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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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.
[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.
[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.

[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.

[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.

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