October  2016, 36(10): 5369-5386. doi: 10.3934/dcds.2016036

Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow

1. 

College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received  August 2015 Revised  April 2016 Published  July 2016

A time-dependent Stokes fluid flow problem is studied with nonlinear boundary conditions described by the Clarke subdifferential. We present equivalent weak formulations of the problem, one of them in the form of a hemivariational inequality. The existence of a solution is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Finally, we present a result on the existence of a solution to an optimal control problem for the hemivariational inequality.
Citation: Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036
References:
[1]

N. Ahmed, Optimal control of hydrodynamic flow with possible application to artificial heart, Dynam. Systems Appl., 1 (1992), 103-119.

[2]

J. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin, 1984.

[3]

E. Balder, Necessary and sufficient conditions for $L^1$-strong-weak lower semicontinuity of integral functionals, Nonlinear Anal., 11 (1987), 1399-1404. doi: 10.1016/0362-546X(87)90092-7.

[4]

V. Barbu, Optimal Control of Variational Inequalities, Pitman, London, 1983.

[5]

K. Bartosz, X. Cheng, P. Kalita, Y. Yu and C. Zheng, Rothe method for parabolic variational-hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862. doi: 10.1016/j.jmaa.2014.09.078.

[6]

C. Carstensen and J. Gwinner, A theory of discretisation for nonlinear evolution inequalities applied to parabolic Signorini problems, Ann. Mat. Pura Appl., 177 (1999), 363-394. doi: 10.1007/BF02505918.

[7]

L. Cesari, Optimization: Theory and Applications, Springer, Berlin, 1983.

[8]

F. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.

[9]

Z. Denkowski and S. Migórski, Optimal shape design problems for a class of systems described by hemivariational inequalities, J. Global Optim., 12 (1998), 37-59. doi: 10.1023/A:1008299801203.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic, Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[11]

Z. Denkowski, S. Migórski and N. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic, Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[12]

J. Djoko, On the time approximation of the Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Anal.\ Model. - B, 11 (2014), 34-53.

[13]

H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math., 149 (2002), 57-69. doi: 10.1016/S0377-0427(02)00520-4.

[14]

V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986.

[15]

W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912. doi: 10.1137/140963248.

[16]

J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.

[17]

J. Haslinger and P. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Analysis: Theory, Methods, and Applications, 24 (1995), 105-119. doi: 10.1016/0362-546X(93)E0022-U.

[18]

P. Kalita, Convergence of Rothe scheme for hemivariational inequalities of parabolic type, Int. J. Numer. Anal. Model., 10 (2013), 445-465.

[19]

Y. Li and K. Li, Penalty finite element method for Stokes problem with nonlinear slip boundary conditions, Appl. Math. Comput., 204 (2008), 216-226. doi: 10.1016/j.amc.2008.06.035.

[20]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Heidelberg, Berlin, 1971.

[21]

M. Miettinen and J. Haslinger, Approximation of optimal control problems of hemivariational inequalities, Numer. Funct. Anal. and Optimiz., 13 (1992), 43-68. doi: 10.1080/01630569208816460.

[22]

S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Systems - B, 6 (2006), 1339-1356. doi: 10.3934/dcdsb.2006.6.1339.

[23]

S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - S, (2013), 545-554. doi: 10.3934/proc.2013.2013.545.

[24]

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300. doi: 10.1023/A:1026555014562.

[25]

S. Migórski and A. Ochal, Hemivariational inequalities for stationary Navier-Stokes equations, J. Math. Anal. Appl., 306 (2005), 197-217. doi: 10.1016/j.jmaa.2004.12.033.

[26]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435. doi: 10.1137/080733231.

[27]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013.

[28]

H. Nagase, On an application of Rothe method to nonlinear parabolic variational inequalities, Funkcial. Ekvac., 32 (1989), 273-299.

[29]

Z. Naniewicz and P. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Dekker, New York, 1995. doi: 10.1007/978-1-4612-0873-0.

[30]

P. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin, 1993.

