{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T07:12:06Z","timestamp":1771744326551,"version":"3.50.1"},"reference-count":66,"publisher":"Springer Science and Business Media LLC","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2010,10]]},"DOI":"10.1007\/s00332-010-9066-x","type":"journal-article","created":{"date-parts":[[2010,4,1]],"date-time":"2010-04-01T18:35:52Z","timestamp":1270146952000},"page":"523-567","source":"Crossref","is-referenced-by-count":55,"title":["Analysis of a General Family of Regularized Navier\u2013Stokes and MHD Models"],"prefix":"10.1007","volume":"20","author":[{"given":"Michael","family":"Holst","sequence":"first","affiliation":[]},{"given":"Evelyn","family":"Lunasin","sequence":"additional","affiliation":[]},{"given":"Gantumur","family":"Tsogtgerel","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2010,4,2]]},"reference":[{"key":"9066_CR1","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1016\/j.physd.2008.01.003","volume":"237","author":"C. Bardos","year":"2008","unstructured":"Bardos, C., Linshiz, J., Titi, E.S.: Global regularity for a Birkhoff-Rott-\u03b1 approximation of the dynamics of vortex sheets of the 2D Euler equations. Physica D 237, 1905\u20131911 (2008)","journal-title":"Physica D"},{"issue":"4","key":"9066_CR2","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1137\/030601739","volume":"3","author":"H.S. Bhat","year":"2005","unstructured":"Bhat, H.S., Fetecau, R.C., Marsden, J.E., Mohseni, K., West, M.: Lagrangian averaging for compressible fluids. Multiscale Model. Simul. 3(4), 818\u2013837 (2005)","journal-title":"Multiscale Model. Simul."},{"issue":"6","key":"9066_CR3","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1002\/cpa.3160350604","volume":"35","author":"L. Caffarelli","year":"1982","unstructured":"Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier\u2013Stokes equations. Commun. Pure Appl. Math. 35(6), 771\u2013831 (1982)","journal-title":"Commun. Pure Appl. Math."},{"key":"9066_CR4","unstructured":"Cao, Y., Titi, E.S.: On the rate of convergence of the two-dimensional \u03b1-models of turbulence to the Navier\u2013Stokes equations. arXiv:0902.4247v1 (2009)"},{"issue":"20","key":"9066_CR5","first-page":"1","volume":"6","author":"C. Cao","year":"2005","unstructured":"Cao, C., Holm, D.D., Titi, E.S.: On the Clark-\u03b1 model of turbulence: global regularity and long-time dynamics. J. Turbul. 6(20), 1\u201311 (2005)","journal-title":"J. Turbul."},{"issue":"4","key":"9066_CR6","doi-asserted-by":"crossref","first-page":"823","DOI":"10.4310\/CMS.2006.v4.n4.a8","volume":"4","author":"Y. Cao","year":"2006","unstructured":"Cao, Y., Lunasin, E., Titi, E.S.: Global well-posedness of the viscous and inviscid simplified Bardina model. Commun. Math. Sci. 4(4), 823\u2013848 (2006)","journal-title":"Commun. Math. Sci."},{"key":"9066_CR7","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1088\/0951-7715\/21\/5\/001","volume":"21","author":"Y. Cao","year":"2008","unstructured":"Cao, Y., Musslimani, Z., Titi, E.S.: Nonlinear Schr\u00f6dinger\u2013Helmholtz equation as numerical regularization of the nonlinear Schr\u00f6dinger equation. Nonlinearity 21, 879\u2013898 (2008)","journal-title":"Nonlinearity"},{"issue":"24","key":"9066_CR8","doi-asserted-by":"crossref","first-page":"5338","DOI":"10.1103\/PhysRevLett.81.5338","volume":"81","author":"S. Chen","year":"1998","unstructured":"Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: Camassa\u2013Holm equations as closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81(24), 5338\u20135341 (1998)","journal-title":"Phys. Rev. Lett."