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On existence and uniqueness of solution for a hydrodynamic problem related to water artificial circulation in a lake. (English) Zbl 1447.35244

Summary: In this work we introduce a well-posed mathematical model for the processes involved in the artificial circulation of water, in order to avoid eutrophication phenomena, for instance, in a lake. This novel and general formulation is based on the modified Navier-Stokes equations following the Smagorinsky model of turbulence, and presenting a suitable nonhomogeneous Dirichlet boundary condition. For the analytical study of the problem, we prove several theoretical results related to existence, uniqueness and smoothness for the solution of this recirculation model.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76F02 Fundamentals of turbulence
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B65 Smoothness and regularity of solutions to PDEs
76U05 General theory of rotating fluids
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References:

[1] An, R.; Li, Y.; Zhang, Y., Error estimates of two-level finite element method for Smagorinsky model, Appl. Math. Comput., 274, 786-800 (2016) · Zbl 1410.65442
[2] Beirão da Veiga, H., On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., 11, 127-167 (2009) · Zbl 1155.35073
[3] Borggaard, J.; Iliescu, T.; Lee, H.; Roop, J. P.; Son, H., A two-level discretization method for the Smagorinsky model, Multiscale Model. Simul., 7, 599-621 (2008) · Zbl 1201.76086
[4] Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Semin. Mat. Univ. Padova, 31, 308-340 (1961) · Zbl 0116.18002
[5] Chacón Rebollo, T.; Delgado Avila, E.; Gómez Mármol, M.; Ballarin, F.; Rozza, G., On a certified Smagorinsky reduced basis turbulence model, SIAM J. Numer. Anal., 55, 3047-3067 (2017) · Zbl 1380.65339
[6] Chacón Rebollo, T.; Hecht, F.; Gómez Mármol, M.; Orzetti, G.; Rubino, S., Numerical approximation of the Smagorinsky turbulence model applied to the primitive equations of the ocean, Math. Comput. Simulation, 99, 54-70 (2014) · Zbl 07312580
[7] Fernández-Cara, E.; Límaco, J.; de Menezes, S. B., Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence, J. Math. Fluid Mech., 17, 669-698 (2015) · Zbl 1330.35482
[8] Gunzburger, M. D.; Ladyzhenskaya, O. A.; Peterson, J. S., On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes and Maxwell equations, J. Math. Fluid Mech., 6, 462-482 (2004) · Zbl 1064.76118
[9] Gunzburger, M.; Trenchea, C., Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations, J. Math. Anal. Appl., 333, 295-310 (2007) · Zbl 1158.49003
[10] Hale, J. K., Ordinary Differential Equations (1980), Robert E. Krieger Publishing Co.: Robert E. Krieger Publishing Co. Huntington · Zbl 0433.34003
[11] John, V., Large Eddy Simulation of Turbulent Incompressible Flows (2004), Springer: Springer Berlin · Zbl 1035.76001
[12] Ladyženskaja, O. A.; Solonnikov, V. A.; Uralceva, N. N., Linear and Quasilinear Equations of Parabolic Type (1968), American Mathematical Society: American Mathematical Society Providence · Zbl 0174.15403
[13] Leveque, E.; Toschi, F.; Shao, L.; Bertoglio, J. P., Shear-improved Smagorinsky model for large-eddy simulation of wall-bounded turbulent flows, J. Fluid Mech., 570, 491-502 (2007) · Zbl 1105.76034
[14] Martínez, A.; Fernández, F. J.; Alvarez-Vázquez, L. J., Water artificial circulation for eutrophication control, Math. Control Relat. Fields, 8, 277-313 (2018) · Zbl 1407.49029
[15] Meyers, J.; Geurts, B. J.; Sagaut, P., A computational error-assessment of central finite-volume discretizations in large-eddy simulation using a Smagorinsky model, J. Comput. Phys., 227, 156-173 (2007) · Zbl 1280.76012
[16] Pakzad, A., Damping functions correct over-dissipation of the Smagorinsky model, Math. Methods Appl. Sci., 40, 5933-5945 (2017) · Zbl 1382.76117
[17] Roubíček, T., Nonlinear Partial Differential Equations with Applications (2013), Birkhäuser: Birkhäuser Basel · Zbl 1270.35005
[18] Temam, R., Navier-Stokes Equations (1979), North-Holland: North-Holland Amsterdam · Zbl 0454.35073
[19] Tran, S.; Sahni, O., Finite element-based large eddy simulation using a combination of the variational multiscale method and the dynamic Smagorinsky model, J. Turbul., 18, 391-417 (2017)
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