Startseite Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
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Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control

  • Sultana Ben Aadi ORCID logo EMAIL logo , Khalid Akhlil ORCID logo und Khadija Aayadi
Veröffentlicht/Copyright: 25. September 2021

Abstract

In this paper, we introduce the g-Navier–Stokes equations with time-fractional derivative of order α ( 0 , 1 ) in domains of 2 . We then study the existence and uniqueness of weak solutions by means of the Galerkin approximation. Finally, an optimal control problem is considered and solved.

MSC 2010: 49J52

Acknowledgements

We would like to express our gratitude to the editor for taking time to handle the manuscript and to the anonymous referees whose constructive comments were very helpful for improving the quality of our paper.

References

[1] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A 40 (2007), no. 24, 6287–6303. 10.1088/1751-8113/40/24/003Suche in Google Scholar

[2] A. A. Alikhanov, A priori estimates for solutions of boundary value problems for equations of fractional order, Differ. Equ. 46 (2010), 660–666. 10.1134/S0012266110050058Suche in Google Scholar

[3] C. T. Anh and D. T. Quyet, g-Navier–Stokes equations with infinite delays, Vietnam J. Math. 40 (2012), no. 1, 57–78. Suche in Google Scholar

[4] C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous g-Navier–Stokes equations, Ann. Polon. Math. 103 (2012), no. 3, 277–302. 10.4064/ap103-3-5Suche in Google Scholar

[5] C. T. Anh, N. V. Thanh and N. V. Tuan, On the stability of solutions to stochastic 2D g-Navier–Stokes equations with finite delays, Random Oper. Stoch. Equ. 25 (2017), no. 4, 211–224. 10.1515/rose-2017-0016Suche in Google Scholar

[6] H.-O. Bae and J. Roh, Existence of solutions of the g-Navier–Stokes equations, Taiwanese J. Math. 8 (2004), 85–102. 10.11650/twjm/1500558459Suche in Google Scholar

[7] R. Camassa, D. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water, Phys. D 98 (1996), no. 2–4, 258–286. 10.1016/0167-2789(96)00117-0Suche in Google Scholar

[8] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems, Springer Monogr. Math., Springer, New York, 2011. 10.1007/978-0-387-09620-9Suche in Google Scholar

[9] J. K. Hale and G. Raugel, A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc. 329 (1992), no. 1, 185–219. 10.1090/S0002-9947-1992-1040261-1Suche in Google Scholar

[10] J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33–95. Suche in Google Scholar

[11] J. Jiang and Y. Hou, The global attractor of g-Navier–Stokes equations with linear dampness on 𝐑 2 , Appl. Math. Comput. 215 (2009), no. 3, 1068–1076. 10.1016/j.amc.2009.06.035Suche in Google Scholar

[12] J.-P. Jiang and Y.-R. Hou, Pullback attractor of 2D non-autonomous g-Navier–Stokes equations on some bounded domains, Appl. Math. Mech. (English Ed.) 31 (2010), no. 6, 697–708. 10.1007/s10483-010-1304-xSuche in Google Scholar

[13] J.-P. Jiang, Y.-R. Hou and X.-X. Wang, Pullback attractor of 2D nonautonomous g-Navier–Stokes equations with linear dampness, Appl. Math. Mech. (English Ed.) 32 (2011), no. 2, 151–166. 10.1007/s10483-011-1402-xSuche in Google Scholar

[14] J.-P. Jiang and X.-X. Wang, Global attractor of 2D autonomous g-Navier–Stokes equations, Appl. Math. Mech. (English Ed.) 34 (2013), no. 3, 385–394. 10.1007/s10483-013-1678-7Suche in Google Scholar

[15] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Suche in Google Scholar

[16] M. Kwak, H. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier–Stokes equations, J. Math. Anal. Appl. 315 (2006), no. 2, 436–461. 10.1016/j.jmaa.2005.04.050Suche in Google Scholar

[17] H. Kwean, The H 1 -compact global attractor of two-dimensional g-Navier–Stokes equations, Far East J. Dyn. Syst. 18 (2012), no. 1, 1–20. Suche in Google Scholar

[18] H. Kwean and J. Roh, The global attractor of the 2D g-Navier–Stokes equations on some unbounded domains, Commun. Korean Math. Soc. 20 (2005), no. 4, 731–749. 10.4134/CKMS.2005.20.4.731Suche in Google Scholar