},{"issue":"1\u20134","key":"9066_CR9","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0167-2789(99)00098-6","volume":"133","author":"S. Chen","year":"1999","unstructured":"Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: The Camassa\u2013Holm equations and turbulence. Physica D 133(1\u20134), 49\u201365 (1999a)","journal-title":"Physica D"},{"issue":"8","key":"9066_CR10","doi-asserted-by":"crossref","first-page":"2343","DOI":"10.1063\/1.870096","volume":"11","author":"S. Chen","year":"1999","unstructured":"Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: A connection between the Camassa\u2013Holm equations and turbulent flows in channels and pipes. Phys. Fluids 11(8), 2343\u20132353 (1999b)","journal-title":"Phys. Fluids"},{"issue":"1\u20134","key":"9066_CR11","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/S0167-2789(99)00099-8","volume":"133","author":"S. Chen","year":"1999","unstructured":"Chen, S., Holm, D.D., Margolin, L.G., Zhang, R.: Direct numerical simulation of the Navier\u2013Stokes alpha model. Physica D 133(1\u20134), 66\u201383 (1999c)","journal-title":"Physica D"},{"key":"9066_CR12","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1098\/rspa.2004.1373","volume":"461","author":"A. Cheskidov","year":"2005","unstructured":"Cheskidov, A., Holm, D.D., Olson, E., Titi, E.S.: On a Leray-\u03b1 model of turbulence. R. Soc. A, Math. Phys, Eng. Sci. 461, 629\u2013649 (2005)","journal-title":"R. Soc. A, Math. Phys, Eng. Sci."},{"key":"9066_CR13","first-page":"563","volume":"321","author":"B. Cockburn","year":"1995","unstructured":"Cockburn, B., Jones, D.A., Titi, E.S.: Degr\u00e9s de libert\u00e9 d\u00e9terminants pour \u00e9quations nonlin\u00e9aires dissipatives. C. R. Acad. Sci. Paris, S\u00e9r. I 321, 563\u2013568 (1995)","journal-title":"C. R. Acad. Sci. Paris, S\u00e9r. I"},{"key":"9066_CR14","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1090\/S0025-5718-97-00850-8","volume":"66","author":"B. Cockburn","year":"1997","unstructured":"Cockburn, B., Jones, D.A., Titi, E.S.: Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems. Math. Comput. 66, 1073\u20131087 (1997)","journal-title":"Math. Comput."},{"key":"9066_CR15","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1017\/S0022112085000209","volume":"150","author":"P. Constantin","year":"1985","unstructured":"Constantin, P., Foias, C., Manley, O.P., Temam, R.: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427\u2013440 (1985)","journal-title":"J. Fluid Mech."},{"key":"9066_CR16","volume-title":"The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time","author":"K.J. Devlin","year":"2002","unstructured":"Devlin, K.J.: The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. Basic Books, New York (2002)"},{"key":"9066_CR17","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1017\/S0022112002001386","volume":"467","author":"C. Doering","year":"2002","unstructured":"Doering, C., Foias, C.: Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289\u2013306 (2002)","journal-title":"J. Fluid Mech."},{"key":"9066_CR18","unstructured":"Ebrahimi, M.A., Holst, M., Lunasin, E.: The Navier\u2013Stokes\u2013Voight model for image inpainting. Submitted for publication. Available as arXiv:0901.4548 [math.AP]"},{"key":"9066_CR19","first-page":"1","volume":"39","author":"C. Foias","year":"1967","unstructured":"Foias, C., Prodi, G.: Sur le comportement global des solutions non stationnaires des \u00e9quations de Navier\u2013Stokes en dimension two. Rend. Semin. Mat. Univ. Padova 39, 1\u201334 (1967)","journal-title":"Rend. Semin. Mat. Univ. Padova"},{"key":"9066_CR20","doi-asserted-by":"crossref","unstructured":"Foias, C., Temam, R.: Asymptotic numerical analysis for the Navier\u2013Stokes equations. Nonlinear Dynamics and Turbulence (1983)","DOI":"10.1016\/0167-2789(83)90297-X"},{"key":"9066_CR21","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1090\/S0025-5718-1984-0744927-9","volume":"43","author":"C. Foias","year":"1984","unstructured":"Foias, C., Temam, R.: Determining the solutions of the Navier\u2013Stokes equations by a set of nodal values. Math. Comput. 43, 177\u2013183 (1984)","journal-title":"Math. Comput."