[31]

T. Roubicek, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005.

[32]

Y. Shang, New stabilized finite element method for time-dependent incompressible flow problems, Int. J. Numer. Meth. Fluids , 62 (2010), 166-187. doi: 10.1002/fld.2010.

[33]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

[34]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Math., 1459, Springer, Berlin, 1990.

[35]

F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010.

show all references

References:
[1]

N. Ahmed, Optimal control of hydrodynamic flow with possible application to artificial heart, Dynam. Systems Appl., 1 (1992), 103-119.

[2]

J. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin, 1984.

[3]

E. Balder, Necessary and sufficient conditions for $L^1$-strong-weak lower semicontinuity of integral functionals, Nonlinear Anal., 11 (1987), 1399-1404. doi: 10.1016/0362-546X(87)90092-7.

[4]

V. Barbu, Optimal Control of Variational Inequalities, Pitman, London, 1983.

[5]

K. Bartosz, X. Cheng, P. Kalita, Y. Yu and C. Zheng, Rothe method for parabolic variational-hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862. doi: 10.1016/j.jmaa.2014.09.078.

[6]

C. Carstensen and J. Gwinner, A theory of discretisation for nonlinear evolution inequalities applied to parabolic Signorini problems, Ann. Mat. Pura Appl., 177 (1999), 363-394. doi: 10.1007/BF02505918.

[7]

L. Cesari, Optimization: Theory and Applications, Springer, Berlin, 1983.

[8]

F. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.

[9]

Z. Denkowski and S. Migórski, Optimal shape design problems for a class of systems described by hemivariational inequalities, J. Global Optim., 12 (1998), 37-59. doi: 10.1023/A:1008299801203.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic, Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[11]

Z. Denkowski, S. Migórski and N. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic, Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[12]

J. Djoko, On the time approximation of the Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Anal.\ Model. - B, 11 (2014), 34-53.

[13]

H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math., 149 (2002), 57-69. doi: 10.1016/S0377-0427(02)00520-4.

[14]

V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986.

[15]

W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912. doi: 10.1137/140963248.

[16]

J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.

[17]

J. Haslinger and P. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Analysis: Theory, Methods, and Applications, 24 (1995), 105-119. doi: 10.1016/0362-546X(93)E0022-U.

[18]

P. Kalita, Convergence of Rothe scheme for hemivariational inequalities of parabolic type, Int. J. Numer. Anal. Model., 10 (2013), 445-465.

[19]

Y. Li and K. Li, Penalty finite element method for Stokes problem with nonlinear slip boundary conditions, Appl. Math. Comput., 204 (2008), 216-226. doi: 10.1016/j.amc.2008.06.035.

[20]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Heidelberg, Berlin, 1971.

[21]

M. Miettinen and J. Haslinger, Approximation of optimal control problems of hemivariational inequalities, Numer. Funct. Anal. and Optimiz., 13 (1992), 43-68. doi: 10.1080/01630569208816460.

[22]

S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Systems - B, 6 (2006), 1339-1356. doi: 10.3934/dcdsb.2006.6.1339.

[23]

S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - S, (2013), 545-554. doi: 10.3934/proc.2013.2013.545.

[24]

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300. doi: 10.1023/A:1026555014562.

[25]

S. Migórski and A. Ochal, Hemivariational inequalities for stationary Navier-Stokes equations, J. Math. Anal. Appl., 306 (2005), 197-217. doi: 10.1016/j.jmaa.2004.12.033.

[26]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435. doi: 10.1137/080733231.

[27]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013.

[28]

H. Nagase, On an application of Rothe method to nonlinear parabolic variational inequalities, Funkcial. Ekvac., 32 (1989), 273-299.

[29]

Z. Naniewicz and P. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Dekker, New York, 1995. doi: 10.1007/978-1-4612-0873-0.

[30]

P. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin, 1993.

[31]

T. Roubicek, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005.

[32]

Y. Shang, New stabilized finite element method for time-dependent incompressible flow problems, Int. J. Numer. Meth. Fluids , 62 (2010), 166-187. doi: 10.1002/fld.2010.

[33]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

[34]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Math., 1459, Springer, Berlin, 1990.

[35]

F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010.

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