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On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach
Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models
1. | Indiana University, Department of Mathematics, Bloomington, IN 47405, United States |
2. | Department of Mathematics, Indiana University, Bloomington, IN, 47405, United States |
3. | Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402-5002, United States |
References:
[1] |
G. K. Batchelor, The Theory of Homogeneous Turbulence,, Cambridge Monographs on Mechanics and Applied Mathematics, (1953).
|
[2] |
V. Chepyzhov, E. S. Titi and M. Vishik, On the convergence of solutions of the 3D Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,, \emph{Discr.} & \emph{Cont. Dyn. Systems A}, 17 (2007), 481.
|
[3] |
A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, \emph{Royal Soc. A, 461 (2005), 629.
doi: 10.1098/rspa.2004.1373. |
[4] |
R. Dascaliuc, C. Foias and M. S. Jolly, Relations between energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations,, \emph{J. Dynam. Differential Equations}, 17 (2005), 643.
doi: 10.1007/s10884-005-8269-6. |
[5] |
R. Dascaliuc, C. Foias and M. S. Jolly, Universal bounds on the attractor of the Navier-Stokes equation in the energy, enstrophy plane,, \emph{J. Math. Phys.}, 48 (2007).
doi: 10.1063/1.2710349. |
[6] |
R. Dascaliuc, C. Foias and M. Jolly, Estimates on enstrophy, palinstrophy, and invariant measures for 2-d turbulence,, \emph{J. Differential Eqns}, 248 (2010), 792.
doi: 10.1016/j.jde.2009.11.020. |
[7] |
C. Doering, The 3D Navier-Stokes problem,, \emph{Annu. Rev. Fluid Mech}, 41 (2009), 109.
doi: 10.1146/annurev.fluid.010908.165218. |
[8] |
C. Foias, M. S. Jolly, O. P. Manley and R. Rosa, Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence,, \emph{J. Statist. Phys.}, 108 (2002), 591.
doi: 10.1023/A:1015782025005. |
[9] |
C. Foias, M. S. Jolly, O. P. Manley, R. Rosa and R. Temam, Kolmogorov theory via finite-time averages,, \emph{Phys. D}, 212 (2005), 245.
doi: 10.1016/j.physd.2005.10.002. |
[10] |
C. Foias, M. S. Jolly and M. Yang, On single mode forcing of the 2D-NSE,, \emph{J. Dynam. Diff. Eqns.}, 25 (2013), 393.
doi: 10.1007/s10884-013-9301-x. |
[11] |
C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory,, \emph{J. Dynam. Differential Equations}, 14 (2002), 1.
doi: 10.1023/A:1012984210582. |
[12] |
C. Foias, O. Manley, R. Rosa and R. Temam, Cascade of energy in turbulent flows,, \emph{Comptes Rendus Acad. Sci. Paris, 332 (2001), 509.
doi: 10.1016/S0764-4442(01)01831-6. |
[13] |
C. Foias, O. Manley, R. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows,, \emph{Comptes Rendus Acad. Sci. Paris, 333 (2001), 499.
doi: 10.1016/S0764-4442(01)02008-0. |
[14] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).
doi: 10.1017/CBO9780511546754. |
[15] |
C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes,, \emph{Ann. Mat. Pura Appl.}, 111 (2001), 307.
|
[16] |
D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid,, \emph{Physica D}, 133 (1999), 215.
doi: 10.1016/S0167-2789(99)00093-7. |
[17] |
D. D. Holm, J. E. Marsden and T. Ratiu, Euler-Poincaré Equations in Geophysical Fluid Dynamics,, \emph{In Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, ().
|
[18] |
M. Holst, E. Lunasin and G. Tsotgtgerel, Analytical study of generalized $\alpha$-models of turbulence,, \emph{Journal of Nonlinear Science}, 20 (2010), 523.
doi: 10.1007/s00332-010-9066-x. |
[19] |
A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, \emph{Proc. Roy. Soc. London Ser. A}, 434 (1991), 9.
doi: 10.1098/rspa.1991.0075. |
[20] |
R. H. Kraichnan, Inertial ranges in two-dimensional turbulence,, \emph{Phys. Fluids}, 5 (1962), 1374. Google Scholar |
[21] |
C. E. Leith, Diffusion approximation for two-dimensional turbulence,, \emph{Phys. Fluids}, 11 (1968), 671. Google Scholar |
[22] |
L. Lu and C. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations,, \emph{Indiana Univ. Math J.}, 57 (2008), 2693.
doi: 10.1512/iumj.2008.57.3716. |
[23] |
E. Lunasin, S. Kurien, M. Taylor and E. S. Titi, A study of the Navier-Stokes-$\alpha$ model for two-dimensional turbulence,, \emph{Journal of Turbulence}, 8 (2007), 751.
doi: 10.1080/14685240701439403. |
[24] |
E. Lunasin, S. Kurien and E. S. Titi, Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence,, \emph{Journal of Physics A: Math. Theor.}, 41 (2008).
doi: 10.1088/1751-8113/41/34/344014. |
[25] |
M. Vishik, E. S. Titi and V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0,, \emph{Mathematicheskii Sbornik}, 198 (2007), 3.
