Weighted decay properties for the incompressible Stokes flow and Navier-Stokes equations in a half space. (English) Zbl 1248.35146
Summary: Using the Stokes solution formula and \(L^q\)-\(L^r\) estimates of the Stokes operator semigroup, we establish the weighted decay properties for the Stokes flow and Navier-Stokes equations including their spatial derivatives in half spaces. In addition, the unboundedness of the projection operator \(P: L^\infty(\mathbb{R}^n_+)\to L^\infty_\sigma(\mathbb{R}^n_+)\) is overcome by employing a decomposition for the nonlinear term, and \(L^\infty\)-asymptotic behavior for the second derivatives of Navier-Stokes flows in half spaces is given.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 35B40 | Asymptotic behavior of solutions to PDEs |
References:
| [1] | Bae, H., Temporal decays in \(L^1\) and \(L^\infty\) for the Stokes flow, J. Differential Equations, 222, 1-20 (2006) · Zbl 1091.35055 |
| [2] | Bae, H., Temporal and spatial decays for the Stokes flow, J. Math. Fluid Mech., 10, 503-530 (2008) · Zbl 1188.35127 |
| [3] | Bae, H., Analyticity and asymptotics for the Stokes solutions in a weighted space, J. Math. Anal. Appl., 269, 149-171 (2002) · Zbl 1027.35100 |
| [4] | Bae, H.; Choe, H., Decay rate for the incompressible flows in half spaces, Math. Z., 238, 799-816 (2001) · Zbl 1028.35118 |
| [5] | Bae, H.; Jin, B., Asymptotic behavior for the Navier-Stokes equations in 2D exterior domains, J. Funct. Anal., 240, 508-529 (2006) · Zbl 1115.35095 |
| [6] | Bae, H.; Jin, B., Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc., 44, 547-567 (2007) · Zbl 1146.35070 |
| [7] | Bae, H.; Jin, B., Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations, 209, 365-391 (2005) · Zbl 1062.35058 |
| [8] | Bae, H.; Jin, B., Temporal and spatial decays for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 135, 461-477 (2005) · Zbl 1076.35089 |
| [9] | Bae, H.; Roh, J., Optimal weighted estimates of the flows in exterior domains, Nonlinear Anal., 73, 1350-1363 (2010) · Zbl 1193.35122 |
| [10] | Brandolese, L., Space-time decay of Navier-Stokes flows invariant under rotations, Math. Ann., 329, 4, 685-706 (2004) · Zbl 1080.35062 |
| [11] | Borchers, W.; Miyakawa, T., \(L^2\) decay for the Navier-Stokes flow in half spaces, Math. Ann., 282, 139-155 (1988) · Zbl 0627.35076 |
| [12] | Choe, H.; Jin, B., Decay rate of solutions of Navier-Stokes equations in 3-dimensional half space, J. Math. Anal. Appl., 339, 1485-1496 (2008) · Zbl 1130.35102 |
| [13] | Desch, W.; Hieber, M.; Pruss, J., \(L^p\)-theory of the Stokes equation in a half-space, J. Evol. Equ., 1, 115-142 (2001) · Zbl 0983.35102 |
| [14] | Farwig, R., Partial regularity and weighted energy estimates of global weak solutions of the Navier-Stokes system, (Chipot, M.; Shafrir, I., Pitman Notes in Math. Ser., vol. 345 (1996), Longman: Longman New York) · Zbl 0865.35100 |
| [15] | Fujigaki, Y.; Miyakawa, T., Asymptotic profiles of non stationary incompressible Navier-Stokes flows in the half-space, Methods Appl. Anal., 8, 121-158 (2001) · Zbl 1027.35089 |
| [16] | Fröhlich, A., Solutions of the Navier-Stokes initial value problem in weighted \(L^q\)-spaces, Math. Nachr., 269, 150-166 (2004) · Zbl 1050.35068 |
| [17] | Giga, Y.; Matsui, S.; Shimizu, Y., On estimates in Hardy spaces for the Stokes flow in a half space, Math. Z., 231, 383-396 (1999) · Zbl 0928.35118 |
