June  2020, 9(2): 301-339. doi: 10.3934/eect.2020007

On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

*Corresponding author: Manil T. Mohan

Received  December 2018 Revised  February 2019 Published  June 2020 Early access  August 2019

In this work, we consider the three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt fluids in bounded and unbounded domains. We investigate the global solvability results, asymptotic behavior and also address some control problems of such viscoelastic fluid flow equations with "fading memory" and "memory of length $ \tau $". A local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique are used to obtain global solvability results. Since we are not using compactness arguments in the proofs, the global solvability results are also valid in unbounded domains like Poincaré domains. We also remark that using an $ m $-accretive quantization of the linear and nonlinear operators, one can establish the existence and uniqueness of strong solutions for the Navier-Stokes-Voigt equations and avoid the tedious Galerkin approximation scheme. We examine the asymptotic behavior of the stationary solutions and also establish the exponential stability results. Finally, under suitable assumptions on the Galerkin basis, we consider the controlled Galerkin approximated 3D Kelvin-Voigt fluid flow equations. Using the Hilbert uniqueness method combined with Schauder's fixed point theorem, the exact controllability of the finite dimensional Galerkin approximated system is established.

Citation: Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations and Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007
References:
[1]

F. D. ArarunaF. W. Chaves-Silva and M. A. Rojas-Medar, Exact controllability of Galerkin's approximations of micropolar fluids, Proceedings of the American Mathematical Society, 138 (2010), 1361-1370.  doi: 10.1090/S0002-9939-09-10154-5.

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[5]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Zeitschrift für angewandte Mathematik und Physik, 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. 
[8]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.

[9]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[10]

T. Caraballo and J. Real, Asymptotic behavior of Navier-Stokes equations with delays, R. Soc. London A, 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.

[11]

F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.

[12]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.

[13]

B. P. W. FernandoS. S. Sritharan and M. Xu, A simple proof of global solvability for 2-D Navier-Stokes equations in unbounded domains, Differential and Integral Equations, 23 (2010), 223-235. 

[14]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.

[15]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math., 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[16]

S. Kundu and A. K. Pani, Stabilization of Kelvin-Voigt viscoelastic fluid flow model, Applicable Analysis, 2018, https://doi.org/10.1080/00036811.2018.1460810. doi: 10.1080/00036811.2018.1460810.

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[18]

J.-L. Lions and E. Zuazua, Contrôlabilité exacte des approximations de Galerkin des èquations de Navier-Stokes, C. R. Acad. Sci. Paris, Ser. I, 324 (1997), 1015-1021.  doi: 10.1016/S0764-4442(97)87878-0.

[19]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 605-621. 

[20]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.

[21]

J. Lukkarinen and M. S. Pakkanen, On the positivity of Riemann-Stieltjes integrals, Bull. Aust. Math. Soc., 87 (2013), 400-405.  doi: 10.1017/S0004972712000639.

[22]

A. O. MarinhoA. T. Lourêdo and M. Milla Miranda, Exact controllability of Galerkin's approximations for the Oldroyd fluid system, International Journal of Modeling and Optimization, 4 (2014), 126-132.  doi: 10.7763/IJMO.2014.V4.359.

[23]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Communications on Stochastic Analysis, 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.

[24]

M. T. Mohan, An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity, Accepted in Pure and Applied Functional Analysis, (2019).

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: stationary solution and stability, Applicable Analysis, (2018), https://doi.org/10.1080/00036811.2018.1546002. doi: 10.1080/00036811.2018.1546002.

[26]

M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Accepted in Stochastic Analysis and Applications, (2019).

[27]

M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted.

[28]

M. T. Mohan and S. S. Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evolution Equations and Control Theory, 5 (2016), 273-302.  doi: 10.3934/eect.2016005.

[29]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear symmetric hyperbolic system perturbed by Lévy noise, Pure and Applied Functional Analysis, 3 (2018), 137-178. 

[30]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equation perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[32]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136. 

[33]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.

[34]

A. P. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Proceedings of Steklov Institute of Mathematics, 2 (1989), 137-182. 

[35]

A. P. Oskolkov, Nonlocal problems for the equations of Kelvin- Voight fluids and their $\varepsilon$approximations in classes of smooth functions, Zap. Nauchn. Sem. POMI, 230 (1995), 214-242.  doi: 10.1007/BF02433999.

