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Navier-Stokes flow past a translating body: attainability of the steady state as limit of an Lp weak solution
Abstract
Consider a viscous, incompressible, Newtonian fluid that fills the region of space exterior to a rigid body. Let the fluid be activated by a perturbation u0∈ Lp(OO), p∈ (2,3) and let the body start to translate with a prescribed velocity −h(t)u∞, t>0, u∞∈R3∖ {0}. Suppose that the body reaches the stationary regime after some finite time. We show that, under suitable assumptions on u∞, a steady state is always reachable by the fluid, asymptotically in time, as limit of a suitable weak solution to the Navier-Stokes flow in suitable metrics. Our result seems to be a further contribution related to the {\it Starting Problem}, posed by R. Finn in 1965.
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BibTeX
@misc{palma2026navierstokes,
title = {{Navier-Stokes flow past a translating body: attainability of the steady state as limit of an $L^p$ weak solution}},
author = {Filippo Palma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28150},
eprint = {2610.28150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28150},
}
RIS
TY - UNPB
TI - Navier-Stokes flow past a translating body: attainability of the steady state as limit of an L^p weak solution
AU - Filippo Palma
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28150
UR - https://arxiv.org/abs/2610.28150
AB - Consider a viscous, incompressible, Newtonian fluid that fills the region of space exterior to a rigid body. Let the fluid be activated by a perturbation u_0∈ L^p(OO), p∈ (2,3) and let the body start to translate with a prescribed velocity −h(t)u_∞, t>0, u_∞∈R^3∖ {0}. Suppose that the body reaches the stationary regime after some finite time. We show that, under suitable assumptions on u_∞, a steady state is always reachable by the fluid, asymptotically in time, as limit of a suitable weak solution to the Navier-Stokes flow in suitable metrics. Our result seems to be a further contribution related to the {\it Starting Problem}, posed by R. Finn in 1965.
ER -