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September  2018, 7(3): 417-445. doi: 10.3934/eect.2018021

Control problems and invariant subspaces for sabra shell model of turbulence

Tania Biswas and  Sheetal Dharmatti ,

School of Mathematics, IISER Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram, Kerala, 695 551, India

* Corresponding author

Received  November 2017 Revised  March 2018 Published  July 2018

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. Another interesting problem studied in this work is to establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space, under certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set.

Keywords: Shell models, turbulence, Navier Stokes Equations, optimal control, control of fluid flow, turbulence control, maximal monotone operators, quasi m-accretive operator, nonlinear semigroups, flow invariance, state constraints.
Mathematics Subject Classification: Primary: 35Q93, 76F70; Secondary: 76D05.
Citation: Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations & Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.  Google Scholar

[2]

V. Barbu, Stabilization of Navier Stokes Flows, Stabilization of Navier-Stokes Flows, Springer London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Bucharest-Noordhoff, Leyden: Editura Academiei, 1976.  Google Scholar

[4]

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

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Science and Business Media, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[6]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998. doi: 10.1017/CBO9780511599972.  Google Scholar

[7]

H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[8]

L. Chong and X.-Q. Jin, Nonlinearly constrained best approximation in Hilbert spaces: The strong CHIP and the basic constraint qualification, SIAM Journal on Optimization, 13 (2002), 228-239.  doi: 10.1137/S1052623401385600.  Google Scholar

[9]

C. K. Chui, F. Deutsch and J. D. Ward, Constrained best approximation in Hilbert space, Constructive Approximation, 6 (1990), 35-64.  doi: 10.1007/BF01891408.  Google Scholar

[10]

C. K. Chui, F. Deutsch and J. D. Ward, Constrained best approximation in Hilbert space, Ⅱ, Journal of Approximation Theory, 71 (1992), 213-238.  doi: 10.1016/0021-9045(92)90117-7.  Google Scholar

[11]

P. Constantin, B. Levant and E. S. Titi, Analytic study of shell models of turbulence, Physica D: Nonlinear Phenomena, 219 (2006), 120-141.  doi: 10.1016/j.physd.2006.05.015.  Google Scholar

[12]

P. Constantin, B. Levant and E. S. Titi, Regularity of inviscid shell models of turbulence, Physical Review E, 75 (2007), 016304, 10pp. doi: 10.1103/PhysRevE.75.016304.  Google Scholar

[13]

P. D. Ditlevsen, Turbulence and Shell Models, Cambridge University Press, Cambridge, 2011.  Google Scholar

[14]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.  Google Scholar

[15]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.   Google Scholar

[16]

L. P. Kadanoff, A Model of Turbulence, Reference Frame, Physics Today, September, 1995. Google Scholar

[17]

V. S. L'vov, E. Podivilov, A. Pomyalov, I. ocaccia and D. Vandembroucq, Improved shell model of turbulence, Physical Review E, 58 (1998), 1811-1822.  doi: 10.1103/PhysRevE.58.1811.  Google Scholar

[18]

J. M. McDonough, Introductory lectures on turbulence physics, mathematics and modeling. 2004. Google Scholar

[19]

K. Ohkitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 81 (1989), 329-341.  doi: 10.1143/PTP.81.329.  Google Scholar

[20]

J. P. Raymond, Optimal Control of Partial Differential Equations, Lecture Notes. Google Scholar

[21]

S. Sritharan, Optimal Control of Viscous Flow, SIAM, 1998. doi: 10.1137/1.9781611971415.  Google Scholar

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.  Google Scholar

[2]

V. Barbu, Stabilization of Navier Stokes Flows, Stabilization of Navier-Stokes Flows, Springer London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Bucharest-Noordhoff, Leyden: Editura Academiei, 1976.  Google Scholar

[4]

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

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Science and Business Media, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[6]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998. doi: 10.1017/CBO9780511599972.  Google Scholar

[7]

H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[8]

L. Chong and X.-Q. Jin, Nonlinearly constrained best approximation in Hilbert spaces: The strong CHIP and the basic constraint qualification, SIAM Journal on Optimization, 13 (2002), 228-239.  doi: 10.1137/S1052623401385600.  Google Scholar

[9]

C. K. Chui, F. Deutsch and J. D. Ward, Constrained best approximation in Hilbert space, Constructive Approximation, 6 (1990), 35-64.  doi: 10.1007/BF01891408.  Google Scholar

[10]

C. K. Chui, F. Deutsch and J. D. Ward, Constrained best approximation in Hilbert space, Ⅱ, Journal of Approximation Theory, 71 (1992), 213-238.  doi: 10.1016/0021-9045(92)90117-7.  Google Scholar

[11]

P. Constantin, B. Levant and E. S. Titi, Analytic study of shell models of turbulence, Physica D: Nonlinear Phenomena, 219 (2006), 120-141.  doi: 10.1016/j.physd.2006.05.015.  Google Scholar

[12]

P. Constantin, B. Levant and E. S. Titi, Regularity of inviscid shell models of turbulence, Physical Review E, 75 (2007), 016304, 10pp. doi: 10.1103/PhysRevE.75.016304.  Google Scholar

[13]

P. D. Ditlevsen, Turbulence and Shell Models, Cambridge University Press, Cambridge, 2011.  Google Scholar

[14]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.  Google Scholar

[15]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.   Google Scholar

[16]

L. P. Kadanoff, A Model of Turbulence, Reference Frame, Physics Today, September, 1995. Google Scholar

[17]

V. S. L'vov, E. Podivilov, A. Pomyalov, I. ocaccia and D. Vandembroucq, Improved shell model of turbulence, Physical Review E, 58 (1998), 1811-1822.  doi: 10.1103/PhysRevE.58.1811.  Google Scholar

[18]

J. M. McDonough, Introductory lectures on turbulence physics, mathematics and modeling. 2004. Google Scholar

[19]

K. Ohkitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 81 (1989), 329-341.  doi: 10.1143/PTP.81.329.  Google Scholar

[20]

J. P. Raymond, Optimal Control of Partial Differential Equations, Lecture Notes. Google Scholar

[21]

S. Sritharan, Optimal Control of Viscous Flow, SIAM, 1998. doi: 10.1137/1.9781611971415.  Google Scholar

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