrift Angew Math Mech,2020, 100: e201900112
[27] Migórski S, Dudek S. Steady flow with unilateral and leak/slip boundary conditions by the Stokes variational-hemivariational inequality. Appl Anal,2020, 101: 2949-2965
[28] Migórski S, Ochal A. Hemivariational inequalities for stationary Navier-Stokes equations. J Math Anal Appl,2005, 306: 197-217
[29] Migórski S, Ochal A. Quasi-static hemivariational inequality via vanishing acceleration approach. SIAM J Math Anal,2009, 41: 1415-1435
[30] Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. New York: Springer, 2013
[31] Migórski S, Paczka D. On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces. Nonlinear Anal,2018, 39: 337-361
[32] Migórski S Paczka D. Frictional contact problems for steady flow of incompressible fluids in Orlicz spaces//Dutta H, Ko$\breve{\rm c}$inac L, Srivastava H. Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Basle: Birkháuser, 2019: 1-53
[33] Morimoto H. Time periodic Navier-Stokes flow with nonhomogeneous boundary condition. J Math Sci Univ Tokyo, 2009, 16: 113-123
[34] Naniewicz Z, Panagiotopoulos P D.Mathematical Theory of Hemivariational Inequalities and Applications. New York: Dekker, 1995
[35] Panagiotopoulos P D.Hemivariational Inequalities: Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993
[36] Saito N. On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions. Pub RIMS, Kyoto University, 2004, 40: 345-383
[37] Saito N, Fujita H. Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition. Lecture Notes in Pure Appl Math, 2001, 223: 73-86
[38] Sofonea M, Migórski S. Variational-Hemivariational Inequalities with Applications. New York: Chapman and Hall, 2017
[39] Zeng S D, Migórski S, Liu Z H. Well-posedness, optimal control and sensitivity analysis for a class of differential variational-hemivariational inequalities. SIAM J Optim,2021, 31: 1158-1183
[40] Zeng S D, Bai Y R, Gasi̤ski L, Winkert P. Existence results for double phase implicit obstacle problems involving multivalued operators. Calc Var Partial Differential Equations,2020, 59: Art 176
[41] Zeng S D,Rãdulescu V D, Winkert P. Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions. SIAM J Math Anal, 2022, 54: 1898-1926
[42] Zeng S D, Migórski S, Liu Z H. Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation
(in Chinese). Sci Sin Math, 2022, 52: 331-354
[43] Zeng S D, Migórski S, Khan A A. Nonlinear quasi-hemivariational inequalities: existence and optimal control. SIAM J Control Optim,2021, 59: 1246-1274
[44] Zeng S D, Rãdulescu V D, Winkert P. Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control. Bull Sci Math,2022, 176: 103131
[45] Zeidler E.Nonlinear Functional Analysis and Applications. II: A/B. New York: Springer, 1990

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

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