March  2017, 6(1): 135-154. doi: 10.3934/eect.2017008

The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework

Department of Mathematics and Economics, Virginia State University, Petersburg, VA 23806, USA

Received  October 2016 Revised  August 2016 Published  December 2016

In this paper, we study a fluid-structure interaction model of Stokes-wave equation coupling system with Kelvin-Voigt type of damping. We show that this damped coupling system generates an analyticity semigroup and thus the semigroup solution, which also satisfies variational framework of weak solution, decays to zero at exponential rate.

Citation: Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008
References:
[1]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction. Part Ⅰ: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-55.  doi: 10.1090/conm/440/08475.  Google Scholar

[2]

G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J. , Special issue dedicated to the memory of J. L. Lions, 15 (2008), 403-437.  Google Scholar

[3]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discr. Cont. Dynam. Sys., 22 (2008), 817-833.  doi: 10.3934/dcds.2008.22.817.  Google Scholar

[4]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[5]

G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evolution Equations and Control Theory, 2 (2013), 563-598.  doi: 10.3934/eect.2013.2.563.  Google Scholar

[6]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

V. Barbu, Nonlinear Semigroup and Differential Equations in Banach Spaces, Springer, 1976.  Google Scholar

[8]

V. BarbuZ. GrujicI. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1773-1207.  doi: 10.1512/iumj.2008.57.3284.  Google Scholar

[9]

V. BarbuZ. GrujicI. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82.  doi: 10.1090/conm/440/08476.  Google Scholar

[10]

S. CanicB. Muha and M. Bukac, Stability of the Kinematically Coupled $β$-Scheme for fluid-structure interaction problems in hemodynamics, International Journal for Numerical Analysis and Modeling, 12 (2015), 54-80.   Google Scholar

[11]

S. Chen and R. Triggiani, Proof of the extensions of two conjectures on structural damping for elastic system, Pacific Journal of Mathematics, 36 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15.  Google Scholar

[12]

S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $ 0 < \alpha < \frac{1}{2}$, Proc. Amer. Math. Soc., 110 (1990), 401-415.  doi: 10.2307/2048084.  Google Scholar

[13]

C. ClasonB. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation, J. Math. Anal. Appl., 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[14]

D. Coutand and S. Shkoller, Motion of an elastic inside an incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 25-102.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[15]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type Memoirs Amer. Math. Soc. 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[16]

W. DeschM. Hieber and J. Pruss, $L_p$ theory of the Stokes equation in a half space, J. Evolution Eqns, 1 (2001), 115-142.  doi: 10.1007/PL00001362.  Google Scholar

[17]

W. Desch and W. Schappacher, Some perturbation results for analytic semigroups, Mathematische Annalen, 281 (1988), 157-162.  doi: 10.1007/BF01449222.  Google Scholar

[18]

Q. DuM. D. GunzburgerL. S. Hou and J. Lee, Analysis of a linear-fluid structure interaction model, Discr. Dynam. Sys., 9 (2003), 633-650.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[19]

Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ space, Mathematische Zeiscrift, 178 (1981), 297-329.  doi: 10.1007/BF01214869.  Google Scholar

[20]

Y. Giga, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. RIMS, Tokyo Univ., 19 (1983), 887-910.  doi: 10.2977/prims/1195182014.  Google Scholar

[21]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[22]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discr. Cont. Dynam. Sys., Series S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[23]

B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations, Appl. Math. & Opti., 62 (2010), 381-410.  doi: 10.1007/s00245-010-9108-7.  Google Scholar

[24]

I. KukavicaA. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Diff. Eq., 247 (2009), 1452-1478.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[25]

I. KukavicaA. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, Adv. Diff. Eq., 15 (2010), 231-254.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[26]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs SIAM, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[27]

I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction, Semigroup Forum, 82 (2011), 61-82.  doi: 10.1007/s00233-010-9281-7.  Google Scholar

[28]

I. Lasiecka and Y. Lu, Interface feedback control stabilization to a nonlinear fluid-structure interaction model, Nonlinear Anal., 75 (2012), 1449-1460.  doi: 10.1016/j.na.2011.04.018.  Google Scholar

[29]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, I: Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74 Cambridge University Press, 2000.  Google Scholar

[30]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Communications on Pure and Applied Analysis, 15 (2016).  doi: 10.3934/cpaa.2016001.  Google Scholar

[31]

K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and Gyroscopitc forces, J. Diff. Eq., 141 (1997), 340-355.  doi: 10.1006/jdeq.1997.3331.  Google Scholar

[32]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems Chapman & Hall/ CRC Research Notes in Mathematics, 1999.  Google Scholar

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. and Opti., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[34]

B. Muha and S. Canic, 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, Archives for Rational Mechanics and Analysis, 207 (2013), 919-296871.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[35]

B. Muha and S. Canic, Existence of a solution to a fluid-multi-layered-structure interaction problem, Journal of Differential Equations, 256 (2014), 658-706.  doi: 10.1016/j.jde.2013.09.016.  Google Scholar

[36]

N. $\ddot{O}$zkaya, M. Nordin, D. Goldsheyder and D. Leger, Fundamentals of Biomechanics-Equilibrium, Motion, and Deformation Springer-Verlag, New York, 2012. Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

J. Pruss, On the spectrum of $C_0$ semigroup, Transactions of American Mathematics Society, 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[39]

G. Simonett and M. Wilke, Well-posedness and long-time behaviour for the Westervelt equation with absorbing boundary conditions of order zero, To appear in in J. of Evol. Eqns. Google Scholar

