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Long-time behavior for the nonstationary Navier-Stokes flows in $$L^1(\mathbb R_+^n)$$. (English) Zbl 1293.35205
Summary: The large time decay for the Navier-Stokes flows in $$L^1(\mathbb R_+^n)$$ is a long-standing unsolved question. The main difficulties are that: usual $$L^q - L^r$$ estimates for the Stokes flow fail in this case; and the projection operator $$P : L^1(\mathbb R_+^n) \to L_{\sigma}^1(\mathbb R_+^n)$$ becomes unbounded. Using the Stokes solution formula, we find a crucial and new estimate for the Stokes flow in $$L^1(\mathbb R_+^n)$$, which plays a fundamental role in studying the time $$L^1$$-behavior for the Navier-Stokes equations. In addition, we decompose the operator $$P$$ into two parts, and reduce its unboundedness to establish an $$L^1$$ estimate for an elliptic problem with Neumann boundary condition, which is overcome by using the weighted estimates of the Gaussian kernel’s convolution. The main results in this article are motivated by H.-O. Bae’s works [J. Differ. Equations 222, No. 1, 1–20 (2006; Zbl 1091.35055); J. Math. Fluid Mech. 10, No. 4, 503–530 (2008; Zbl 1188.35127)], and L. Brandolese’s work [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 2, 125–130 (2001; Zbl 0973.35149)].

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76D07 Stokes and related (Oseen, etc.) flows 35D35 Strong solutions to PDEs
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##### 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.; Choe, H., Decay rate for the incompressible flows in half spaces, Math. Z., 238, 799-816, (2001) · Zbl 1028.35118 [4] Bae, H.; Choe, H., A regularity criterion for the Navier-Stokes equations, Comm. Partial Differential Equations, 32, 1173-1187, (2007) · Zbl 1220.35111 [5] 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 [6] 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 [7] 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 [8] 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 [9] Bae, H.; Jin, B., Existence of strong mild solution of the Navier-Stokes equations in the half space with nondecaying initial data, J. Korean Math. Soc., 49, 113-138, (2012) · Zbl 1234.35176 [10] Borchers, W.; Miyakawa, T., $$L^2$$ decay for the Navier-Stokes flow in half spaces, Math. Ann., 282, 139-155, (1988) · Zbl 0627.35076 [11] Brandolese, L., On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in $$R^n$$, C. R. Acad. Sci. Paris Sér. I Math., 332, 125-130, (2001) · Zbl 0973.35149 [12] Brandolese, L., Space-time decay of Navier-Stokes flows invariant under rotations, Math. Ann., 329, 685-706, (2004) · Zbl 1080.35062 [13] Brandolese, L.; Vigneron, F., New asymptotic profiles of nonstationary solutions of the Navier-Stokes system, J. Math. Pures Appl., 88, 64-86, (2007) · Zbl 1127.35033 [14] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831, (1982) · Zbl 0509.35067 [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] Galdi, G., An introduction to the mathematical theory of the Navier-Stokes equations, vol. I. linearized steady problems, Springer Tracts Nat. Philos., vol. 38, (1994), Springer-Verlag New York · Zbl 0949.35004 [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] Han, P., Weighted decay properties for the incompressible Stokes flow and Navier-Stokes equations in a half space, J. Differential Equations, 253, 1744-1778, (2012) · Zbl 1248.35146 [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., Nonstationary Navier-Stokes flows in a two-dimensional exterior domain with rotational symmetries, Indiana Univ. Math. J., 55, 1483-1555, (2006) · Zbl 1122.35095 [23] Lin, F., A new proof of the caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51, 241-257, (1998) · Zbl 0958.35102 [24] Schonbek, M. E., $$L^2$$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031 [25] 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 [26] 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 [27] 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 [28] 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 [29] Ukai, S., A solution formula for the Stokes equation in $$\mathbb{R}^N$$, Comm. Pure Appl. Math., XL, 611-621, (1987) · Zbl 0638.76040
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