| [18] | Han, P., Asymptotic behavior for the Stokes flow and Navier-Stokes equations in half spaces, J. Differential Equations, 249, 1817-1852 (2010) · Zbl 1200.35216 |
| [19] | Han, P., Decay results of solutions to the incompressible Navier-Stokes flows in a half space, J. Differential Equations, 250, 3937-3959 (2011) · Zbl 1211.35215 |
| [20] | He, C.; Miyakawa, T., Nonstationary Navier-Stokes flows in a two-dimensional exterior domain with rotational symmetries, Indiana Univ. Math. J., 55, 1483-1555 (2006) · Zbl 1122.35095 |
| [21] | He, C.; Miyakawa, T., On \(L^1\)-summability and asymptotic profiles for smooth solutions to Navier-Stokes equations in a 3D exterior domain, Math. Z., 245, 387-417 (2003) · Zbl 1042.35046 |
| [22] | He, C.; Miyakawa, T., On weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain, J. Differential Equations, 246, 2355-2386 (2009) · Zbl 1162.35061 |
| [23] | He, C.; Wang, L., Moment estimates for weak solutions to the Navier-Stokes equations in half-space, Math. Methods Appl. Sci., 32, 1878-1892 (2009) · Zbl 1175.35102 |
| [24] | Jin, B., Weighted \(L^q-L^1\) estimate of the Stokes flow in the half space, Nonlinear Anal., 72, 1031-1043 (2010) · Zbl 1182.35181 |
| [25] | Jin, B., Spatial and temporal decay estimate of the Stokes flow of weighted \(L^1\) initial data in the half space, Nonlinear Anal., 73, 1394-1407 (2010) · Zbl 1193.35133 |
| [26] | Kozono, H., Global \(L^n\)-solution and its decay property for the Navier-Stokes equations in half-space \(R_+^n\), J. Differential Equations, 79, 79-88 (1989) · Zbl 0715.35062 |
| [27] | Kozono, H., \(L^1\)-solutions of the Navier-Stokes equations in exterior domains, Math. Ann., 312, 319-340 (1998) · Zbl 0920.35108 |
| [28] | Kozono, H.; Ogawa, T., Some \(L^p\) estimate for the exterior Stokes flow and an application to the non-stationary Navier-Stokes equations, Indiana Univ. Math. J., 41, 789-808 (1992) · Zbl 0759.35035 |
| [29] | Miyakawa, T., Hardy spaces of solenoidal vector fields, with application to the Navier-Stokes equations, Kyushu J. Math., 50, 1-64 (1997) · Zbl 0883.35088 |
| [30] | Mizumachi, R., On the asymptotic behavior of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 36, 498-522 (1984) · Zbl 0578.76026 |
| [31] | 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 |
| [32] | 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 |
| [33] | Shimizu, Y., \(L^\infty \)-estimate of first-order space derivatives of Stokes flow in a half space, Funkcial. Ekvac., 42, 291-309 (1999) · Zbl 1141.35435 |
| [34] | Solonnikov, V. A., Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator, Uspekhi Mat. Nauk, 58, 123-156 (2003) · Zbl 1059.35101 |
| [35] | Solonnikov, V. A., On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci., 114, 1726-1740 (2003) · Zbl 1054.35059 |
| [36] | Stein, E. M., Harmonic Analysis (1993), Princeton University Press: Princeton University Press Princeton · Zbl 0821.42001 |
| [37] | Torchinsky, A., Real-Variable Methods in Harmonic Analysis (1986), Academic Press · Zbl 0621.42001 |
| [38] | Ukai, S., A solution formula for the Stokes equation in \(R^N\), Comm. Pure Appl. Math., XL, 611-621 (1987) · Zbl 0638.76040 |
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