[36]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.

[37]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1992), 2699-2723.  doi: 10.1007/BF01102639.

[38]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2, Journal of Mathematical Sciences, 59 (1992), 1206-1214.  doi: 10.1007/BF01374082.

[39]

A. P. Oskolkov and R. D. Shadiev, Towards a theory of global solvability on $[0, \infty)$ of initial-boundary value problems for the equations of motion of oldroyd and Kelvin-Voight fluids, Journal of Mathematical Sciences, 68 (1994), 240-253.  doi: 10.1007/BF01249338.

[40]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[42]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.

show all references

References:
[1]

F. D. ArarunaF. W. Chaves-Silva and M. A. Rojas-Medar, Exact controllability of Galerkin's approximations of micropolar fluids, Proceedings of the American Mathematical Society, 138 (2010), 1361-1370.  doi: 10.1090/S0002-9939-09-10154-5.

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[5]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Zeitschrift für angewandte Mathematik und Physik, 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. 
[8]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.

[9]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[10]

T. Caraballo and J. Real, Asymptotic behavior of Navier-Stokes equations with delays, R. Soc. London A, 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.

[11]

F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.

[12]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.

[13]

B. P. W. FernandoS. S. Sritharan and M. Xu, A simple proof of global solvability for 2-D Navier-Stokes equations in unbounded domains, Differential and Integral Equations, 23 (2010), 223-235. 

[14]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.

[15]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math., 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[16]

S. Kundu and A. K. Pani, Stabilization of Kelvin-Voigt viscoelastic fluid flow model, Applicable Analysis, 2018, https://doi.org/10.1080/00036811.2018.1460810. doi: 10.1080/00036811.2018.1460810.

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[18]

J.-L. Lions and E. Zuazua, Contrôlabilité exacte des approximations de Galerkin des èquations de Navier-Stokes, C. R. Acad. Sci. Paris, Ser. I, 324 (1997), 1015-1021.  doi: 10.1016/S0764-4442(97)87878-0.

[19]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 605-621. 

[20]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.

[21]

J. Lukkarinen and M. S. Pakkanen, On the positivity of Riemann-Stieltjes integrals, Bull. Aust. Math. Soc., 87 (2013), 400-405.  doi: 10.1017/S0004972712000639.

[22]

A. O. MarinhoA. T. Lourêdo and M. Milla Miranda, Exact controllability of Galerkin's approximations for the Oldroyd fluid system, International Journal of Modeling and Optimization, 4 (2014), 126-132.  doi: 10.7763/IJMO.2014.V4.359.

[23]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Communications on Stochastic Analysis, 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.

[24]

M. T. Mohan, An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity, Accepted in Pure and Applied Functional Analysis, (2019).

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: stationary solution and stability, Applicable Analysis, (2018), https://doi.org/10.1080/00036811.2018.1546002. doi: 10.1080/00036811.2018.1546002.

[26]

M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Accepted in Stochastic Analysis and Applications, (2019).

[27]

M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted.

[28]

M. T. Mohan and S. S. Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evolution Equations and Control Theory, 5 (2016), 273-302.  doi: 10.3934/eect.2016005.

[29]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear symmetric hyperbolic system perturbed by Lévy noise, Pure and Applied Functional Analysis, 3 (2018), 137-178. 

[30]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equation perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[32]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136. 

[33]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.

[34]

A. P. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Proceedings of Steklov Institute of Mathematics, 2 (1989), 137-182. 

[35]

A. P. Oskolkov, Nonlocal problems for the equations of Kelvin- Voight fluids and their $\varepsilon$approximations in classes of smooth functions, Zap. Nauchn. Sem. POMI, 230 (1995), 214-242.  doi: 10.1007/BF02433999.

[36]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.

[37]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1992), 2699-2723.  doi: 10.1007/BF01102639.

[38]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2, Journal of Mathematical Sciences, 59 (1992), 1206-1214.  doi: 10.1007/BF01374082.

[39]

A. P. Oskolkov and R. D. Shadiev, Towards a theory of global solvability on $[0, \infty)$ of initial-boundary value problems for the equations of motion of oldroyd and Kelvin-Voight fluids, Journal of Mathematical Sciences, 68 (1994), 240-253.  doi: 10.1007/BF01249338.

[40]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[42]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.

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