# American Institute of Mathematical Sciences

June  2019, 39(6): 3291-3313. doi: 10.3934/dcds.2019136

## Existence of time-periodic strong solutions to a fluid–structure system

 1 Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne F-31062 Toulouse Cedex 9, France 2 School of Mathematical Sciences, Monash University, Melbourne, Australia

Received  June 2018 Revised  October 2018 Published  February 2019

Fund Project: The author is partially supported by the ANR-Project IFSMACS (ANR 15-CE40.0010)

We study a nonlinear coupled fluid–structure system modelling the blood flow through arteries. The fluid is described by the incompressible Navier–Stokes equations in a 2D rectangular domain where the upper part depends on a structure satisfying a damped Euler–Bernoulli beam equation. The system is driven by time-periodic source terms on the inflow and outflow boundaries. We prove the existence of time-periodic strong solutions for this problem under smallness assumptions for the source terms.

Citation: Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52. doi: 10.1007/s00021-003-0082-5. Google Scholar [3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6. Google Scholar [4] M. Bostan, Periodic Solutions for Evolution Equations, vol. 3 of Electronic Journal of Differential Equations. Monograph, Southwest Texas State University, San Marcos, TX, 2002, Available from: https://ejde.math.txstate.edu/Monographs/03/bostan.pdf. Google Scholar [5] J.-J. Casanova, Fluid structure system with boundary conditions involving the pressure, 2017, arXiv: 1707.06382.Google Scholar [6] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55, Available from: http://projecteuclid.org/euclid.pjm/1102650841. doi: 10.2140/pjm.1989.136.15. Google Scholar [7] G. Da Prato and A. Ichikawa, Quadratic control for linear time-varying systems, SIAM J. Control Optim., 28 (1990), 359-381. doi: 10.1137/0328019. Google Scholar [8] D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, vol. 279 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. Google Scholar [9] G. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9. Google Scholar [10] G. P. Galdi, Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1237-1257. doi: 10.3934/dcdss.2013.6.1237. Google Scholar [11] C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737. doi: 10.1137/070699196. Google Scholar [12] C. Grandmont and M. Hillairet, Existence of global strong solutions to a beam-fluid interaction system, Arch. Ration. Mech. Anal., 220 (2016), 1283-1333. doi: 10.1007/s00205-015-0954-y. Google Scholar [13] C. Grandmont, M. Hillairet and J. Lequeurre, Existence of local strong solutions to fluidbeam and fluidrod interaction systems, Ann. Inst. H. Poincaré C, Anal. non linéaire, Available from: http://www.sciencedirect.com/science/article/pii/S0294144918301148.Google Scholar [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar [15] V. I. Judovič, Periodic motions of a viscous incompressible fluid, Soviet Math. Dokl., 1 (1960), 168-172. Google Scholar [16] S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes equations, Arch. Rational Mech. Anal., 24 (1967), 363-369. doi: 10.1007/BF00253153. Google Scholar [17] T. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition, Tokyo J. Math., 32 (2009), 409-424. doi: 10.3836/tjm/1264170239. Google Scholar [18] H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411. Google Scholar [19] M. Kyed, Time-Periodic Solutions to the Navier-Stokes Equations, Habilitation, Technische Universit t, Darmstadt, 2012, Available from: http://tuprints.ulb.tu-darmstadt.de/3309/.Google Scholar [20] J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410. doi: 10.1137/10078983X. Google Scholar [21] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Google Scholar [22] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. Google Scholar [23] A. Lunardi, Bounded solutions of linear periodic abstract parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988), 135-159. doi: 10.1017/S0308210500024926. Google Scholar [24] A. Lunardi, Stability of the periodic solutions to fully nonlinear parabolic equations in Banach spaces, Differential Integral Equations, 1 (1988), 253-279. Google Scholar [25] P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529, Available from: http://stacks.iop.org/0951-7715/4/503. doi: 10.1088/0951-7715/4/2/013. Google Scholar [26] P. Maremonti and M. Padula, Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 233 (1996), 142-182. doi: 10.1007/BF02366850. Google Scholar [27] V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, vol. 162 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162. Google Scholar [28] H. Morimoto, Survey on time periodic problem for fluid flow under inhomogeneous boundary condition, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 631-639. doi: 10.3934/dcdss.2012.5.631. Google Scholar [29] B. Muha and S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919-968. doi: 10.1007/s00205-012-0585-5. Google Scholar [30] A. Pazy, Semi-groups of Linear Operators and Applications to Partial Differential Equations, Department of Mathematics, University of Maryland, College Park, Md., 1974, Department of Mathematics, University of Maryland, Lecture Note, No. 10. Google Scholar [31] G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso bidimensionale, Rend. Sem. Mat. Univ. Padova, 30 (1960), 1-15, Available from: http://www.numdam.org/item?id=RSMUP_1960__30__1_0. Google Scholar [32] G. Prouse, Soluzioni periodiche dell'equazione di Navier-Stokes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 35 (1963), 443-447. Google Scholar [33] J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443. doi: 10.1137/080744761. Google Scholar [34] J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122. doi: 10.1007/BF00284169. Google Scholar [35] A. Takeshita, On the reproductive property of the $2$-dimensional Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1969), 297-311. Google Scholar [36] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418. Google Scholar