},{"key":"9066_CR22","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1088\/0951-7715\/4\/1\/009","volume":"4","author":"C. Foias","year":"1991","unstructured":"Foias, C., Titi, E.S.: Determining nodes, finite difference schemes and inertial manifolds. Nonlinearity 4, 135\u2013153 (1991)","journal-title":"Nonlinearity"},{"key":"9066_CR23","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0167-2789(83)90297-X","volume":"9","author":"C. Foias","year":"1983","unstructured":"Foias, C., Manley, O.P., Temam, R., Treve, Y.: Asymptotic analysis of the Navier\u2013Stokes equations. Physica D 9, 157\u2013188 (1983)","journal-title":"Physica D"},{"key":"9066_CR24","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0022-0396(88)90110-6","volume":"73","author":"C. Foias","year":"1988","unstructured":"Foias, C., Sell, G., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differ. Equ. 73, 309\u2013353 (1988)","journal-title":"J. Differ. Equ."},{"key":"9066_CR25","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/S0167-2789(01)00191-9","volume":"152\/153","author":"C. Foias","year":"2001","unstructured":"Foias, C., Holm, D.D., Titi, E.S.: The Navier\u2013Stokes-alpha model of fluid turbulence. Advances in nonlinear mathematics and science. Physica D 152\/153, 505\u2013519 (2001)","journal-title":"Physica D"},{"key":"9066_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1012984210582","volume":"14","author":"C. Foias","year":"2002","unstructured":"Foias, C., Holm, D.D.. Titi, E.S.: The three-dimensional viscous Camassa\u2013Holm equations, and their relation to the Navier\u2013Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14, 1\u201335 (2002)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9066_CR27","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0167-2789(99)00093-7","volume":"133","author":"B. Geurts","year":"1999","unstructured":"Geurts, B., Holm, D.D.: Fluctuation effects on 3d-Lagrangian mean and Eulerian mean fluid motion. Physica D 133, 215\u2013269 (1999)","journal-title":"Physica D"},{"key":"9066_CR28","doi-asserted-by":"crossref","first-page":"L13","DOI":"10.1063\/1.1529180","volume":"15","author":"B. Geurts","year":"2003","unstructured":"Geurts, B., Holm, D.D.: Regularization modeling for large eddy simulation. Phys. Fluids 15, L13\u2013L16 (2003)","journal-title":"Phys. Fluids"},{"issue":"1","key":"9066_CR29","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.physd.2006.06.012","volume":"220","author":"J.D. Gibbon","year":"2006","unstructured":"Gibbon, J.D., Holm, D.D.: Length-scale estimates for the LANS-\u03b1 equations in terms of the Reynolds number. Physica D: Nonlinear Phenom. 220(1), 69\u201378 (2006)","journal-title":"Physica D: Nonlinear Phenom."},{"issue":"6","key":"9066_CR30","doi-asserted-by":"crossref","first-page":"2761","DOI":"10.1512\/iumj.2008.57.3701","volume":"57","author":"J.D. Gibbon","year":"2008","unstructured":"Gibbon, J.D., Holm, D.D.: Estimates for the LANS-\u03b1, Leray-\u03b1 and Bardina models in terms of a Navier\u2013Stokes Reynolds number. Indiana Univ. Math. J. 57(6), 2761\u20132774 (2008)","journal-title":"Indiana Univ. Math. J."},{"issue":"1\u20132","key":"9066_CR31","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.physd.2005.05.014","volume":"207","author":"J.-L. Guermond","year":"2005","unstructured":"Guermond, J.-L., Prudhomme, S.: On the construction of suitable solutions to the Navier\u2013Stokes equations and questions regarding the definition of large eddy simulation. Physica D 207(1\u20132), 64\u201378 (2005)","journal-title":"Physica D"},{"key":"9066_CR32","doi-asserted-by":"crossref","first-page":"5691","DOI":"10.1016\/j.jcp.2008.02.018","volume":"227","author":"M. Hecht","year":"2008","unstructured":"Hecht, M., Holm, D.D., Wingate, B.: Implementation of the LANS-\u03b1 turbulence model in a primitive equation ocean model. J. Comput. Phys. 227, 5691 (2008a)","journal-title":"J. Comput. Phys."