×

zbMATH — the first resource for mathematics

Initial profile for the slow decay of the Navier-Stokes flow in the half-space. (English) Zbl 1326.35246
This paper is concerned with the study of the lower bound of the weak solutions of the Navier-Stokes equations in the half-space with boundary conditions and initial data. The author investigate the application for the energy concentration phenomenon into the lower spectral bounds.
MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Borchers, W.; Miyakawa, T., \(L\)\^{2} decay for the Navier-Stokes flow in halfspaces, Math. Ann., 282, 139-155, (1988) · Zbl 0627.35076
[2] Borchers, W.; Miyakawa, T., Algebraic \(L\)\^{2} decay for Navier-Stokes flows in exterior domains, Acta Math., 165, 189-227, (1990) · Zbl 0722.35014
[3] Carpio, A., Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal., 27, 449-475, (1996) · Zbl 0845.76019
[4] Fujigaki, Y.; Miyakawa, T., Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the half-space, Methods Appl. Anal., 8, 121-157, (2001) · Zbl 1027.35089
[5] Fujigaki, Y.; Miyakawa, T., Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space, SIAM J. Math. Anal., 33, 523-544, (2001) · Zbl 0995.35046
[6] Y. Fujigaki and T. Miyakawa, On solutions with fast decay of nonstationary Navier-Stokes system in the half-space, In: Nonlinear problems in mathematical physics and related topics, I, Int. Math. Ser. (N. Y.), 1, 91-120, 2002. · Zbl 1290.35183
[7] Kajikiya, R.; Miyakawa, T., On \(L\)\^{2} decay of weak solutions of the Navier-Stokes equations in R\^{\(n\)}, Math. Z., 192, 135-148, (1986) · Zbl 0607.35072
[8] T. Kobayashi and T. Kubo, Weighted\(L\)\^{\(p\)}-\(L\)\^{\(q\)}estimates of Stokes semigroup in half-space and its application to the Navier-Stokes equations, preprint. · Zbl 1341.35101
[9] Kozono, H., Global \(L\)\^{\(n\)} -solution and its decay property for the Navier-Stokes equations in half-space \({\textbf{R}^n_+}\), J. Differential Equations, 79, 79-88, (1989) · Zbl 0715.35062
[10] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05
[11] Masuda, K., Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2), 36, 623-646, (1984) · Zbl 0568.35077
[12] T. Miyakawa and M. E. Schonbek, On optimal decay rates for weak solutions to the Navier-Stokes equations in\({\mathbb{R}^n}\) , in “Proceedings of Partial Differential Equations and Applications (Olomouc, 1999),” Math. Bohem., 126 (2001), 443-455. · Zbl 0981.35048
[13] Okabe, T., Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations, J. Differential Equations, 246, 895-908, (2009) · Zbl 1155.35069
[14] Okabe, T., Lower bound of \(L\)\^{2} decay of the Navier-Stokes flow in the half-space \({\mathbb{R}^n_+}\) and its asymptotic behavior in the frequency space, J. Math. Anal. Appl., 401, 534-547, (2013) · Zbl 1307.35199
[15] T. Okabe, Space-time asymptotics of the two dimensional Navier-Stokes flow in the whole plane, preprint. · Zbl 1378.35225
[16] Schonbek, M. E., \(L\)\^{2} decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031
[17] Schonbek, M. E., Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763, (1986) · Zbl 0607.35071
[18] Schonbek, M. E., Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc., 4, 423-449, (1991) · Zbl 0739.35070
[19] Schonbek, M. E., Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 41, 809-823, (1992) · Zbl 0759.35036
[20] Skalák, Z., Large time behavior of energy in exponentially decreasing solutions of the Navier-Stokes equations, Nonlinear Anal., 71, 593-603, (2009) · Zbl 1170.35496
[21] Skalák, Z., Asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations in R\^{3}, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55, 377-394, (2009) · Zbl 1205.35211
[22] Skalák, Z., Conditions for asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations, Nonlinear Anal., 71, e2070-e2081, (2009) · Zbl 1239.76022
[23] Skalák, Z., On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R\^{3}, Discrete Contin. Dyn. Syst. Ser. S, 3, 361-370, (2010) · Zbl 1193.35136
[24] Skalák, Z., Some aspects of the asymptotic dynamics of solutions of the homogeneous Navier-Stokes equations in general domains, J. Math. Fluid Mech., 12, 503-535, (2010) · Zbl 1270.35355
[25] Skalák, Z., On large-time energy concentration in solutions to the Navier-Stokes equations in general domains, ZAMM Z. Angew. Math. Mech., 91, 724-732, (2011) · Zbl 1239.76027
[26] Z. Skalák, A note on lower bounds of decay rates for solutions to the Navier-Stokes equations in the norms of Besov spaces, preprint. · Zbl 0638.76040
[27] Ukai, S., A solution formula for the Stokes equation in \({\textbf{R}^n_+}\), Comm. Pure Appl. Math., 40, 611-621, (1987) · Zbl 0638.76040
[28] Wiegner, M., Decay results for weak solutions of the Navier-Stokes equations on R\^{\(n\)}, J. London Math. Soc. (2), 35, 303-313, (1987) · Zbl 0652.35095
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.