show all references

##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52. doi: 10.1007/s00021-003-0082-5. Google Scholar [3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6. Google Scholar [4] M. Bostan, Periodic Solutions for Evolution Equations, vol. 3 of Electronic Journal of Differential Equations. Monograph, Southwest Texas State University, San Marcos, TX, 2002, Available from: https://ejde.math.txstate.edu/Monographs/03/bostan.pdf. Google Scholar [5] J.-J. Casanova, Fluid structure system with boundary conditions involving the pressure, 2017, arXiv: 1707.06382.Google Scholar [6] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55, Available from: http://projecteuclid.org/euclid.pjm/1102650841. doi: 10.2140/pjm.1989.136.15. Google Scholar [7] G. Da Prato and A. Ichikawa, Quadratic control for linear time-varying systems, SIAM J. Control Optim., 28 (1990), 359-381. doi: 10.1137/0328019. Google Scholar [8] D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, vol. 279 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. Google Scholar [9] G. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9. Google Scholar [10] G. P. Galdi, Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1237-1257. doi: 10.3934/dcdss.2013.6.1237. Google Scholar [11] C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737. doi: 10.1137/070699196. Google Scholar [12] C. Grandmont and M. Hillairet, Existence of global strong solutions to a beam-fluid interaction system, Arch. Ration. Mech. Anal., 220 (2016), 1283-1333. doi: 10.1007/s00205-015-0954-y. Google Scholar [13] C. Grandmont, M. Hillairet and J. Lequeurre, Existence of local strong solutions to fluidbeam and fluidrod interaction systems, Ann. Inst. H. Poincaré C, Anal. non linéaire, Available from: http://www.sciencedirect.com/science/article/pii/S0294144918301148.Google Scholar [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar [15] V. I. Judovič, Periodic motions of a viscous incompressible fluid, Soviet Math. Dokl., 1 (1960), 168-172. Google Scholar [16] S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes equations, Arch. Rational Mech. Anal., 24 (1967), 363-369. doi: 10.1007/BF00253153. Google Scholar [17] T. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition, Tokyo J. Math., 32 (2009), 409-424. doi: 10.3836/tjm/1264170239. Google Scholar [18] H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411. Google Scholar [19] M. Kyed, Time-Periodic Solutions to the Navier-Stokes Equations, Habilitation, Technische Universit t, Darmstadt, 2012, Available from: http://tuprints.ulb.tu-darmstadt.de/3309/.Google Scholar [20] J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410. doi: 10.1137/10078983X. Google Scholar [21] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Google Scholar [22] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. Google Scholar [23] A. Lunardi, Bounded solutions of linear periodic abstract parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988), 135-159. doi: 10.1017/S0308210500024926. Google Scholar [24] A. Lunardi, Stability of the periodic solutions to fully nonlinear parabolic equations in Banach spaces, Differential Integral Equations, 1 (1988), 253-279. Google Scholar [25] P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529, Available from: http://stacks.iop.org/0951-7715/4/503. doi: 10.1088/0951-7715/4/2/013. Google Scholar [26] P. Maremonti and M. Padula, Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 233 (1996), 142-182. doi: 10.1007/BF02366850. Google Scholar [27] V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, vol. 162 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162. Google Scholar [28] H. Morimoto, Survey on time periodic problem for fluid flow under inhomogeneous boundary condition, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 631-639. doi: 10.3934/dcdss.2012.5.631. Google Scholar [29] B. Muha and S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919-968. doi: 10.1007/s00205-012-0585-5. Google Scholar [30] A. Pazy, Semi-groups of Linear Operators and Applications to Partial Differential Equations, Department of Mathematics, University of Maryland, College Park, Md., 1974, Department of Mathematics, University of Maryland, Lecture Note, No. 10. Google Scholar [31] G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso bidimensionale, Rend. Sem. Mat. Univ. Padova, 30 (1960), 1-15, Available from: http://www.numdam.org/item?id=RSMUP_1960__30__1_0. Google Scholar [32] G. Prouse, Soluzioni periodiche dell'equazione di Navier-Stokes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 35 (1963), 443-447. Google Scholar [33] J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443. doi: 10.1137/080744761. Google Scholar [34] J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122. doi: 10.1007/BF00284169. Google Scholar [35] A. Takeshita, On the reproductive property of the $2$-dimensional Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1969), 297-311. Google Scholar [36] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418. Google Scholar
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