Global well-posedness of a 2D fluid-structure interaction problem with free surface
Thomas Alazard, Chengyang Shao, Haocheng Yang
TL;DR
This work studies the global well-posedness of a 2D fluid–structure interaction with a free surface, where a 2D incompressible, irrotational fluid occupies a time-dependent domain beneath a linearly elastic boundary. The authors transform the coupled Euler–beam problem into a nonlinear Schrödinger-type evolution on the free boundary by exploiting the Craig–Sulem–Zakharov formulation and a refined Dirichlet-to-Neumann operator analysis, enabling a low-regularity global theory. They develop low-regularity paralinearization, shape-derivative estimates, and Strichartz-type dispersive bounds to control the nonlinear and nonlocal terms, proving global well-posedness for finite-energy data and propagation of regularity, without any dissipative effects. Energy conservation for smooth solutions and tameness in high-regularity settings underpin the long-time well-posedness and stability results, highlighting dispersive mechanisms as a replacement for damping in this fluid–structure system. The approach advances the mathematical understanding of boundary-reduced dispersive models for fluid–structure interactions and furnishes new tools for Dirichlet-to-Neumann analysis at low regularity, with potential implications for 2D free-surface dynamics.
Abstract
This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally well-posed in time for any irrotational initial data in the energy space, without any smallness assumption. We also prove continuity with respect to the initial data and the propagation of regularity. The main novelty is that no dissipative effect is assumed in the system. In the absence of parabolic regularization, the key observation is that the system can be transformed into a nonlinear Schrödinger-type equation, to which dispersive estimates are applied. This allows us to construct solutions that are very rough from the point of view of fluid dynamics-the initial fluid velocity has merely one-half derivative in $L^2$. The main difficulty is that the problem is critical in the energy space with respect to several key inequalities from harmonic analysis. The proof incorporates new estimates for the Dirichlet-to-Neumann operator in the low-regularity regime, including refinements of paralinearization formulas and shape derivative formulas, which played a key role in the analysis of water waves.