On questions of decay and existence for the viscous Camassa-Holm equations
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 5, pp. 907-936.
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author = {Bjorland, Clayton and Schonbek, Maria E.},
title = {On questions of decay and existence for the viscous {Camassa-Holm} equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Bjorland, Clayton; Schonbek, Maria E. On questions of decay and existence for the viscous Camassa-Holm equations. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 5, pp. 907-936. doi : 10.1016/j.anihpc.2007.07.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2007.07.003/

[1] Ben-Artzi M., Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (4) (1994) 329-358. | MR | Zbl

[2] Caffarelli L., Kohn R., Nirenberg L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771-831. | MR | Zbl

[3] Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661-1664. | MR | Zbl

[4] Chen S., Foias C., Holm D.D., Olson E., Titi E.S., Wynne S., Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett. 81 (24) (1998) 5338-5341. | MR | Zbl

[5] Chen S., Foias C., Holm D.D., Olson E., Titi E.S., Wynne S., The Camassa-Holm equations and turbulence, Phys. D 133 (1-4) (1999) 49-65. | MR

[6] Chen S., Foias C., Holm D.D., Olson E., Titi E.S., Wynne S., A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids 11 (8) (1999) 2343-2353. | MR | Zbl

[7] Constantin P., Foias C., Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. | MR | Zbl

[8] J.A. Domaradzki, D.D. Holm, Navier-Stokes-alpha model: Les equations with nonlinear dispersion, 2001.

[9] Foias C., Holm D.D., Titi E.S., The Navier-Stokes-alpha model of fluid turbulence, Phys. D 152/153 (2001) 505-519. | MR | Zbl

[10] Foias C., Holm D.D., Titi E.S., The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations 14 (1) (2002) 1-35. | MR | Zbl

[11] Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1) (1998) 1-81. | MR | Zbl

[12] Holm D.D., Titi E.S., Computational models of turbulence: The lans-α model and the role of global analysis, SIAM News 38 (2005).

[13] Hopf E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951) 213-231. | MR | Zbl

[14] Ilyin A.A., Titi E.S., Attractors for the two-dimensional Navier-Stokes-α model: an α-dependence study, J. Dynam. Differential Equations 15 (4) (2003) 751-778. | MR | Zbl

[15] Leray J., Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934) 193-248. | JFM | MR

[16] Marsden J.E., Shkoller S., Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (1784) (2001) 1449-1468. | MR | Zbl

[17] Masuda K., Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2) 36 (4) (1984) 623-646. | MR | Zbl

[18] Ogawa T., Rajopadhye S.V., Schonbek M.E., Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces, J. Funct. Anal. 144 (2) (1997) 325-358. | MR | Zbl

[19] Prodi G., Teoremi di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie, Rend. Sem. Mat. Univ. Padova 32 (1962) 374-397. | Numdam | MR | Zbl

[20] Schonbek M., The Fourier splitting method, in: Advances in Geometric Analysis and Continuum Mechanics, Stanford, CA, 1993, Internat. Press, Cambridge, MA, 1995, pp. 269-274. | MR | Zbl

[21] Schonbek M.E., Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (5) (1980) 449-473. | MR | Zbl

[22] Schonbek M.E., Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations, Comm. Partial Differential Equations 7 (1) (1980) 449-473. | Zbl

[23] Schonbek M.E., ${L}^{2}$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (3) (1985) 209-222. | MR | Zbl

[24] Schonbek M.E., Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (7) (1986) 733-763. | MR | Zbl

[25] Schonbek M.E., Large time behaviour of solutions to the Navier-Stokes equations in ${H}^{m}$ spaces, Comm. Partial Differential Equations 20 (1-2) (1995) 103-117. | Zbl

[26] Schonbek M.E., Schonbek T.P., Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discrete Contin. Dyn. Syst. 13 (5) (2005) 1277-1304. | Zbl

[27] Schonbek M.E., Wiegner M., On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 126 (3) (1996) 677-685. | Zbl

[28] Temam R., Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. | Zbl

[29] Wiegner M., Decay results for weak solutions of the Navier-Stokes equations on ${R}^{n}$, J. London Math. Soc. (2) 35 (2) (1987) 303-313. | Zbl

[30] Wiegner M., Higher order estimates in further dimensions for the solutions of Navier-Stokes equations, in: Evolution Equations, Warsaw, 2001, Banach Center Publ., vol. 60, Polish Acad. Sci., Warsaw, 2003, pp. 81-84. | MR | Zbl

[31] Zhang L.H., Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations, Comm. Partial Differential Equations 20 (1-2) (1995) 119-127. | MR | Zbl

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