Global strong solutions without contact for a viscous fluid interacting with a purely elastic beam
Abstract
We consider a two-dimensional incompressible viscous fluid confined in a periodic channel between a fixed flat wall and a purely elastic periodic beam. For compatible initial fluid velocity in H1, beam height in H3, and beam velocity in H1, with strictly positive initial height, we establish global existence and uniqueness of strong solutions. In particular, there is no contact of the beam and bottom, which is built on a uniform distance bound derived by the mechanical energy and fluid dissipation through a height-dependent degenerate elliptic regularization of the classical distance test. Combining tangential differentiation, a fractional estimate for the height, and a negative-order tangential estimate for the accelerations, we establish a global regularity estimate. Moreover, quadratic time-weighted estimates that quantify the higher regularity at positive times are justified. As a consequence, the work gives an affirmative answer to the problem proposed by Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220(3): 1283--1333, 2016) of whether fluid dissipation alone is sufficient to derive the distance estimate.
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BibTeX
@misc{liu2026global,
title = {{Global strong solutions without contact for a viscous fluid interacting with a purely elastic beam}},
author = {Yadong Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22090},
eprint = {2610.22090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22090},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global strong solutions without contact for a viscous fluid interacting with a purely elastic beam
AU - Yadong Liu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22090
UR - https://arxiv.org/abs/2610.22090
AB - We consider a two-dimensional incompressible viscous fluid confined in a periodic channel between a fixed flat wall and a purely elastic periodic beam. For compatible initial fluid velocity in H^1, beam height in H^3, and beam velocity in H^1, with strictly positive initial height, we establish global existence and uniqueness of strong solutions. In particular, there is no contact of the beam and bottom, which is built on a uniform distance bound derived by the mechanical energy and fluid dissipation through a height-dependent degenerate elliptic regularization of the classical distance test. Combining tangential differentiation, a fractional estimate for the height, and a negative-order tangential estimate for the accelerations, we establish a global regularity estimate. Moreover, quadratic time-weighted estimates that quantify the higher regularity at positive times are justified. As a consequence, the work gives an affirmative answer to the problem proposed by Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220(3): 1283--1333, 2016) of whether fluid dissipation alone is sufficient to derive the distance estimate.
ER -