doi: 10.1070/SM2007v198n12ABEH003902. |
show all references
References:
[1] |
G. K. Batchelor, The Theory of Homogeneous Turbulence,, Cambridge Monographs on Mechanics and Applied Mathematics, (1953).
|
[2] |
V. Chepyzhov, E. S. Titi and M. Vishik, On the convergence of solutions of the 3D Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,, \emph{Discr.} & \emph{Cont. Dyn. Systems A}, 17 (2007), 481.
|
[3] |
A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, \emph{Royal Soc. A, 461 (2005), 629.
doi: 10.1098/rspa.2004.1373. |
[4] |
R. Dascaliuc, C. Foias and M. S. Jolly, Relations between energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations,, \emph{J. Dynam. Differential Equations}, 17 (2005), 643.
doi: 10.1007/s10884-005-8269-6. |
[5] |
R. Dascaliuc, C. Foias and M. S. Jolly, Universal bounds on the attractor of the Navier-Stokes equation in the energy, enstrophy plane,, \emph{J. Math. Phys.}, 48 (2007).
doi: 10.1063/1.2710349. |
[6] |
R. Dascaliuc, C. Foias and M. Jolly, Estimates on enstrophy, palinstrophy, and invariant measures for 2-d turbulence,, \emph{J. Differential Eqns}, 248 (2010), 792.
doi: 10.1016/j.jde.2009.11.020. |
[7] |
C. Doering, The 3D Navier-Stokes problem,, \emph{Annu. Rev. Fluid Mech}, 41 (2009), 109.
doi: 10.1146/annurev.fluid.010908.165218. |
[8] |
C. Foias, M. S. Jolly, O. P. Manley and R. Rosa, Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence,, \emph{J. Statist. Phys.}, 108 (2002), 591.
doi: 10.1023/A:1015782025005. |
[9] |
C. Foias, M. S. Jolly, O. P. Manley, R. Rosa and R. Temam, Kolmogorov theory via finite-time averages,, \emph{Phys. D}, 212 (2005), 245.
doi: 10.1016/j.physd.2005.10.002. |
[10] |
C. Foias, M. S. Jolly and M. Yang, On single mode forcing of the 2D-NSE,, \emph{J. Dynam. Diff. Eqns.}, 25 (2013), 393.
doi: 10.1007/s10884-013-9301-x. |
[11] |
C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory,, \emph{J. Dynam. Differential Equations}, 14 (2002), 1.
doi: 10.1023/A:1012984210582. |
[12] |
C. Foias, O. Manley, R. Rosa and R. Temam, Cascade of energy in turbulent flows,, \emph{Comptes Rendus Acad. Sci. Paris, 332 (2001), 509.
doi: 10.1016/S0764-4442(01)01831-6. |
[13] |
C. Foias, O. Manley, R. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows,, \emph{Comptes Rendus Acad. Sci. Paris, 333 (2001), 499.
doi: 10.1016/S0764-4442(01)02008-0. |
[14] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).
doi: 10.1017/CBO9780511546754. |
[15] |
C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes,, \emph{Ann. Mat. Pura Appl.}, 111 (2001), 307.
|
[16] |
D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid,, \emph{Physica D}, 133 (1999), 215.
doi: 10.1016/S0167-2789(99)00093-7. |
[17] |
D. D. Holm, J. E. Marsden and T. Ratiu, Euler-Poincaré Equations in Geophysical Fluid Dynamics,, \emph{In Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, ().
|
[18] |
M. Holst, E. Lunasin and G. Tsotgtgerel, Analytical study of generalized $\alpha$-models of turbulence,, \emph{Journal of Nonlinear Science}, 20 (2010), 523.
doi: 10.1007/s00332-010-9066-x. |
[19] |
A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, \emph{Proc. Roy. Soc. London Ser. A}, 434 (1991), 9.
doi: 10.1098/rspa.1991.0075. |
[20] |
R. H. Kraichnan, Inertial ranges in two-dimensional turbulence,, \emph{Phys. Fluids}, 5 (1962), 1374. Google Scholar |
[21] |
C. E. Leith, Diffusion approximation for two-dimensional turbulence,, \emph{Phys. Fluids}, 11 (1968), 671. Google Scholar |
[22] |
L. Lu and C. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations,, \emph{Indiana Univ. Math J.}, 57 (2008), 2693.
doi: 10.1512/iumj.2008.57.3716. |
[23] |
E. Lunasin, S. Kurien, M. Taylor and E. S. Titi, A study of the Navier-Stokes-$\alpha$ model for two-dimensional turbulence,, \emph{Journal of Turbulence}, 8 (2007), 751.
doi: 10.1080/14685240701439403. |
[24] |
E. Lunasin, S. Kurien and E. S. Titi, Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence,, \emph{Journal of Physics A: Math. Theor.}, 41 (2008).
doi: 10.1088/1751-8113/41/34/344014. |
[25] |
M. Vishik, E. S. Titi and V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0,, \emph{Mathematicheskii Sbornik}, 198 (2007), 3.
doi: 10.1070/SM2007v198n12ABEH003902. |
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