On questions of decay and existence for the viscous Camassa–Holm equations

Clayton Bjorland; Maria E. Schonbek

Annales de l'I.H.P. Analyse non linéaire (2008)

  • Volume: 25, Issue: 5, page 907-936
  • ISSN: 0294-1449

How to cite

top

Bjorland, Clayton, and Schonbek, Maria E.. "On questions of decay and existence for the viscous Camassa–Holm equations." Annales de l'I.H.P. Analyse non linéaire 25.5 (2008): 907-936. <http://eudml.org/doc/78819>.

@article{Bjorland2008,
author = {Bjorland, Clayton, Schonbek, Maria E.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {regularity; decay without a rate; algebraic rate},
language = {eng},
number = {5},
pages = {907-936},
publisher = {Elsevier},
title = {On questions of decay and existence for the viscous Camassa–Holm equations},
url = {http://eudml.org/doc/78819},
volume = {25},
year = {2008},
}

TY - JOUR
AU - Bjorland, Clayton
AU - Schonbek, Maria E.
TI - On questions of decay and existence for the viscous Camassa–Holm equations
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2008
PB - Elsevier
VL - 25
IS - 5
SP - 907
EP - 936
LA - eng
KW - regularity; decay without a rate; algebraic rate
UR - http://eudml.org/doc/78819
ER -

References

top
  1. [1] Ben-Artzi M., Global solutions of two-dimensional Navier–Stokes and Euler equations, Arch. Rational Mech. Anal.128 (4) (1994) 329-358. Zbl0837.35110MR1308857
  2. [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. Zbl0509.35067MR673830
  3. [3] Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett.71 (11) (1993) 1661-1664. Zbl0972.35521MR1234453
  4. [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. Zbl1042.76525MR1745983
  5. [5] Chen S., Foias C., Holm D.D., Olson E., Titi E.S., Wynne S., The Camassa–Holm equations and turbulence, Phys. D133 (1–4) (1999) 49-65. Zbl1194.76069MR1721139
  6. [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. Fluids11 (8) (1999) 2343-2353. Zbl1147.76357MR1719962
  7. [7] Constantin P., Foias C., Navier–Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. Zbl0687.35071MR972259
  8. [8] J.A. Domaradzki, D.D. Holm, Navier–Stokes-alpha model: Les equations with nonlinear dispersion, 2001. 
  9. [9] Foias C., Holm D.D., Titi E.S., The Navier–Stokes-alpha model of fluid turbulence, Phys. D152/153 (2001) 505-519. Zbl1037.76022MR1837927
  10. [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 Equations14 (1) (2002) 1-35. Zbl0995.35051MR1878243
  11. [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. Zbl0951.37020MR1627802
  12. [12] Holm D.D., Titi E.S., Computational models of turbulence: The lans-α model and the role of global analysis, SIAM News38 (2005). 
  13. [13] Hopf E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr.4 (1951) 213-231. Zbl0042.10604MR50423
  14. [14] Ilyin A.A., Titi E.S., Attractors for the two-dimensional Navier–Stokes-α model: an α-dependence study, J. Dynam. Differential Equations15 (4) (2003) 751-778. Zbl1039.35078MR2031580
  15. [15] Leray J., Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math.63 (1934) 193-248. MR1555394JFM60.0726.05
  16. [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. Zbl1006.35074MR1853633
  17. [17] Masuda K., Weak solutions of Navier–Stokes equations, Tohoku Math. J. (2)36 (4) (1984) 623-646. Zbl0568.35077MR767409
  18. [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. Zbl0873.35064MR1432588
  19. [19] Prodi G., Teoremi di tipo locale per il sistema di Navier–Stokes e stabilità delle soluzioni stazionarie, Rend. Sem. Mat. Univ. Padova32 (1962) 374-397. Zbl0108.28602MR189354
  20. [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. Zbl0842.35142MR1356749
  21. [21] Schonbek M.E., Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations5 (5) (1980) 449-473. Zbl0476.35012MR571048
  22. [22] Schonbek M.E., Sharp rate of decay of solutions to 2-dimensional Navier–Stokes equations, Comm. Partial Differential Equations7 (1) (1980) 449-473. Zbl0476.35012
  23. [23] Schonbek M.E., L 2 decay for weak solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal.88 (3) (1985) 209-222. Zbl0602.76031MR775190
  24. [24] Schonbek M.E., Large time behaviour of solutions to the Navier–Stokes equations, Comm. Partial Differential Equations11 (7) (1986) 733-763. Zbl0607.35071MR837929
  25. [25] Schonbek M.E., Large time behaviour of solutions to the Navier–Stokes equations in H m spaces, Comm. Partial Differential Equations20 (1–2) (1995) 103-117. Zbl0831.35132
  26. [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. Zbl1091.35070
  27. [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. A126 (3) (1996) 677-685. Zbl0862.35086
  28. [28] Temam R., Navier–Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. Zbl0981.35001
  29. [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. Zbl0652.35095
  30. [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. Zbl1029.35198MR1993060
  31. [31] Zhang L.H., Sharp rate of decay of solutions to 2-dimensional Navier–Stokes equations, Comm. Partial Differential Equations20 (1–2) (1995) 119-127. Zbl0823.35145MR1312702

NotesEmbed ?

top

You must be logged in to post comments.