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February  2022, 11(1): 325-346. doi: 10.3934/eect.2021018

Improved boundary regularity for a Stokes-Lamé system

Francesca Bucci

Università degli Studi di Firenze, Dipartimento di Matematica e Informatica, 50139 Firenze, Italy

Received  July 2020 Revised  January 2021 Published  February 2022 Early access  April 2021

This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates are geared toward ensuring the well-posedness of the Riccati equations which arise from the associated optimal boundary control problems on a finite as well as infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

Keywords: Fluid-structure interaction, linear elasticity, boundary regularity, Riccati equations, optimal feedback control.
Mathematics Subject Classification: Primary: 35B65, 35Q74, 35Q93; Secondary: 49N10.
Citation: Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2022, 11 (1) : 325-346. doi: 10.3934/eect.2021018
References:
[1]

P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs, Adv. Differential Equations, 10 (2005), 1389-1436.   Google Scholar

[2]

P. Acquistapace, F. Bucci and I. Lasiecka, A trace regularity result for thermoelastic equations with application to optimal boundary control, J. Math. Anal. Appl., 310 (2005), 262-277.  doi: 10.1016/j.jmaa.2005.02.008.  Google Scholar

[3]

P. Acquistapace, F. Bucci and I. Lasiecka, A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control, SIAM J. Math. Anal., 45 (2013), 1825-1870.  doi: 10.1137/120867433.  Google Scholar

[4]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, (eds. A. Favini, G. Fragnelli and R. Mininni), Springer INdAM Ser., 10, Springer, Cham, (2014), 49–78. doi: 10.1007/978-3-319-11406-4_3.  Google Scholar

[5]

G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423.  doi: 10.1016/j.jde.2015.01.037.  Google Scholar

[6]

G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics, J. Optim. Theory Appl., 91 (1996), 695-728.  doi: 10.1007/BF02190128.  Google Scholar

[7]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Series S, 2 (2009), 417-447.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[8]

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

[9]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and Waves, Contemp. Math., Amer. Math. Soc., Providence, RI, 440 (2007), 55–72, .  Google Scholar

[10]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^{nd}$ edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. xxviii+575 pp. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[11]

L. Bociu, L. Castle, I. Lasiecka and A. Tuffaha, Minimizing drag in a moving boundary fluid-elasticity interaction, Nonlinear Anal., 197 (2020), 111837, 44 pp. doi: 10.1016/j.na.2020.111837.  Google Scholar

[12]

F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results, Appl. Math., 35 (2008), 305-321.  doi: 10.4064/am35-3-4.  Google Scholar

[13]

F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 (2010), 217-235.  doi: 10.1007/s00526-009-0259-9.  Google Scholar

[14]

F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505-521.  doi: 10.3934/dcdss.2011.4.505.  Google Scholar

[15]

N. V. Chemetov, Š. Nečasová and B. Muha, Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition, J. Math. Phys., 60 (2019), 011505, 13 pp. doi: 10.1063/1.5007824.  Google Scholar

[16]

I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500.  doi: 10.1007/s00245-016-9349-1.  Google Scholar

[17]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656.  doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[18]

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

[19]

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

[20]

L. de Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine (Italian), Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223.   Google Scholar

[21]

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

[22]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3 (2003), 419-441.  doi: 10.1007/s00028-003-0110-1.  Google Scholar

[23]

M. Ignatova, I. Kukavica, I. 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

[24]

M. Ignatova, I. Kukavica, I. 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

[25]

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

[26]

I. Lasiecka, Unified theory for abstract parabolic boundary problems–a semigroup approach, Appl. Math. Optim., 6 (1980), 287-333.  doi: 10.1007/BF01442900.  Google Scholar

[27]

I. Lasiecka, Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators. Applications to Boundary and Point control problems, in: Functional Analytic Methods for Evolution Equations, 313–369, Lecture Notes in Math., 1855, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-44653-8_3.  Google Scholar

[28]

I. Lasiecka, J. -L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[29]

I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Vol. 164, Springer-Verlag, Berlin, 1991. xii+160 pp. doi: 10.1007/BFb0006880.  Google Scholar

[30]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅰ. Abstract Parabolic Systems; Ⅱ. Abstract Hyperbolic-like Systems over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, 75. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511574801.002.  Google Scholar

[31]

I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems and Control Letters, 58 (2009), 499-509.  doi: 10.1016/j.sysconle.2009.02.010.  Google Scholar

[32]

I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system, Control Cybernet., 38 (2009), 1429-1460.   Google Scholar

[33]

I. Lasiecka and J. Webster, Flow-plate interactions: Well-posedness and long time behavior, in: Mathematical Theory of Evolutionary Fluid-flow Structure Interactions, 173–268, Oberwolfach Semin., 48, Birkhäuser/Springer, Cham, 2018.  Google Scholar

[34]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (French), Dunod; Gauthier-Villars, Paris, 1969, xx+554 pp.  Google Scholar

[35]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vols. Ⅰ and Ⅱ, Springer Verlag, 1972.  Google Scholar

[36]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, [2013 reprint of the 1995 original] Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. xviii+424 pp.  Google Scholar

[37]

B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations, 260 (2016), 8550-8589.  doi: 10.1016/j.jde.2016.02.029.  Google Scholar

[38]

A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods, Comput. Vis. Sci., 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[39]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl., 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[40]

