Global well-posedness for a damped two dimensional fluid--beam system
Abstract
We study the normalized surface system of an irrotational incompressible fluid of infinite depth beneath an elastic Euler--Bernoulli beam with structural damping. For every fixed damping coefficient γ>0, we establish global well-posedness for arbitrary initial data in H2(R)×(Ḣ1/2(R)∩Ḣ1(R)). The main novelty is that the admissible data include finite energy initial potentials for which neither ψ0 nor u0=ψ0+G(η0)ψ0 has an L2 representative. We also prove the exact energy dissipation identity at the regularity of the constructed solutions. The main analytical difficulties are the possible failure of L2 integrability at low frequencies and the justification of energy differentiation at the solution regularity. To handle the low-frequency difficulty, we fix the low-frequency part of the initial potential and evolving an L2 remainder. Homogeneous Dirichlet--Neumann estimates and a high-frequency Fourier symmetrizer yield local well-posedness with H−1 forcing. Continuous Fréchet differentiability of the energy in these affine coordinates, together with a chain rule on H1⊂ L2⊂ H−1, justifies the dissipation identity. The Energy identity and coercivity provide uniform bounds that yields global continuation tharough the local theory. Iterating the local uniqueness and stability estimates gives global uniqueness within the stated affine solution class and Lipschitz dependence on bounded sets over every finite time interval.
Related
- GLOBAL ATTRACTOR FOR TWO-DIMENSIONAL WEAKLY DAMPED NONLINEAR PARAMETRIC KLEIN-GORDON SCHRODINGER SYSTEM
- Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations
- Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
- Global attractor for weakly damped nonlinear parametric Klein-Gordon Schr{ö}dinger system
- The Global Existence and Energy Decay of a Partially Logarithmically Damped Wave System
BibTeX
@misc{wang2026global,
title = {{Global well-posedness for a damped two dimensional fluid--beam system}},
author = {Weihua Wang and Lu Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18310},
eprint = {2610.18310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18310},
}
RIS
TY - UNPB
TI - Global well-posedness for a damped two dimensional fluid--beam system
AU - Weihua Wang
AU - Lu Yang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.18310
UR - https://arxiv.org/abs/2610.18310
AB - We study the normalized surface system of an irrotational incompressible fluid of infinite depth beneath an elastic Euler--Bernoulli beam with structural damping. For every fixed damping coefficient γ>0, we establish global well-posedness for arbitrary initial data in H^2(R)×(Ḣ^1/2(R)∩Ḣ^1(R)). The main novelty is that the admissible data include finite energy initial potentials for which neither ψ_0 nor u_0=ψ_0+G(η_0)ψ_0 has an L^2 representative. We also prove the exact energy dissipation identity at the regularity of the constructed solutions. The main analytical difficulties are the possible failure of L^2 integrability at low frequencies and the justification of energy differentiation at the solution regularity. To handle the low-frequency difficulty, we fix the low-frequency part of the initial potential and evolving an L^2 remainder. Homogeneous Dirichlet--Neumann estimates and a high-frequency Fourier symmetrizer yield local well-posedness with H^−1 forcing. Continuous Fréchet differentiability of the energy in these affine coordinates, together with a chain rule on H^1⊂ L^2⊂ H^−1, justifies the dissipation identity. The Energy identity and coercivity provide uniform bounds that yields global continuation tharough the local theory. Iterating the local uniqueness and stability estimates gives global uniqueness within the stated affine solution class and Lipschitz dependence on bounded sets over every finite time interval.
ER -