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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

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.

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

C. K. ChuiF. 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.

[11]

P. ConstantinB. 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.

[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.

[13]

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

[14]

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

[15]

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

[16]

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

[17]

V. S. L'vovE. PodivilovA. PomyalovI. ocaccia and D. Vandembroucq, Improved shell model of turbulence, Physical Review E, 58 (1998), 1811-1822. doi: 10.1103/PhysRevE.58.1811.

[18]

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

[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.

[20]

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

[21]

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

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

C. K. ChuiF. 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.

[11]

P. ConstantinB. 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.

[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.

[13]

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

[14]

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

[15]

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

[16]

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

[17]

V. S. L'vovE. PodivilovA. PomyalovI. ocaccia and D. Vandembroucq, Improved shell model of turbulence, Physical Review E, 58 (1998), 1811-1822. doi: 10.1103/PhysRevE.58.1811.

[18]

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

[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.

[20]

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

[21]

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

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