Abstract
We study incompressible Navier–Stokes flows in \({\mathbb R^d}\) with small and well localized data and external force f. We establish pointwise estimates for large |x| of the form \({c_t|x|^{-d}\le |u(x,t)|\le c^\prime_t|x|^{-d}}\), where c t > 0 whenever \({\int_0^t\int f(x,s)\,dx\,ds\not= {\bf 0}}\) . This sharply contrasts with the case of the Navier–Stokes equations without force, studied in Brandolese and Vigneron (J Math Pures Appl 88:64–86, 2007) where the spatial asymptotic behavior was \({|u(x,t)|\simeq C_t|x|^{-d-1}}\) . In particular, this shows that external forces with non-zero mean, no matter how small and well localized (say, compactly supported in space-time), increase the velocity of fluid particles at all times t and at at all points x in the far-field. As an application of our analysis on the pointwise behavior, we deduce sharp upper and lower bounds of weighted L p-norms for strong solutions, extending the results obtained in Bae et al. (to appear) for weak solutions, by considering here a larger (and in fact, optimal) class of weight functions.
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References
Amrouche C., Girault V., Schonbek M.E., Schonbek T.P.: Pointwise decay of solutions and of higher derivatives to Navier–Stokes equations. SIAM J. Math. Anal. 31(4), 740–753 (2000)
Bae, H.-O., Brandolese, L., Jin, B.J.: Asymptotic behavior for the Navier–Stokes equations with nonzero external forces. Nonlinear analysis, doi:10.1016/j.na.2008.10.074 (to appear)
Bae H.-O., Jin B.J.: Upper and lower bounds of temporal and spatial decays for the Navier–Stokes equations. J. Diff. Eq 209, 365–391 (2006)
Bergh J., Löfsrtŏm J.: Interpolation Spaces, an Introduction. Springer, Heidelberg (1976)
Brandolese L., Vigneron F.: New asymptotic profiles of nonstationnary solutions of the Navier–Stokes system. J. Math. Pures Appl. 88, 64–86 (2007)
Brandolese L.: Fine properties of self-similar solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 192(3), 375–401 (2009)
Choe H.J., Jin B.J.: Weighted estimates of the asymptotic profiles of the Navier–Stokes flow in \({\mathbb R^n}\) . J. Math. Anal. Appl. 344(1), 353–366 (2008)
He C., Xin Z.: On the decay properties of solutions to the nonstationary Navier–Stokes equations in \({\mathbb R^3}\) . Proc. Roy. Edinbourgh Soc. Sect. A. 131(3), 597–619 (2001)
Kato T.: Strong L p-solutions of the Navier–Stokes equations in \({\mathbb R^m}\) , with applications to weak solutions. Math. Z 187, 471–480 (1984)
Kukavica I., Torres J.J.: Weighted bounds for velocity and vorticity for the Navier–Stokes equations. Nonlinearity 19, 293–303 (2006)
Kukavica I., Torres J.J.: Weighted L p decay for solutions of the Navier–Stokes equations. Comm. Part. Diff. Eq. 32, 819–831 (2007)
Lemarié-Rieusset P.G.: Recent developments in the Navier–Stokes problem. Chapman & Hall, CRC Press, Boca Raton (2002)
Miyakawa T.: On space time decay properties of nonstationary incompressible Navier–Stokes flows in \({\mathbb R^n}\) . Funkcial. Ekvac. 32(2), 541–557 (2000)
Oliver M., Titi E.: Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in \({\mathbb R^n}\) . J. Funct. Anal. 172, 1–18 (2000)
Schonbek M.E.: L 2 decay for weak solutions of the Navier–Stokes equations. Arch. Rat. Mech. Anal. 88, 209–222 (1985)
Schonbek M.E.: Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Amer. Math. Soc. 4(3), 423–449 (1991)
Skalák, Z.: Asymptotic decay of higher-order norms of solutions to the Navier–Stokes equations in \({\mathbb R^3}\) . Preprint (2009), http://mat.fsv.cvut.cz/nales/preprints/
Takahashi S.: A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. 37, 751–789 (1999)
Zhang Qi S.: Global solutions of the Navier–Stokes equations with large L 2 norms in a new function space. Adv. Diff. Eq. 9(5–6), 587–624 (2004)
Zhou Y.: A remark on the decay of solutions to the 3-D Navier–Stokes equations. Math. Meth. Appl. Sci. 30, 1223–1229 (2007)
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This research was supported by EGIDE, through the program Huber-Curien “Star” N. 16560RK.
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Bae, HO., Brandolese, L. On the effect of external forces on incompressible fluid motions at large distances. Ann. Univ. Ferrara 55, 225–238 (2009). https://doi.org/10.1007/s11565-009-0079-z
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DOI: https://doi.org/10.1007/s11565-009-0079-z
Keywords
- Navier–Stokes
- Asymptotic profiles
- Asymptotic behavior
- Far-field
- Spatial infinity
- Strong solutions
- Incompressible viscous flows
- Decay estimates