Global existence and convergence near equilibrium for the moving interface problem between Navier-Stokes and the linear wave equation

We first establish existence for all positive time near equilibrium for the moving interface problem between the Navier-Stokes equations for the evolving fluid phase (moved by the fluid velocity) and an elastic body modelled by the linear wave equation. This problem has an infinite number of simple solutions with a flat interface (with zero velocity in the fluid, and zero horizontal velocity in the solid), that we call flat interface solutions. We then show that if the initial data is close enough to the canonical equilibrium, the solution converges towards a flat interface solution in large time, showing that these flat interface solutions capture the long time behaviour of this fluid-structure problem near the canonical equilibrium. This result is established with gravity (which can be set to zero or not). It is established for the case where the solid has initial volume close to the volume of its reference configuration (where the linear wave equation is naturally written).

💡 Research Summary

This paper addresses the mathematically challenging fluid‑structure interaction (FSI) problem in which a three‑dimensional incompressible Navier‑Stokes fluid is coupled with an elastic solid modeled by the linear wave equation. The fluid occupies a time‑dependent domain Ω_f(t) and the solid occupies Ω_s(t); the two subdomains share a moving interface Γ(t) that is advected by the fluid velocity. The authors consider periodic boundary conditions in the horizontal directions and allow for a constant gravity term g≥0. The solid’s displacement η satisfies
 η_tt = λΔη – g e_3 in Ω_s⁰,
with homogeneous Dirichlet conditions on the interface and prescribed initial data (η₀, η̇₀). The fluid velocity u and pressure p satisfy the incompressible Navier‑Stokes equations in Ω_f(t) together with the usual no‑slip condition on the fixed outer boundary and continuity of velocity and normal stress across Γ(t).

A key difficulty is that the linear wave equation provides no intrinsic dissipation; consequently, standard energy methods do not yield global‑in‑time control for the coupled system. Previous global results for fluid‑elastic interaction have either required higher‑order elastic operators (e.g., fourth‑order plate or shell models) or introduced artificial damping terms in the solid or on the interface. The present work shows that, even without any damping, global existence and asymptotic stability can be achieved provided the initial data are sufficiently close to a flat equilibrium configuration.

Main contributions

1. Arbitrary Lagrangian–Eulerian (ALE) formulation – Instead of the classical Lagrangian map for the fluid, the authors introduce an ALE map based on a Stokes‑extension of the solid displacement η from the interface Γ into the whole fluid domain. This extension, denoted E(η), yields a deformation X̃ = Id + E(η) that transports the fluid equations while preserving the divergence‑free condition. The ALE framework is crucial because it allows the authors to treat η as a “hidden” variable in the fluid energy, even though η does not appear directly in the dissipative part of the energy.

2. Dissipative norm and energy hierarchy – The basic energy functional combines the kinetic energy of the fluid and the elastic energy of the solid:
    E(t) = ½‖u‖²_{L²(Ω_f)} + ½λ‖∇η‖²_{L²(Ω_s)}.
   The viscous term ν‖∇u‖²_{L²(Ω_f)} provides the only source of dissipation. The authors augment the norm to include first‑order horizontal derivatives of η, which are shown to be controlled by the fluid dissipation (Section 6). This “surprising” estimate establishes that the interface regularity is indirectly damped by the fluid viscosity.

3. A priori estimates and bootstrap – The analysis proceeds through three levels of differentiated problems: (i) the original (0‑th order) system, (ii) the first time‑differentiated system, and (iii) the second time‑differentiated system (the highest time order). For each level the authors derive uniform Sobolev estimates. The first‑order time‑differentiated system is particularly favorable because the nonlinear terms cancel, allowing a clean energy inequality. The highest‑order system requires careful handling of commutators and the smallness of the initial data; the authors also need λ to be large relative to g when gravity is present.

4. Global existence near equilibrium (Theorem 1) – Assuming the initial displacement, velocity, and fluid data are sufficiently small in the dissipative norm and that the solid’s initial volume is close to that of the reference configuration, the authors prove the existence of a unique strong solution for all t>0. The proof relies on a Grönwall‑type argument that closes the hierarchy of estimates.

5. Asymptotic convergence to flat interface solutions (Theorem 2) – For the same class of small data, the solution converges as t→∞ to one of the infinitely many flat‑interface equilibria (u≡0, η≡0, constant pressure). The fluid velocity decays to zero in L², the interface displacement tends to zero in H^{3/2}(Γ), and the vertical component of the solid’s displacement satisfies a weak limit that solves a one‑dimensional wave equation with homogeneous Dirichlet boundary conditions. The convergence proof uses the decay of the dissipative energy, the interface regularity estimate, and continuity of velocity and normal stress across Γ(t).

6. Physical interpretation and extensions – The result shows that even without any artificial damping, the viscous fluid alone can stabilize the coupled system provided the deformation is small. The requirement that the solid’s average height h_e be close to the reference height h_s reflects a small‑strain regime; the volume constraint is a consequence of fluid incompressibility. The methodology—ALE extension plus a carefully designed dissipative norm—offers a template for tackling more complex FSI problems, such as those involving nonlinear elasticity, plasticity, or multi‑phase interfaces.

Conclusion
The paper delivers the first global‑in‑time existence and long‑time stability results for the Navier‑Stokes–linear‑wave interface problem without any added damping. By exploiting an ALE representation and establishing that the fluid’s viscous dissipation indirectly controls the solid’s interface regularity, the authors overcome the lack of intrinsic solid damping. This work closes a notable gap in the mathematical theory of fluid‑elastic interaction and opens avenues for future research on undamped hyperbolic‑parabolic coupled systems.