American Institute of Mathematical Sciences

Advanced
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 & 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

Figure 1.  Fluid-structure system.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325
[2]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133
[3]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
[4]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[5]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813
[6]

George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817
[7]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[8]

Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419
[9]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[10]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[11]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355
[12]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[13]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[14]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189
[15]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[16]

Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269
[17]

George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
[18]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783
[19]

Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
[20]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (45)
  • HTML views (80)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences