zbMATH — the first resource for mathematics

Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces. (English) Zbl 1205.35200
Summary: We derive an asymptotic expansion for smooth solutions of the Navier-Stokes equations in weighted spaces. This result removes previous restrictions on the number of terms of the asymptotics, as well as on the range of the polynomial weights. We also write the expansion in terms of expressions involving non-linear quantities.

MSC:
 35Q30 Navier-Stokes equations 35C20 Asymptotic expansions of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 35A22 Transform methods (e.g., integral transforms) applied to PDEs
Full Text:
References:
 [1] 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) · Zbl 0986.35085 [2] Bae, H.-O.; Jin, B.J., Temporal and spatial decays for the Navier-Stokes equations, Proc. roy. soc. Edinburgh sect. A, 135, 3, 461-477, (2005) · Zbl 1076.35089 [3] Brandolese, L., Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations, Rev. mat. iberoamericana, 20, 1, 223-256, (2004) · Zbl 1057.35026 [4] Brandolese, L., Space-time decay of Navier-Stokes flows invariant under rotations, Math. ann., 329, 4, 685-706, (2004) · Zbl 1080.35062 [5] Brandolese, L.; Meyer, Y., On the instantaneous spreading for the Navier-Stokes system in the whole space, ESAIM control optim. calc. var., 8, 273-285, (2002), a tribute to J.L. Lions · Zbl 1080.35063 [6] Brandolese, L.; Vigneron, F., New asymptotic profiles of nonstationary solutions of the Navier-Stokes system, J. math. pures appl. (9), 88, 1, 64-86, (2007) · Zbl 1127.35033 [7] Carpio, A., Large-time behavior in incompressible Navier-Stokes equations, SIAM J. math. anal., 27, 2, 449-475, (1996) · Zbl 0845.76019 [8] Dobrokhotov, S.Yu.; Shafarevich, A.I., Some integral identities and remarks on the decay at infinity of the solutions to the Navier-Stokes equations in the entire space, Russ. J. math. phys., 2, 1, 133-135, (1994) · Zbl 0976.35508 [9] Fujigaki, Y.; Miyakawa, T., Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space, SIAM J. math. anal., 33, 3, 523-544, (2001) · Zbl 0995.35046 [10] Foias, C.; Saut, J.-C., Asymptotic behavior, as $$t \rightarrow + \infty$$, of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana univ. math. J., 33, 3, 459-477, (1984) · Zbl 0565.35087 [11] Foias, C.; Saut, J.-C., On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations, Indiana univ. math. J., 33, 6, 911-926, (1984) · Zbl 0572.35081 [12] Foias, C.; Saut, J.-C., Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. inst. H. Poincaré anal. non linéaire, 4, 1, 1-47, (1987) · Zbl 0635.35075 [13] Foias, C.; Saut, J.-C., Asymptotic integration of Navier-Stokes equations with potential forces. I, Indiana univ. math. J., 40, 1, 305-320, (1991) · Zbl 0739.35066 [14] Grujić, Z.; Kukavica, I., A remark on time-analyticity for the Kuramoto-Sivashinsky equation, Nonlinear anal., 52, 1, 69-78, (2003) · Zbl 1020.35095 [15] T. Gallay, C.E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on $$\mathbb{R}^3$$, in: Recent Developments in the Mathematical Theory of Water Waves, Oberwolfach, 2001. · Zbl 1048.35055 [16] Kato, T., Strong $$L^p$$-solutions of the Navier-Stokes equation in $$\mathbb{R}^m$$, with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [17] Kukavica, I., Space-time decay for solutions of the Navier-Stokes equations, Indiana univ. math. J., 50, 205-222, (2001), (special issue dedicated to Professors Ciprian Foias and Roger Temam, Bloomington, IN, 2000) · Zbl 1006.35078 [18] Kukavica, I., On the weighted decay for solutions of the Navier-Stokes system, Nonlinear anal., 70, 6, 2466-2470, (2009) · Zbl 1166.35358 [19] Kajikiya, R.; Miyakawa, T., On $$L^2$$ decay of weak solutions of the Navier-Stokes equations in $$R^n$$, Math. Z., 192, 1, 135-148, (1986) · Zbl 0607.35072 [20] Kenig, C.E.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. amer. math. soc., 4, 2, 323-347, (1991) · Zbl 0737.35102 [21] Kukavica, I.; Torres, J.J., Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19, 2, 293-303, (2006) · Zbl 1106.35052 [22] Kukavica, I.; Torres, J.J., Weighted $$L^p$$ decay for solutions of the Navier-Stokes equations, Comm. partial differential equations, 32, 4-6, 819-831, (2007) · Zbl 1121.35101 [23] Lemarié-Rieusset, P.G., Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC res. notes math., vol. 431, (2002), Chapman & Hall/CRC Boca Raton, FL · Zbl 1034.35093 [24] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta math., 63, 1, 193-248, (1934) · JFM 60.0726.05 [25] Miyakawa, T., Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flows in $$R^n$$, Funkcial. ekvac., 45, 2, 271-289, (2002) · Zbl 1141.35434 [26] Miyakawa, T.; Schonbek, M.E., On optimal decay rates for weak solutions to the Navier-Stokes equations, Math. bohem., 126, 443-455, (2001) · Zbl 0981.35048 [27] Schonbek, M.E., $$L^2$$ decay for weak solutions of the Navier-Stokes equations, Arch. ration. mech. anal., 88, 3, 209-222, (1985) · Zbl 0602.76031 [28] Schonbek, M.E., Large time behavior of solutions to the Navier-Stokes equations, Comm. partial differential equations, 11, 733-763, (1986) · Zbl 0607.35071 [29] Schonbek, M.E., Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations, Indiana univ. math. J., 41, 3, 809-823, (1992) · Zbl 0759.35036 [30] 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, 677-685, (1996) · Zbl 0862.35086 [31] Takahashi, S., A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear anal., 37, 6, 751-789, (1999) · Zbl 0941.35066 [32] Wiegner, M., Decay and stability in $$L^p$$ for strong solutions of the Cauchy problem for the Navier-Stokes equations, (), 95-99
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.