},{"key":"9066_CR33","doi-asserted-by":"crossref","DOI":"10.1088\/1751-8113\/41\/34\/344009","volume":"41","author":"M. Hecht","year":"2008","unstructured":"Hecht, M., Holm, D.D., Wingate, B.: The LANS-\u03b1 and Leray turbulence parametrizations in primitive equation ocean modeling. J. Phys. A: Math. Theor. 41, 344009 (2008b)","journal-title":"J. Phys. A: Math. Theor."},{"key":"9066_CR34","doi-asserted-by":"crossref","first-page":"4173","DOI":"10.1103\/PhysRevLett.80.4173","volume":"80","author":"D.D. Holm","year":"1998","unstructured":"Holm, D.D., Marsden, J., Ratiu, T.: Euler\u2013Poincar\u00e9 models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, 4173\u20134176 (1998)","journal-title":"Phys. Rev. Lett."},{"key":"9066_CR35","unstructured":"Holst, M.: Some bounds on the number of determining nodes for weak solutions of the Navier\u2013Stokes equations. Technical Report CRPC-95-1, Applied Mathematics and CRPC, California Institute of Technology (1995)"},{"key":"9066_CR36","doi-asserted-by":"crossref","unstructured":"Holst, M., Titi, E.: Determining projections and functionals for weak solutions of the Navier\u2013Stokes equations. Technical Report CRPC-96-4, Applied Mathematics and CRPC, California Institute of Technology (1996)","DOI":"10.1090\/conm\/204\/02626"},{"key":"9066_CR37","series-title":"Contemporary Mathematics","volume-title":"Recent Developments in Optimization Theory and Nonlinear Analysis","author":"M. Holst","year":"1997","unstructured":"Holst, M., Titi, E.: Determining projections and functionals for weak solutions of the Navier\u2013Stokes equations. In: Censor, Y., Reich, S. (eds.) Recent Developments in Optimization Theory and Nonlinear Analysis. Contemporary Mathematics, vol. 204. American Mathematical Society, Providence (1997)"},{"key":"9066_CR38","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1088\/0951-7715\/19\/4\/006","volume":"19","author":"A. Ilyin","year":"2006","unstructured":"Ilyin, A., Lunasin, E., Titi, E.S.: A modified-Leray-\u03b1 subgrid scale model of turbulence. Nonlinearity 19, 879\u2013897 (2006)","journal-title":"Nonlinearity"},{"key":"9066_CR39","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/0167-2789(92)90233-D","volume":"60","author":"D.A. Jones","year":"1992","unstructured":"Jones, D.A., Titi, E.S.: Determining finite volume elements for the 2D Navier\u2013Stokes equations. Physica D 60, 117\u2013133 (1992a)","journal-title":"Physica D"},{"key":"9066_CR40","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/0022-247X(92)90190-O","volume":"168","author":"D.A. Jones","year":"1992","unstructured":"Jones, D.A., Titi, E.S.: On the number of determining nodes for the 2D Navier\u2013Stokes equations. J. Math. Anal. Appl. 168, 72\u201388 (1992b)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"9066_CR41","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1512\/iumj.1993.42.42039","volume":"42","author":"D.A. Jones","year":"1993","unstructured":"Jones, D.A., Titi, E.S.: Upper bounds on the number of determining modes, nodes, and volume elements for the Navier\u2013Stokes equations. Indiana Univ. Math. J. 42(3), 875\u2013887 (1993)","journal-title":"Indiana Univ. Math. J."},{"issue":"6","key":"9066_CR42","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1007\/s11401-009-0205-3","volume":"30","author":"V.K. Kalantarov","year":"2009","unstructured":"Kalantarov, V.K., Titi, E.S.: Global attractors and determining modes for the 3D Navier\u2013Stokes\u2013Voight equations. Chin. Ann. Math. B 30(6), 697\u2013714 (2009)","journal-title":"Chin. Ann. Math. B"},{"issue":"2","key":"9066_CR43","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s00332-008-9029-7","volume":"19","author":"V.K. Kalantarov","year":"2009","unstructured":"Kalantarov, V.K., Levant, B., Titi, E.S.: Gevrey regularity for the attractor of the 3D Navier\u2013Stokes\u2013Voight equations. J. Nonlinear Sci. 19(2), 133\u2013152 (2009)","journal-title":"J. Nonlinear Sci."