[19] P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC, Boca Raton, 2002. 10.1201/9781420035674Suche in Google Scholar

[20] C. D. Levermore, M. Oliver and E. S. Titi, Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J. 45 (1996), no. 2, 479–510. 10.1512/iumj.1996.45.1199Suche in Google Scholar

[21] C. D. Levermore, M. Oliver and E. S. Titi, Global well-posedness for the lake equations, Phys. D 98 (1996), 492–596. 10.1016/0167-2789(96)00108-XSuche in Google Scholar

[22] J.-L. Lions, Sur l’existence de solutions des équations de Navier–Stokes, C. R. Acad. Sci. Paris 248 (1959), 2847–2849. Suche in Google Scholar

[23] G. Ł ukaszewicz and P. Kalita, Navier–Stokes Equations. An Introduction with Applications, Adv. Mech. Math. 34, Springer, Cham, 2016. Suche in Google Scholar

[24] H. Mahdioui, S. Ben Aadi and K. Akhlil, Hemivariational inequality for Navier–Stokes equations: Existence, dependence, and optimal control, Bull. Iran. Math. Soc. (2020), 10.1007/s41980-020-00470-x. 10.1007/s41980-020-00470-xSuche in Google Scholar

[25] D. T. Quyet, Asymptotic behavior of strong solutions to 2D g-Navier–Stokes equations, Commun. Korean Math. Soc. 29 (2014), no. 4, 505–518. 10.4134/CKMS.2014.29.4.505Suche in Google Scholar

[26] D. T. Quyet, Pullback attractors for strong solutions of 2D non-autonomous g-Navier–Stokes equations, Acta Math. Vietnam. 40 (2015), no. 4, 637–651. 10.1007/s40306-014-0073-0Suche in Google Scholar

[27] D. T. Quyet, Pullback attractors for 2D g-Navier–Stokes equations with infinite delays, Commun. Korean Math. Soc. 31 (2016), no. 3, 519–532. 10.4134/CKMS.c150186Suche in Google Scholar

[28] D. T. Quyet and N. V. Tuan, On the stationary solutions to 2D g-Navier–Stokes equations, Acta Math. Vietnam. 42 (2017), no. 2, 357–367. 10.1007/s40306-016-0180-1Suche in Google Scholar

[29] G. Raugel and G. R. Sell, Navier–Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), no. 3, 503–568. 10.1090/S0894-0347-1993-1179539-4Suche in Google Scholar

[30] J. Roh, g-Navier–Stokes equations, Ph.D. Thesis, University of Minnesota, 2001. Suche in Google Scholar

[31] J. Roh, Dynamics of the g-Navier–Stokes equations, J. Differential Equations 211 (2005), no. 2, 452–484. 10.1016/j.jde.2004.08.016Suche in Google Scholar

[32] J. Roh, Geometry of L 2 ( Ω ; g ) , J. Chungcheong Math. Soc. 19 (2006), no. 3, 283–289. Suche in Google Scholar

[33] T. Tachim Medjo, A note on the regularity of weak solutions to the coupled 2D Allen–Cahn–Navier–Stokes system, J. Appl. Anal. 25 (2019), no. 1, 111–117. 10.1515/jaa-2019-0012Suche in Google Scholar

[34] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, 3rd ed., Stud. Math. Appl. 2, North-Holland, Amsterdam, 1984. Suche in Google Scholar

[35] D. Wu, The finite-dimensional uniform attractors for the nonautonomous g-Navier–Stokes equations, J. Appl. Math. 2009 (2009), Article ID 150420. 10.1155/2009/150420Suche in Google Scholar

[36] D. Wu, On the dimension of the pullback attractors for g-Navier–Stokes equations, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 893240. 10.1155/2010/893240Suche in Google Scholar

[37] W. M. Zaja̧czkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition, J. Appl. Anal. 4 (1998), no. 2, 167–204. 10.1515/JAA.1998.167Suche in Google Scholar

[38] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Hackensack, 2014. 10.1142/9069Suche in Google Scholar

[39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016. 10.1016/B978-0-12-804277-9.50002-XSuche in Google Scholar

[40] Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier–Stokes equations and optimal control, Comput. Math. Appl. 73 (2017), no. 6, 1016–1027. 10.1016/j.camwa.2016.07.007Suche in Google Scholar

Received: 2020-06-05
Revised: 2021-01-02
Accepted: 2021-01-04
Published Online: 2021-09-25
Published in Print: 2022-06-01

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