T. Richter, Fluid-structure Interactions. Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, vol. 118, Springer, 2017. doi: 10.1007/978-3-319-63970-3.  Google Scholar

[41]

T. Richter and T. Wick, Optimal control and parameter estimation for stationary fluid-structure interaction problems, SIAM J. Sci. Comput., 35 (2013), B1085–B1104. doi: 10.1137/120893239.  Google Scholar

[42]

J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147.  doi: 10.1007/s002050100172.  Google Scholar

show all references

References:
[1]

P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs, Adv. Differential Equations, 10 (2005), 1389-1436.   Google Scholar

[2]

P. Acquistapace, F. Bucci and I. Lasiecka, A trace regularity result for thermoelastic equations with application to optimal boundary control, J. Math. Anal. Appl., 310 (2005), 262-277.  doi: 10.1016/j.jmaa.2005.02.008.  Google Scholar

[3]

P. Acquistapace, F. Bucci and I. Lasiecka, A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control, SIAM J. Math. Anal., 45 (2013), 1825-1870.  doi: 10.1137/120867433.  Google Scholar

[4]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, (eds. A. Favini, G. Fragnelli and R. Mininni), Springer INdAM Ser., 10, Springer, Cham, (2014), 49–78. doi: 10.1007/978-3-319-11406-4_3.  Google Scholar

[5]

G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423.  doi: 10.1016/j.jde.2015.01.037.  Google Scholar

[6]

G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics, J. Optim. Theory Appl., 91 (1996), 695-728.  doi: 10.1007/BF02190128.  Google Scholar

[7]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Series S, 2 (2009), 417-447.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[8]

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

[9]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and Waves, Contemp. Math., Amer. Math. Soc., Providence, RI, 440 (2007), 55–72, .  Google Scholar

[10]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^{nd}$ edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. xxviii+575 pp. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[11]

L. Bociu, L. Castle, I. Lasiecka and A. Tuffaha, Minimizing drag in a moving boundary fluid-elasticity interaction, Nonlinear Anal., 197 (2020), 111837, 44 pp. doi: 10.1016/j.na.2020.111837.  Google Scholar

[12]

F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results, Appl. Math., 35 (2008), 305-321.  doi: 10.4064/am35-3-4.  Google Scholar

[13]

F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 (2010), 217-235.  doi: 10.1007/s00526-009-0259-9.  Google Scholar

[14]

F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505-521.  doi: 10.3934/dcdss.2011.4.505.  Google Scholar

[15]

N. V. Chemetov, Š. Nečasová and B. Muha, Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition, J. Math. Phys., 60 (2019), 011505, 13 pp. doi: 10.1063/1.5007824.  Google Scholar

[16]

I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500.  doi: 10.1007/s00245-016-9349-1.  Google Scholar

[17]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656.  doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[18]

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

[19]

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

[20]

L. de Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine (Italian), Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223.   Google Scholar

[21]

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

[22]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3 (2003), 419-441.  doi: 10.1007/s00028-003-0110-1.  Google Scholar

[23]

M. Ignatova, I. Kukavica, I. 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

[24]

M. Ignatova, I. Kukavica, I. 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

[25]

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

[26]

I. Lasiecka, Unified theory for abstract parabolic boundary problems–a semigroup approach, Appl. Math. Optim., 6 (1980), 287-333.  doi: 10.1007/BF01442900.  Google Scholar

[27]

I. Lasiecka, Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators. Applications to Boundary and Point control problems, in: Functional Analytic Methods for Evolution Equations, 313–369, Lecture Notes in Math., 1855, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-44653-8_3.  Google Scholar

[28]

I. Lasiecka, J. -L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[29]

I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Vol. 164, Springer-Verlag, Berlin, 1991. xii+160 pp. doi: 10.1007/BFb0006880.  Google Scholar

[30]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅰ. Abstract Parabolic Systems; Ⅱ. Abstract Hyperbolic-like Systems over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, 75. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511574801.002.  Google Scholar

[31]

I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems and Control Letters, 58 (2009), 499-509.  doi: 10.1016/j.sysconle.2009.02.010.  Google Scholar

[32]

I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system, Control Cybernet., 38 (2009), 1429-1460.   Google Scholar

[33]

I. Lasiecka and J. Webster, Flow-plate interactions: Well-posedness and long time behavior, in: Mathematical Theory of Evolutionary Fluid-flow Structure Interactions, 173–268, Oberwolfach Semin., 48, Birkhäuser/Springer, Cham, 2018.  Google Scholar

[34]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (French), Dunod; Gauthier-Villars, Paris, 1969, xx+554 pp.  Google Scholar

[35]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vols. Ⅰ and Ⅱ, Springer Verlag, 1972.  Google Scholar

[36]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, [2013 reprint of the 1995 original] Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. xviii+424 pp.  Google Scholar

[37]

B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations, 260 (2016), 8550-8589.  doi: 10.1016/j.jde.2016.02.029.  Google Scholar

[38]

A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods, Comput. Vis. Sci., 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[39]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl., 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[40]

T. Richter, Fluid-structure Interactions. Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, vol. 118, Springer, 2017. doi: 10.1007/978-3-319-63970-3.  Google Scholar

[41]

T. Richter and T. Wick, Optimal control and parameter estimation for stationary fluid-structure interaction problems, SIAM J. Sci. Comput., 35 (2013), B1085–B1104. doi: 10.1137/120893239.  Google Scholar

[42]

J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147.  doi: 10.1007/s002050100172.  Google Scholar

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