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 $\gamma>0$, we establish global well-posedness for arbitrary initial data in $H^2(\R)\times\bigl(\dot H^{1/2}(\R)\cap\dot H^1(\R)\bigr)$. The main novelty is that the admissible data include finite energy initial potentials for which neither $\psi_0$ nor $u_0=\psi_0+G(\eta_0)\psi_0$ 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 Fr\'echet differentiability of the energy in these affine coordinates, together with a chain rule on $H^1\subset L^2\subset H^{-1}$, 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.