an:06299470
Zbl 1293.35205
Han, Pigong
Long-time behavior for the nonstationary Navier-Stokes flows in \(L^1(\mathbb R_+^n)\)
EN
J. Funct. Anal. 266, No. 3, 1511-1546 (2014).
00332991
2014
j
35Q30 76D05 76D07 35D35
Navier-Stokes flows; solution formula; long-time behavior; strong solution
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 \textit{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 \textit{L. Brandolese}'s work [C. R. Acad. Sci., Paris, S??r. I, Math. 332, No. 2, 125--130 (2001; Zbl 0973.35149)].
Zbl 1091.35055; Zbl 1188.35127; Zbl 0973.35149