Regularity and Decay for Navier-Slip Fluid-Structure Interaction in Exterior Domains

Technical Summary: Regularity and Decay for Navier-Slip Fluid-Structure Interaction in Exterior Domains

Problem Statement
This paper investigates the linearized motion of a spherical rigid body immersed in an incompressible, viscous fluid occupying the entire three-dimensional space ($\mathbb{R}^3$). The system is governed by the Navier-Stokes equations coupled with ordinary differential equations describing the rigid body's translational and rotational dynamics. A critical feature of this study is the imposition of Navier slip boundary conditions with friction at the fluid-structure interface, as opposed to the classical no-slip (Dirichlet) conditions. The authors focus on the linearized system around the rest state, where the fluid velocity, pressure, and the body's translational and angular velocities evolve according to a coupled system in an exterior domain $\Omega = \mathbb{R}^3 \setminus B$, where $B$ is a ball.

Methodology
The analysis is conducted within the framework of semigroup theory and maximal $L^q$-regularity. The authors define a fluid-structure operator $A_{\alpha,q}$ acting on a Banach space $X_q$ that encapsulates the fluid velocity, translational velocity, and angular velocity. The methodology proceeds through several key stages:

1. Operator Formulation: The system is reformulated on the whole space $\mathbb{R}^3$ by extending the rigid body velocity into the interior of the sphere. This allows the use of a Helmholtz-Weyl decomposition adapted to the fluid-structure interaction, facilitating the definition of the generator $A_{\alpha,q}$.
2. $L^2$-Theory: The authors first establish the self-adjointness and positive definiteness of the operator in the Hilbert space setting ($q=2$). This relies on a specific bilinear form involving the strain rate tensor and the friction term on the boundary.
3. Localization and A Priori Estimates: To extend results to general $L^q$ spaces ($1 < q < \infty$), the authors employ a localization technique. They patch together known results for the Stokes operator in the whole space and in bounded domains. Crucially, they derive a generalized Korn-type inequality for the Navier-slip condition with friction, which bounds the full gradient of the velocity by the symmetric gradient and the slip term on the boundary.
4. Sectoriality and Analyticity: The proof of the semigroup being bounded and analytic relies on establishing the sectoriality of the operator. For $q=2$, this follows from self-adjointness. For $1 < q < 3/2$, it is derived from uniform-in-time maximal regularity estimates. The results for the remaining ranges of $q$ are obtained via interpolation and duality arguments.
5. Decay Estimates: The decay rates are derived by combining the established regularity with energy methods. A key step involves closing the energy estimate using the Korn-type inequality and the Gagliardo-Nirenberg inequality, followed by a Gronwall-type argument to determine the time-decay rates for large times.

Key Contributions and Results
The paper establishes the following main results for the linearized fluid-structure system under Navier-slip conditions in an exterior domain:

• Strong $L^q$-Regularity: The authors prove the existence of unique strong solutions to the linearized system with maximal $L^q$-regularity. Specifically, for initial data satisfying appropriate compatibility conditions and external forces in $L^q$, the solution $(v, \pi, \ell, \omega)$ satisfies estimates in spaces such as $L^q(0, T; W^{2,q}(\Omega))$ and $W^{1,q}(0, T; L^q(\Omega))$.
• Bounded Analytic Semigroup: The fluid-structure operator generates a bounded analytic semigroup on the subspace $X_q$ for all $1 < q < \infty$. This is a significant extension, as previous results for slip conditions were largely limited to bounded domains.
• Sharp $L^q-L^r$ Decay Estimates: The paper derives optimal decay rates for the solution as $t \to \infty$. For initial data in $L^r$, the velocity and its derivatives decay with rates depending on the difference between $1/r$ and $1/q$. Specifically, the authors show:
  • $\|v(t)\|_{L^q} \sim t^{-\frac{3}{2}(\frac{1}{r} - \frac{1}{q})}$ for short times and specific large-time regimes.
  • Refined estimates for the translational and angular velocities $(\ell, \omega)$ and their time derivatives, showing decay rates of $(1+t)^{-\frac{1}{2} - \frac{3}{2}(\frac{1}{r} - \frac{1}{m})}$ for large $m$.
  • Estimates for the pressure gradient and second-order spatial derivatives are also provided, distinguishing between short-time ($t \le 1$) and large-time ($t \ge 1$) behaviors.

Significance
The authors state that these results provide a rigorous foundation for constructing strong solutions to the corresponding nonlinear system. By establishing strong $W^{2,1}_q$ regularity and Kato-type solutions for the linearized problem, the work enables the application of fixed-point arguments to the full nonlinear fluid-structure interaction problem. The paper notes that while semigroup regularity and decay are well-understood for the no-slip case in exterior domains, such results have been limited in the slip setting, particularly regarding the unbounded nature of the domain. This work bridges that gap by exploiting the geometric regularity of the sphere to handle the exterior domain setting under Navier-slip conditions.