},{"issue":"2","key":"9066_CR44","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s00039-002-8250-z","volume":"12","author":"N.H. Katz","year":"2002","unstructured":"Katz, N.H., Pavlovi\u0107, N.: A cheap Caffarelli\u2013Kohn\u2013Nirenberg inequality for the Navier\u2013Stokes equations with hyper-dissipation. Geom. Funct. Anal. 12(2), 355\u2013379 (2002)","journal-title":"Geom. Funct. Anal."},{"issue":"10","key":"9066_CR45","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1002\/cpa.20218","volume":"61","author":"B. Khouider","year":"2008","unstructured":"Khouider, B., Titi, E.S.: An inviscid regularization for the surface quasi-geostrophic equation. Commun. Pure Appl. Math. 61(10), 1331\u20131346 (2008)","journal-title":"Commun. Pure Appl. Math."},{"key":"9066_CR46","doi-asserted-by":"crossref","first-page":"111","DOI":"10.3934\/dcdsb.2006.6.111","volume":"6","author":"W. Layton","year":"2006","unstructured":"Layton, W., Lewandowski, R.: On a well-posed turbulence model. Discrete Contin. Dyn. Syst. B 6, 111\u2013128 (2006)","journal-title":"Discrete Contin. Dyn. Syst. B"},{"key":"9066_CR47","doi-asserted-by":"crossref","unstructured":"Levant, B., Ramos, F., Titi, E.S.: On the statistical properties of the 3d incompressible Navier\u2013Stokes-Voight model. Commun. Math. Sci. (2009)","DOI":"10.4310\/CMS.2010.v8.n1.a14"},{"key":"9066_CR48","doi-asserted-by":"crossref","unstructured":"Linshiz, J., Titi, E.S.: Analytical study of certain magnetohydrodynamic-alpha models. J. Math. Phys. 48(6) (2007)","DOI":"10.1063\/1.2360145"},{"key":"9066_CR49","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1080\/14685240701439403","volume":"8","author":"E. Lunasin","year":"2007","unstructured":"Lunasin, E., Kurien, S., Taylor, M.A., Titi, E.S.: A study of the Navier\u2013Stokes-\u03b1 model for two-dimensional turbulence. J. Turbul. 8, 751\u2013778 (2007)","journal-title":"J. Turbul."},{"issue":"34","key":"9066_CR50","doi-asserted-by":"crossref","DOI":"10.1088\/1751-8113\/41\/34\/344014","volume":"41","author":"E. Lunasin","year":"2008","unstructured":"Lunasin, E., Kurien, S., Titi, E.S.: Spectral scaling of the Leray-\u03b1 model for two-dimensional turbulence. J. Phys. A: Math. Theor. 41(34), 344014 (2008)","journal-title":"J. Phys. A: Math. Theor."},{"issue":"1784","key":"9066_CR51","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1098\/rsta.2001.0852","volume":"359","author":"J.E. Marsden","year":"2001","unstructured":"Marsden, J.E., Shkoller, S.: Global well-posedness for the Lagrangian averaged Navier\u2013Stokes (LANS-\u03b1) equations on bounded domains. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359(1784), 1449\u20131468 (2001). Topological methods in the physical sciences (London, 2000)","journal-title":"R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci."},{"issue":"1","key":"9066_CR52","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s00205-002-0207-8","volume":"166","author":"J.E. Marsden","year":"2003","unstructured":"Marsden, J.E., Shkoller, S.: The anisotropic Lagrangian averaged Euler and Navier\u2013Stokes equations. Arch. Ration. Mech. Anal. 166(1), 27\u201346 (2003)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9066_CR53","unstructured":"Maxwell, D.: Initial Data for Black Holes and Rough Spacetimes. Ph.D. Thesis, University of Washington (2004)"},{"key":"9066_CR54","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.71.046304","volume":"71","author":"P.D. Mininni","year":"2005","unstructured":"Mininni, P.D., Montgomery, D.C., Pouquet, A.: Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model. Phys. Rev. E 71, 046304 (2005a)","journal-title":"Phys. Rev. E"},{"key":"9066_CR55","doi-asserted-by":"crossref","DOI":"10.1063\/1.1863260","volume":"17","author":"P.D. Mininni","year":"2005","unstructured":"Mininni, P.D., Montgomery, D.C., Pouquet, A.: A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows. Phys. Fluids 