# American Institute of Mathematical Sciences

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  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 & Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007
##### References:
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Sritharan, Navier-Stokes equation with hereditary viscosity, Zeitschrift für angewandte Mathematik und Physik, 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.  Google Scholar [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.  Google Scholar [7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993.   Google Scholar [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.  Google Scholar [9] Y. P. Cao, E. 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.  Google Scholar [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.  Google Scholar [11] F. Di Plinio, A. Giorgini, V. 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.  Google Scholar [12] L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.  Google Scholar [13] B. P. W. Fernando, S. 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. O. Marinho, A. 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.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [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). Google Scholar [27] M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar [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.   Google Scholar [33] A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] F. D. Araruna, F. 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.  Google Scholar [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.  Google Scholar [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993.   Google Scholar [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.  Google Scholar [9] Y. P. Cao, E. 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.  Google Scholar [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.  Google Scholar [11] F. Di Plinio, A. Giorgini, V. 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.  Google Scholar [12] L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.  Google Scholar [13] B. P. W. Fernando, S. 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. O. Marinho, A. 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.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [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). Google Scholar [27] M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar [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.   Google Scholar [33] A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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