Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions

Technical Summary: Weak Solutions and Singular Limits for a Compressible Fluid-Structure Interaction Problem with Slip Boundary Conditions

Problem Statement
This paper investigates a mathematical model describing the interaction between a three-dimensional compressible barotropic fluid and a two-dimensional viscoelastic solid shell (plate). The fluid is governed by the compressible Navier–Stokes equations, while the elastic structure is modeled by a linear fourth-order (bending) equation in Lagrangian coordinates. A defining feature of this study is the imposition of Navier-slip boundary conditions at the fluid-structure interface, allowing for tangential slip and normal velocity continuity, rather than the standard no-slip (adherence) condition. The study addresses two primary challenges:

1. The existence of global weak solutions for the coupled system, particularly in the presence of moving boundaries and slip conditions.
2. The rigorous justification of the singular limit where the fluid transitions from compressible/viscous to incompressible/inviscid (low Mach number and high Reynolds number regimes), converging to an Euler-plate interaction system.

The authors distinguish between two geometric settings: a general reference geometry (curved boundary) and a flat reference geometry (slab-like domain), noting that the latter allows for stronger regularity results due to the specific form of the kinematic coupling.

Methodology
The analysis employs a combination of advanced techniques from the theory of compressible fluids and fluid-structure interaction (FSI):

• Existence of Weak Solutions:

  • Domain Extension and Regularization: The authors extend the fluid domain to a fixed large domain $B$ and approximate the viscosity coefficients and pressure (using an artificial pressure term $\delta \varrho^\beta$) to handle the moving boundary and nonlinearity.
  • Time Discretization and Operator Splitting: Following the scheme in [45], the problem is decoupled into fluid and structure subproblems, solved iteratively, and then passed to the limit in the time step.
  • Equi-integrability of Density: A critical technical hurdle is establishing the equi-integrability of the density near the moving boundary under slip conditions. The authors utilize the specific structure of the Navier-slip condition and, in the flat geometry case, the resulting relation $\partial_t w = (u \cdot n_w) \circ \Phi_w$ to gain additional spatial regularity for the displacement $w$. This allows for a relaxation of the adiabatic exponent $\gamma$ requirements compared to no-slip cases without damping.
  • Effective Viscous Flux: Standard techniques involving the effective viscous flux identity are adapted to identify the weak limit of the pressure term.

• Singular Limits (Incompressible Inviscid Limit):

  • Modified Relative Entropy Method: To compare the weak solution of the compressible system (defined on a time-dependent domain $\Omega_{w^\varepsilon}$) with the strong solution of the limit Euler-plate system (defined on $\Omega_\eta$), the authors employ a modified relative energy inequality.
  • Piola Transformation: A naive coordinate transformation fails because it destroys the divergence-free property of the limit velocity and alters the boundary normal vectors. The authors introduce a Piola transform based on the composition of the deformation maps. This transformation preserves the solenoidal property of the velocity and correctly maps the normal vectors, ensuring the test functions satisfy the necessary boundary conditions.
  • Error Control: The analysis requires controlling error terms arising from the transformation (involving the Jacobian and deformation gradient). This necessitates additional regularity assumptions on the weak solution's displacement, which are justified by the presence of structural damping ($\nu_s > 0$) and specific geometric settings.

Key Contributions and Results

1. Existence of Weak Solutions (Theorem 1.4):

   • The authors prove the existence of global weak solutions for the compressible FSI system with Navier-slip boundary conditions.
   • Conditions:
     • With structural damping ($\nu_s > 0$), existence is shown for $\gamma > 3/2$.
     • Without structural damping ($\nu_s = 0$), existence is shown for the flat reference geometry ($\Omega \in \mathcal{G}_2$) provided $\gamma > 12/7$.
   • Significance: This extends previous results (e.g., [7], [45]) which primarily focused on no-slip conditions or required higher $\gamma$ for slip conditions. The proof highlights that in general geometries without damping, the lack of regularity for the normal velocity prevents closing the estimates, necessitating either damping or a flat geometry.

2. Incompressible Inviscid Limit (Theorem 1.7):

   • The paper rigorously justifies the convergence of the compressible viscous fluid-structure system to the incompressible inviscid (Euler) system coupled with a viscoelastic plate.
   • Regime: Low Mach number ($\varepsilon \to 0$) and high Reynolds number ($\nu \to 0$).
   • Requirements: The limit is established for well-prepared initial data, a flat reference geometry, structural damping ($\nu_s > 0$), and $\gamma > 3/2$. An additional regularity assumption on the weak solution's displacement (specifically regarding $\nabla w$) is required to handle the error terms in the relative entropy estimate.
   • Convergence Rate: The authors derive a convergence rate depending on the initial data perturbation and the parameters $\varepsilon$ and $\nu$.

3. Incompressible Limit (Theorem 4.14):

   • As a byproduct, the paper also addresses the limit where only the Mach number tends to zero (keeping viscosity fixed). In this case, the system converges to the incompressible Navier–Stokes equations coupled with a viscoelastic plate. Notably, this result holds for $\gamma > 3$ without the extra regularity assumptions required for the inviscid limit.

Significance and Claims
The authors claim that this work presents the first result concerning the weak solutions of compressible fluids interacting with elastic structures under Navier-slip boundary conditions. Furthermore, it provides the first rigorous justification of the low Mach number and high Reynolds number singular limit for such fluid-structure interaction problems.

The paper emphasizes that the interplay between the slip condition and the moving boundary geometry creates significant analytical difficulties, particularly in recovering the kinematic coupling condition and establishing the equi-integrability of the density. The use of the Piola transform to handle the domain mismatch in the relative entropy method is presented as a crucial technical innovation for deriving singular limits in time-dependent FSI domains.

The authors remain modest regarding the necessity of additional regularity assumptions for the inviscid limit, noting that the uniform regularity of weak solutions in the inviscid regime is insufficient to justify the limit without these extra constraints, a limitation shared by similar problems in the literature.