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

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

Jean-Jérôme Casanova 1,2,

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.

Keywords: Fluid–structure interaction, Navier–Stokes equations, beam equation, mixed boundary conditions, periodic system.
Mathematics Subject Classification: Primary: 74F10, 35B10; Secondary: 76D03, 35Q30, 76D05.
Citation: Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

J.-J. Casanova, Fluid structure system with boundary conditions involving the pressure, 2017, arXiv: 1707.06382.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[15]

V. I. Judovič, Periodic motions of a viscous incompressible fluid, Soviet Math. Dokl., 1 (1960), 168-172. 

[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.

[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.

[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.

[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/.

[20]

J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410.  doi: 10.1137/10078983X.

[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.

[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.

[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.

[24]

A. Lunardi, Stability of the periodic solutions to fully nonlinear parabolic equations in Banach spaces, Differential Integral Equations, 1 (1988), 253-279. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[33]

J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.  doi: 10.1137/080744761.

[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.

[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. 

[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.

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.

[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.

[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.

[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.

[5]

J.-J. Casanova, Fluid structure system with boundary conditions involving the pressure, 2017, arXiv: 1707.06382.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[15]

V. I. Judovič, Periodic motions of a viscous incompressible fluid, Soviet Math. Dokl., 1 (1960), 168-172. 

[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.

[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.

[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.

[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/.

[20]

J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410.  doi: 10.1137/10078983X.

[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.

[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.

[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.

[24]

A. Lunardi, Stability of the periodic solutions to fully nonlinear parabolic equations in Banach spaces, Differential Integral Equations, 1 (1988), 253-279. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[33]

J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.  doi: 10.1137/080744761.

[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.

[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. 

[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.

Figure 1.  Fluid-structure system.
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