Large time decay of the Oseen flow in exterior domains subject to the Navier slip-with-friction boundary condition

Technical Summary: Large Time Decay of the Oseen Flow in Exterior Domains Subject to the Navier Slip-with-Friction Boundary Condition

Problem Statement
The paper investigates the asymptotic behavior of the Oseen flow, which serves as the linearization of the Navier-Stokes equations, in a three-dimensional exterior domain $\Omega$ (the complement of a bounded obstacle). The fluid is viscous and incompressible. The study focuses on the system subject to the Navier slip-with-friction boundary condition on the obstacle's boundary $\partial\Omega$ and a prescribed outflow velocity $u_\infty$ at spatial infinity.

The governing equations for the perturbation velocity $u$ and pressure $p$ around the steady state $u_\infty$ are given by the Oseen initial value problem:
$$\partial_t u = \Delta u + \eta \cdot \nabla u - \nabla p, \quad \text{div } u = 0 \quad \text{in } \Omega \times (0, \infty),$$
where $\eta = -u_\infty$. The boundary conditions are:
$$\nu \cdot u|_{\partial\Omega} = 0, \quad [(2Du)\nu]_\tau + \alpha u_\tau|_{\partial\Omega} = 0, \quad \lim_{|x|\to\infty} u = 0,$$
where $\nu$ is the outward unit normal, $Du$ is the symmetric gradient, $\alpha(x) \geq 0$ is the friction coefficient, and the subscript $\tau$ denotes the tangential component.

The primary objectives are twofold:

1. To characterize the resolvent set of the associated Oseen operator $L$ in the $L^q$-framework, specifically analyzing the spectral properties near the origin in the complex plane under the influence of the boundary geometry, the friction coefficient $\alpha$, and the outflow $\eta$.
2. To establish $L^q$-$L^r$ decay estimates for the Oseen semigroup $e^{-tL}$ as $t \to \infty$, which are crucial for proving the global well-posedness of the nonlinear Navier-Stokes system near a steady state.

Methodology
The analysis employs a combination of spectral theory, functional analysis, and energy methods, structured as follows:

1. Resolvent Analysis and Uniqueness: The paper first establishes the uniqueness of solutions to the resolvent system $(\lambda + L)u = f$ within specific classes of functions. This involves deriving energy estimates by multiplying the equation by the solution and integrating over the domain. A critical step is handling the boundary terms arising from the Navier condition. The authors utilize the Weingarten map of the boundary $\partial\Omega$ to relate the deformation tensor $Du$ to the gradient $\nabla u$. Specifically, they use the identity:
   $$2\|Du\|_2^2 = \|\nabla u\|_2^2 - \int_{\partial\Omega} [(\nabla N)u] \cdot u \, d\sigma,$$
   where $-\nabla N$ is the Weingarten map and $\kappa(x)$ represents its minimum eigenvalue (related to principal curvatures). This geometric term allows for the derivation of conditions under which the real part of the energy identity is non-negative, ensuring uniqueness.

2. Parametrix Construction: To construct the resolvent in the exterior domain, the authors use a cut-off procedure. They combine the solution operator for the whole space $\mathbb{R}^3$ (Oseen resolvent) with the solution operator for a bounded interior domain (subject to the same boundary condition). A Bogovskii operator is employed in an annular region to correct the divergence of the patched solution, ensuring the solenoidal condition is met. This yields a representation of the resolvent involving a compact perturbation operator $T_\eta(\lambda)$.

3. Regularity near the Origin: The core of the decay analysis relies on the behavior of the resolvent $(\lambda + L)^{-1}$ near $\lambda = 0$, particularly along the imaginary axis. The authors prove that under specific conditions on $\alpha$ and the curvature of $\partial\Omega$, the resolvent is analytic in a neighborhood of the origin (excluding the origin itself for the Stokes case) and satisfies specific Hölder continuity estimates in the $L^q$-norm.

4. Local Energy Decay: Using the regularity of the resolvent along the imaginary axis, the paper derives local energy decay estimates. This is achieved by expressing the semigroup via the Dunford integral and applying a lemma relating the decay rate of the inverse Fourier transform to the regularity of the function in the frequency domain (specifically, the modulus of continuity of the resolvent).

Key Contributions and Results

• Resolvent Set Characterization:
  The paper establishes that the resolvent set $\rho(-L)$ contains the right half-plane (excluding the origin) under the condition:
  $$\alpha(x) \geq \frac{\eta \cdot \nu(x)}{2} \quad \text{for all } x \in \partial\Omega.$$
  Furthermore, if a stronger condition involving the boundary curvature is met:
  $$\alpha(x) + \kappa(x) \geq \frac{\eta \cdot \nu(x)}{2},$$
  where $\kappa(x)$ is the minimum of the principal curvatures, the resolvent set contains the region $\mathbb{C} \setminus S_\eta$, where $S_\eta = \{ \lambda \in \mathbb{C} : |\eta|^2 \text{Re } \lambda + (\text{Im } \lambda)^2 \leq 0 \}$. This extends previous results known for the no-slip condition to the Navier slip-with-friction case.

• $L^q$-$L^r$ Decay Estimates:
  The main result (Theorem 2.2) provides decay estimates for the Oseen semigroup. For initial data $f \in L^q_\sigma(\Omega)$, the solution satisfies:
  $$\|\nabla^j e^{-tL}f\|_r \leq C t^{-\frac{j}{2} - \frac{3}{2}(\frac{1}{q} - \frac{1}{r})} \|f\|_q$$
  for $j=0$, and a slightly modified rate involving $(1+t)$ for $j=1$. These estimates hold for $t \to \infty$ under the condition $\alpha(x) \geq \frac{\eta \cdot \nu(x)}{2}$.

• Recovery of Known Results:
  The results recover the known decay estimates for the Stokes semigroup ($\eta=0$) under full slip ($\alpha=0$) and the Oseen semigroup under the no-slip condition ($\alpha \to \infty$) as particular cases.

Significance and Claims
The paper claims to fill a gap in the existing literature regarding the exterior problem with non-zero outflow ($u_\infty \neq 0$) under Navier boundary conditions, as previous works largely assumed zero outflow for such boundary conditions.

The significance lies in:

1. Bridging the Gap: The results interpolate between the full slip ($\alpha=0$) and no-slip ($\alpha \to \infty$) regimes, providing a unified framework for large-time decay estimates.
2. Geometric Insight: The analysis highlights the role of the boundary geometry (via the Weingarten map and principal curvatures) in determining the spectral properties of the Oseen operator, specifically the shape of the resolvent set near the origin.
3. Foundation for Nonlinear Stability: The established $L^q$-$L^r$ decay estimates are presented as a necessary tool for investigating the global well-posedness and stability of the nonlinear Navier-Stokes system in exterior domains with slip boundary conditions, particularly for initial data close to a steady state.

The authors note that while the decay rates for the gradient ($j=1$) are optimal for the Stokes case under no-slip, the optimality for the Oseen case with non-zero outflow is not claimed due to the different spatial decay structure (wake structure) of the steady Oseen flow. Additionally, the dependence of the decay constant on the friction coefficient $\alpha$ remains an open issue due to a lack of uniform a priori estimates in bounded domains for small $\alpha$.