[40]

R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A. V. Balakrishnan, 73(3) (2016), 571-594. doi: 10.1007/s00245-016-9348-2.  Google Scholar

[41]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction, Arch. Rat. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

show all references

References:
[1]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction. Part Ⅰ: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-55.  doi: 10.1090/conm/440/08475.  Google Scholar

[2]

G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J. , Special issue dedicated to the memory of J. L. Lions, 15 (2008), 403-437.  Google Scholar

[3]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discr. Cont. Dynam. Sys., 22 (2008), 817-833.  doi: 10.3934/dcds.2008.22.817.  Google Scholar

[4]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[5]

G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evolution Equations and Control Theory, 2 (2013), 563-598.  doi: 10.3934/eect.2013.2.563.  Google Scholar

[6]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

V. Barbu, Nonlinear Semigroup and Differential Equations in Banach Spaces, Springer, 1976.  Google Scholar

[8]

V. BarbuZ. GrujicI. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1773-1207.  doi: 10.1512/iumj.2008.57.3284.  Google Scholar

[9]

V. BarbuZ. GrujicI. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82.  doi: 10.1090/conm/440/08476.  Google Scholar

[10]

S. CanicB. Muha and M. Bukac, Stability of the Kinematically Coupled $β$-Scheme for fluid-structure interaction problems in hemodynamics, International Journal for Numerical Analysis and Modeling, 12 (2015), 54-80.   Google Scholar

[11]

S. Chen and R. Triggiani, Proof of the extensions of two conjectures on structural damping for elastic system, Pacific Journal of Mathematics, 36 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15.  Google Scholar

[12]

S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $ 0 < \alpha < \frac{1}{2}$, Proc. Amer. Math. Soc., 110 (1990), 401-415.  doi: 10.2307/2048084.  Google Scholar

[13]

C. ClasonB. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation, J. Math. Anal. Appl., 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[14]

D. Coutand and S. Shkoller, Motion of an elastic inside an incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 25-102.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[15]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type Memoirs Amer. Math. Soc. 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[16]

W. DeschM. Hieber and J. Pruss, $L_p$ theory of the Stokes equation in a half space, J. Evolution Eqns, 1 (2001), 115-142.  doi: 10.1007/PL00001362.  Google Scholar

[17]

W. Desch and W. Schappacher, Some perturbation results for analytic semigroups, Mathematische Annalen, 281 (1988), 157-162.  doi: 10.1007/BF01449222.  Google Scholar

[18]

Q. DuM. D. GunzburgerL. S. Hou and J. Lee, Analysis of a linear-fluid structure interaction model, Discr. Dynam. Sys., 9 (2003), 633-650.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[19]

Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ space, Mathematische Zeiscrift, 178 (1981), 297-329.  doi: 10.1007/BF01214869.  Google Scholar

[20]

Y. Giga, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. RIMS, Tokyo Univ., 19 (1983), 887-910.  doi: 10.2977/prims/1195182014.  Google Scholar

[21]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[22]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discr. Cont. Dynam. Sys., Series S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[23]

B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations, Appl. Math. & Opti., 62 (2010), 381-410.  doi: 10.1007/s00245-010-9108-7.  Google Scholar

[24]

I. KukavicaA. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Diff. Eq., 247 (2009), 1452-1478.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[25]

I. KukavicaA. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, Adv. Diff. Eq., 15 (2010), 231-254.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[26]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs SIAM, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[27]

I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction, Semigroup Forum, 82 (2011), 61-82.  doi: 10.1007/s00233-010-9281-7.  Google Scholar

[28]

I. Lasiecka and Y. Lu, Interface feedback control stabilization to a nonlinear fluid-structure interaction model, Nonlinear Anal., 75 (2012), 1449-1460.  doi: 10.1016/j.na.2011.04.018.  Google Scholar

[29]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, I: Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74 Cambridge University Press, 2000.  Google Scholar

[30]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Communications on Pure and Applied Analysis, 15 (2016).  doi: 10.3934/cpaa.2016001.  Google Scholar

[31]

K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and Gyroscopitc forces, J. Diff. Eq., 141 (1997), 340-355.  doi: 10.1006/jdeq.1997.3331.  Google Scholar

[32]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems Chapman & Hall/ CRC Research Notes in Mathematics, 1999.  Google Scholar

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. and Opti., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[34]

B. Muha and S. Canic, 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, Archives for Rational Mechanics and Analysis, 207 (2013), 919-296871.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[35]

B. Muha and S. Canic, Existence of a solution to a fluid-multi-layered-structure interaction problem, Journal of Differential Equations, 256 (2014), 658-706.  doi: 10.1016/j.jde.2013.09.016.  Google Scholar

[36]

N. $\ddot{O}$zkaya, M. Nordin, D. Goldsheyder and D. Leger, Fundamentals of Biomechanics-Equilibrium, Motion, and Deformation Springer-Verlag, New York, 2012. Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

J. Pruss, On the spectrum of $C_0$ semigroup, Transactions of American Mathematics Society, 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[39]

G. Simonett and M. Wilke, Well-posedness and long-time behaviour for the Westervelt equation with absorbing boundary conditions of order zero, To appear in in J. of Evol. Eqns. Google Scholar

[40]

R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A. V. Balakrishnan, 73(3) (2016), 571-594. doi: 10.1007/s00245-016-9348-2.  Google Scholar

[41]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction, Arch. Rat. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

Figure 1.  THE FLUID–STRUCTURE INTERACTION
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