• Previous Article
    Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system
  • DCDS Home
  • This Issue
  • Next Article
    On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points
March  2020, 40(3): 1555-1593. doi: 10.3934/dcds.2020086

Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation

1. 

Department of Mathematical Science, Tsinghua University, Beijing 100084, China

2. 

Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Peng-Fei Yao

Received  February 2019 Revised  September 2019 Published  December 2019

Fund Project: The first and third author are supported by the National Science Foundation of China, grants no. 61473126 and no. 61573342, and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011. The second author is supported by the National Science Foundation of China, grants no. 11771235.

We consider a nonlinear, free boundary fluid-structure interaction model in a bounded domain. The viscous incompressible fluid interacts with a nonlinear elastic body on the common boundary via the velocity and stress matching conditions. The motion of the fluid is governed by incompressible Navier-Stokes equations while the displacement of elastic structure is determined by a nonlinear elastodynamic system with boundary dissipation. The boundary dissipation is inserted in the velocity matching condition. We prove the global existence of the smooth solutions for small initial data and obtain the exponential decay of the energy of this system as well.

Citation: Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086
References:
[1]

P. Cherrier and A. Milani, Linear and Quasi-linear Evolution Equations in Hilbert Spaces, Graduate Studies in Mathematics, 135, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/135.  Google Scholar

[2]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102.  doi: 10.1007/s00205-004-0340-7.  Google Scholar

[3]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[4]

Q. DuM. D. GunzburgerL. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[5]

G. GuidoboniR. GlowwinskiN. Cavallini and S. Canic, Stable loosely coupled type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937.  doi: 10.1016/j.jcp.2009.06.007.  Google Scholar

[6]

G. GuidoboniR. GlowwinskiN. CavalliniS. Canic and S. Lapin, A kinematically coupled time-splitting scheme for fluid-structure interaction in blood flow, Appl. Math. Lett., 22 (2009), 684-688.  doi: 10.1016/j.aml.2008.05.006.  Google Scholar

[7]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. Math. Phys., 53 (2012), 13pp. doi: 10.1063/1.4766724.  Google Scholar

[8]

M. IgnatovaI. KukavicaI. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[9]

M. IgnatovaI. KukavicaI. Lasiecka and A. Tuffaha, Small data global existence for a fluid-structure model, Nonlinearity, 30 (2017), 848-898.  doi: 10.1088/1361-6544/aa4ec4.  Google Scholar

[10]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete. Contin. Dyn. Syst., 32 (2012), 1355-1389.  doi: 10.3934/dcds.2012.32.1355.  Google Scholar

[11]

I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indina. Univ. Math. J., 61 (2012), 1817-1859.  doi: 10.1512/iumj.2012.61.4746.  Google Scholar

[12]

I. Kukavica and A. Tuffaha, Well-posedness for the compressible Navier-Stokes-Lamé system with a free interface, Nonlinearity, 25 (2012), 3111-3137.  doi: 10.1088/0951-7715/25/11/3111.  Google Scholar

[13]

Y. Qin and P. Yao, Energy decay and global solutions for a damped free boundary fluid-elastic structure interface model with variable coefficients in elasticity, Applicable Analysis, (2019). doi: 10.1080/00036811.2018.1551996.  Google Scholar

[14]

R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[15] P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics. A Differential Geometric Approach, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b11042.  Google Scholar
[16]

P. F. Yao, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differential Equations, 241 (2017), 62-93.  doi: 10.1016/j.jde.2007.06.014.  Google Scholar

[17]

Z.-F. Zhang and P.-F. Yao, Global smooth solutions of the quasi-linear wave equation with internal velocity feedback, SIAM J. Control Optim., 47 (2008), 2044-2077.  doi: 10.1137/070679454.  Google Scholar

[18]

Z.-F. Zhang and P.-F. Yao, Global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation, Systems Control Lett., 58 (2009), 491-498.  doi: 10.1016/j.sysconle.2009.02.007.  Google Scholar

show all references

References:
[1]

P. Cherrier and A. Milani, Linear and Quasi-linear Evolution Equations in Hilbert Spaces, Graduate Studies in Mathematics, 135, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/135.  Google Scholar

[2]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102.  doi: 10.1007/s00205-004-0340-7.  Google Scholar

[3]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[4]

Q. DuM. D. GunzburgerL. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[5]

G. GuidoboniR. GlowwinskiN. Cavallini and S. Canic, Stable loosely coupled type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937.  doi: 10.1016/j.jcp.2009.06.007.  Google Scholar

[6]

G. GuidoboniR. GlowwinskiN. CavalliniS. Canic and S. Lapin, A kinematically coupled time-splitting scheme for fluid-structure interaction in blood flow, Appl. Math. Lett., 22 (2009), 684-688.  doi: 10.1016/j.aml.2008.05.006.  Google Scholar

[7]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. Math. Phys., 53 (2012), 13pp. doi: 10.1063/1.4766724.  Google Scholar

[8]

M. IgnatovaI. KukavicaI. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[9]

M. IgnatovaI. KukavicaI. Lasiecka and A. Tuffaha, Small data global existence for a fluid-structure model, Nonlinearity, 30 (2017), 848-898.  doi: 10.1088/1361-6544/aa4ec4.  Google Scholar

[10]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete. Contin. Dyn. Syst., 32 (2012), 1355-1389.  doi: 10.3934/dcds.2012.32.1355.  Google Scholar

[11]

I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indina. Univ. Math. J., 61 (2012), 1817-1859.  doi: 10.1512/iumj.2012.61.4746.  Google Scholar

[12]

I. Kukavica and A. Tuffaha, Well-posedness for the compressible Navier-Stokes-Lamé system with a free interface, Nonlinearity, 25 (2012), 3111-3137.  doi: 10.1088/0951-7715/25/11/3111.  Google Scholar

[13]

Y. Qin and P. Yao, Energy decay and global solutions for a damped free boundary fluid-elastic structure interface model with variable coefficients in elasticity, Applicable Analysis, (2019). doi: 10.1080/00036811.2018.1551996.  Google Scholar

[14]

R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[15] P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics. A Differential Geometric Approach, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b11042.  Google Scholar
[16]

P. F. Yao, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differential Equations, 241 (2017), 62-93.  doi: 10.1016/j.jde.2007.06.014.  Google Scholar

[17]

Z.-F. Zhang and P.-F. Yao, Global smooth solutions of the quasi-linear wave equation with internal velocity feedback, SIAM J. Control Optim., 47 (2008), 2044-2077.  doi: 10.1137/070679454.  Google Scholar

[18]

Z.-F. Zhang and P.-F. Yao, Global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation, Systems Control Lett., 58 (2009), 491-498.  doi: 10.1016/j.sysconle.2009.02.007.  Google Scholar

[1]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[2]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349

[3]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[4]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[5]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[6]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[7]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[8]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[9]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[10]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141

[11]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[12]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[13]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[14]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[15]

Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002

[16]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[17]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[18]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062

[19]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128

[20]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (116)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]