Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-04-30T16:48:17.522Z Has data issue: false hasContentIssue false

11 - On the decay of solutions of the Navier–Stokes system with potential forces

Published online by Cambridge University Press:  05 November 2012

I. Kukavica
Affiliation:
University of Southern California
James C. Robinson
Affiliation:
University of Warwick
José L. Rodrigo
Affiliation:
University of Warwick
Witold Sadowski
Affiliation:
Uniwersytet Warszawski, Poland
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amrouche, C., Girault, V., Schonbek, M.E., & Schonbek, T.P. (2000) Pointwise decay of solutions and of higher derivatives to Navier–Stokes equations. SIAM J. Math. Anal. 31, no. 4, 740–753 (electronic).Google Scholar
Bae, H.-O. & Jin, B.J. (2005) Temporal and spatial decays for the Navier–Stokes equations. Proc. Roy. Soc. Edinburgh Sect.A 135, no. 3, 461–477.Google Scholar
Brandolese, L. (2004) Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier–Stokes equations. Rev. Mat. Iberoamericana 20, no. 1, 223–256.Google Scholar
Brandolese, L. (2004) Space-time decay of Navier-Stokes flows invariant under rotations. Math. Ann. 329, no. 4, 685–706.Google Scholar
Brandolese, L. & Meyer, Y. (2002) On the instantaneous spreading for the Navier-Stokes system in the whole space. ESAIM Control Optim. Calc. Var. 8, 273–285 (electronic).Google Scholar
Brandolese, L. & Vigneron, F. (2007) New asymptotic profiles of nonstationary solutions of the Navier–Stokes system. J. Math. Pures Appl. (9) 88, no. 1, 64–86.Google Scholar
Carpio, A. (1996) Large-time behavior in incompressible Navier–Stokes equations. SIAM J. Math. Anal. 27, no. 2, 449–475.Google Scholar
Caffarelli, L., Kohn, R., & Nirenberg, L. (1982) Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35, no. 6, 771–831.Google Scholar
Dobrokhotov, S.Y. & Shafarevich, A.I. (1994) Some integral identities and remarks on the decay at infinity of the solutions to the Navier–Stokes equations in the entire space. Russian J. Math. Phys. 2, no. 1, 133–135.Google Scholar
Fujigaki, Y. & Miyakawa, T. (2001) Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the whole space. SIAM J. Math. Anal. 33, no. 3, 523–544 (electronic).Google Scholar
Foias, C. & Saut, J.-C. (1984a) Asymptotic behavior, as t → + ∞, of solutions of Navier–Stokes equations and nonlinear spectral manifolds. Indiana Univ. Math. J. 33, no. 3, 459–477.Google Scholar
Foias, C. & Saut, J.-C. (1984b) On the smoothness of the nonlinear spectral manifolds associated to the Navier–Stokes equations, Indiana Univ. Math. J. 33, no. 6, 911–926.Google Scholar
Foias, C. & Saut, J.-C. (1987) Linearization and normal form of the Navier–Stokes equations with potential forces. Ann. Inst. H. Poincaré Anal. Non Linéaire 4, no. 1, 1–47.Google Scholar
Foias, C. & Saut, J.-C. (1991) Asymptotic integration of Navier–Stokes equations with potential forces. IIndiana Univ. Math. J. 40, no. 1, 305–320.Google Scholar
Gallay, T. & Wayne, C.E. (2006) Long-time asymptotics of the Navier–Stokes equation in ℝ2 and ℝ3. [Plenary lecture presented at the 76th Annual GAMM Conference, Luxembourg, 29 March–1 April 2005], ZAMM Z. Angew. Math. Mech. 86, no. 4, 256–267.Google Scholar
Kajikiya, R. & Miyakawa, T. (1986) On L2 decay of weak solutions of the Navier–Stokes equations in Rn. Math. Z. 192, no. 1, 135–148.Google Scholar
Kato, T. (1984) Strong Lp-solutions of the Navier–Stokes equation in Rm, with applications to weak solutions. Math. Z. 187, no. 4, 471–480.Google Scholar
Kukavica, I. (2001) Space-time decay for solutions of the Navier–Stokes equations. Indiana Univ. Math. J. 50, 205–222.Google Scholar
Kukavica, I. (2009) On the weighted decay for solutions of the Navier–Stokes system. Nonlinear Anal. 70, no. 6, 2466–2470.Google Scholar
Kukavica, I. & Reis, E. (2011) Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces. J. Diff. Eq. 250, 607–622.Google Scholar
Kukavica, I. & Torres, J.J. (2006) Weighted bounds for the velocity and the vorticity for the Navier–Stokes equations. Nonlinearity 19, no. 2, 293–303.Google Scholar
Kukavica, I. & Torres, J.J. (2007) Weighted Lp decay for solutions of the Navier–Stokes equations. Comm. Partial Differential Equations 32, no. 4–6, 819–831.Google Scholar
Lemarié-Rieusset, P.G. (2002) Recent developments in the Navier–Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL.
Leray, J. (1934) Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63, no. 1, 193–248.Google Scholar
Miyakawa, T. (2002) Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flows in Rn. Funkcial. Ekvac. 45, no. 2, 271–289.Google Scholar
Miyakawa, T. & Schonbek, M.E. (2001) On optimal decay rates for weak solutions to the Navier–Stokes equations in ℝn. Proceedings of Partial Differential Equations and Applications (Olomouc, 1999) 126, 443–455.Google Scholar
Schonbek, M.E. (1985) L2 decay for weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 88, no. 3, 209–222.Google Scholar
Schonbek, M.E. (1986) Large time behaviour of solutions to the Navier–Stokes equations. Comm. Partial Differential Equations 11, no. 7, 733–763.Google Scholar
Schonbek, M.E. (1992) Asymptotic behavior of solutions to the threedimensional Navier–Stokes equations. Indiana Univ. Math. J. 41, no. 3, 809–823.Google Scholar
Schonbek, M.E. & Wiegner, M. (1996) On the decay of higher-order norms of the solutions of Navier–Stokes equations. Proc. Roy. Soc. Edinburgh Sect.A 126, no. 3, 677–685.Google Scholar
Takahashi, S. (1999) A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. Ser. A: Theory Methods 37, no. 6, 751–789.Google Scholar
Wiegner, M. (1990) Decay and stability in Lp for strong solutions of the Cauchy problem for the Navier–Stokes equations. The Navier–Stokes equations (Oberwolfach, 1988), Lecture Notes in Math., vol. 1431, Springer, Berlin, 1990, 95–99.
Zhou, Y. (2007) A remark on the decay of solutions to the 3-D Navier–Stokes equations. Math. Methods Appl. Sci. 30, no. 10, 1223–1229.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×