17, 035112-17 (2005b)","journal-title":"Phys. Fluids"},{"issue":"2","key":"9066_CR56","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1063\/1.1533069","volume":"15","author":"K. Mohseni","year":"2003","unstructured":"Mohseni, K., Kosovi\u0107, B., Shkoller, S., Marsden, J.E.: Numerical simulations of the Lagrangian averaged Navier\u2013Stokes equations for homogeneous isotropic turbulence. Phys. Fluids 15(2), 524\u2013544 (2003)","journal-title":"Phys. Fluids"},{"issue":"11","key":"9066_CR57","doi-asserted-by":"crossref","first-page":"2427","DOI":"10.1016\/j.na.2006.03.030","volume":"66","author":"E. Olson","year":"2007","unstructured":"Olson, E., Titi, E.S.: Viscosity versus vorticity stretching: global well-posedness for a family of Navier\u2013Stokes-alpha-like models. Nonlinear Anal. 66(11), 2427\u20132458 (2007)","journal-title":"Nonlinear Anal."},{"key":"9066_CR58","first-page":"98","volume":"38","author":"A.P. Oskolkov","year":"1973","unstructured":"Oskolkov, A.P.: The uniqueness and solvability in the large of the boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Naucn. Semin. Leningr. Otd. Mat. Inst. Steklov (LOMI) 38, 98\u2013136 (1973)","journal-title":"Zap. Naucn. Semin. Leningr. Otd. Mat. Inst. Steklov (LOMI)"},{"key":"9066_CR59","first-page":"233","volume":"96","author":"A.P. Oskolkov","year":"1980","unstructured":"Oskolkov, A.P.: On the theory of Voight fluids. Zap. Naucn. Semin. Leningr. Otd. Mat. Inst. Steklov (LOMI) 96, 233\u2013236 (1980)","journal-title":"Zap. Naucn. Semin. Leningr. Otd. Mat. Inst. Steklov (LOMI)"},{"key":"9066_CR60","doi-asserted-by":"crossref","first-page":"5717","DOI":"10.1016\/j.jcp.2008.02.017","volume":"227","author":"M. Petersen","year":"2008","unstructured":"Petersen, M., Hecht, M., Wingate, B.: Efficient form of the lans-alpha turbulence model in a primitive-equation ocean model. J. Comput. Phys. 227, 5717 (2008)","journal-title":"J. Comput. Phys."},{"key":"9066_CR61","volume-title":"Navier\u2013Stokes Equations: Theory and Numerical Analysis","author":"R. Temam","year":"1977","unstructured":"Temam, R.: Navier\u2013Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1977)"},{"key":"9066_CR62","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-0313-8","volume-title":"Infinite-Dimensional Dynamical Systems in Mechanics and Physics","author":"R. Temam","year":"1988","unstructured":"Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)"},{"key":"9066_CR63","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Fan, J.: Global well-posedness for two modified-Leray-\u03b1-MHD models with partial viscous terms. Math. Methods Appl. Sci. (2009a, to appear)","DOI":"10.1002\/mma.1198"},{"key":"9066_CR64","unstructured":"Zhou, Y., Fan, J.: On the Cauchy problem for a Leray-\u03b1-MHD model (2009b, submitted)"},{"key":"9066_CR65","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Fan, J.: Regularity criteria for the viscous Camassa\u2013Holm equations. International Mathematics Research Notices, page ARTICLE ID rnp023 (2009c)","DOI":"10.1093\/imrn\/rnp023"},{"issue":"2","key":"9066_CR66","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1017\/S0308210500009872","volume":"78","author":"J.L. Zolesio","year":"1977","unstructured":"Zolesio, J.L.: Multiplication dans les espaces de Besov. Proc. R. Soc. Edinb. (A) 78(2), 113\u2013117 (1977)","journal-title":"Proc. R. Soc. Edinb. (A)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.springerlink.com\/index\/pdf\/10.1007\/s00332-010-9066-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T17:57:59Z","timestamp":1685555879000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-010-9066-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,2]]},"references-count":66,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,10]]}},"alternative-id":["9066"],"URL":"https:\/\/doi.org\/10.1007\/s00332-010-9066-x","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,4,2]]}}}