Abstract
There is a long standing problem on the nonstationary Navier–Stokes equations which pertains to how to characterize the L1−time asymptotic expansion of the Navier–Stokes flows in the half space. Beyond a few partial results, new progress has not yet to be made on this open question. In this article, we give a confirmed answer to this problem; namely, a thorough characterization on L1−summability is revealed. In order to prove this result, we need to avoid the unboundedness of the projection operator, which is overcome by treating an elliptic Neumann problem. Finally, using the weighted estimates on the heat kernel’s convolution, we obtain the exact profile structure of the asymptotic expansion in \({L^1(\mathbb{R}^n_+)}\). In addition, some crucial estimates on the fractional spatial derivatives of the non-stationary Stokes and Navier–Stokes flows are established for the first time, which allows for a better understanding of the L1−decay problem.
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Han, P. Decay Results of the Nonstationary Navier–Stokes Flows in Half-Spaces. Arch Rational Mech Anal 230, 977–1015 (2018). https://doi.org/10.1007/s00205-018-1263-z
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DOI: https://doi.org/10.1007/s00205-018-1263-z