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Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains
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 |
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.
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. |

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