L1 Behavior of the Stokes System and the Navier--Stokes System in the Half Space
Abstract
In this paper, the detailed behavior of solutions of both the Stokes and the Navier--Stokes system in the half space is investigated when the initial data belongs to L1. We first give a detailed asymptotic expansion of the solution to the Stokes system supplemented with L1 initial data. With the aid of this expansion, we provide two sufficient conditions on the initial data so that the associated solutions of the Navier--Stokes system do not belong to L1, corresponding to long time and short time behaviors, respectively. Moreover, for the n- dimensional Navier--Stokes system, a pointwise spatial lower bound of order n is derived in a conic neighborhood of the xn-axis, provided that the initial data decays faster than order n. One of the main difficulties to get precise form of leading order term is the non-commutativity of the Leray projection operator with the Laplacian. Our strategy is to employ the special structure of the Green tensor observed by Han and a cancellation property of the initial data. This also yields the sufficient conditions on the initial data to get precise asymptotic behavior via comparing the linear and nonlinear parts.
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BibTeX
@misc{li2026behavior,
title = {{$L^1$ Behavior of the Stokes System and the Navier--Stokes System in the Half Space}},
author = {Rao Li and Haitao Wang and Chunjing Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37484},
eprint = {2609.37484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37484},
}
RIS
TY - UNPB
TI - L^1 Behavior of the Stokes System and the Navier--Stokes System in the Half Space
AU - Rao Li
AU - Haitao Wang
AU - Chunjing Xie
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.37484
UR - https://arxiv.org/abs/2609.37484
AB - In this paper, the detailed behavior of solutions of both the Stokes and the Navier--Stokes system in the half space is investigated when the initial data belongs to L^1. We first give a detailed asymptotic expansion of the solution to the Stokes system supplemented with L^1 initial data. With the aid of this expansion, we provide two sufficient conditions on the initial data so that the associated solutions of the Navier--Stokes system do not belong to L^1, corresponding to long time and short time behaviors, respectively. Moreover, for the n- dimensional Navier--Stokes system, a pointwise spatial lower bound of order n is derived in a conic neighborhood of the x_n-axis, provided that the initial data decays faster than order n. One of the main difficulties to get precise form of leading order term is the non-commutativity of the Leray projection operator with the Laplacian. Our strategy is to employ the special structure of the Green tensor observed by Han and a cancellation property of the initial data. This also yields the sufficient conditions on the initial data to get precise asymptotic behavior via comparing the linear and nonlinear parts.
ER -