Stationary solutions to the spherically symmetric compressible fluid with capillarity effect

Technical Summary: Stationary Solutions to the Spherically Symmetric Compressible Fluid with Capillarity Effect

Problem Statement
This paper investigates the existence, uniqueness, and asymptotic behavior of stationary solutions to the spherically symmetric Navier–Stokes–Korteweg (NSK) system on an exterior domain $\Omega = \{x \in \mathbb{R}^n \mid |x| > 1\}$ with $n \ge 2$. The system models a viscous compressible fluid with capillarity effects, governed by the conservation of mass and momentum equations including a capillarity term $\kappa \rho \nabla \Delta \rho$. The study focuses on three distinct boundary condition scenarios at the inner boundary $r=1$:

1. Impermeable wall: Zero velocity at the boundary ($u_- = 0$).
2. Inflow: Positive velocity entering the domain ($u_- > 0$).
3. Outflow: Negative velocity exiting the domain ($u_- < 0$).

The primary goal is to establish the existence of smooth stationary solutions $(\tilde{\rho}, \tilde{u})$ satisfying specific far-field conditions $(\rho_+, u_+)$ and to analyze the decay rates of these solutions as $r \to \infty$. Additionally, the paper examines the "vanishing capillarity limit" ($\kappa \to 0$) for the impermeable wall problem, comparing the NSK solutions to the standard Navier–Stokes (NS) limit.

Methodology
The analysis proceeds by reducing the partial differential equations (PDEs) to ordinary differential equations (ODEs) under the assumption of spherical symmetry.

• Reduction to ODEs: The system is transformed into equations for density $\rho(r)$ and radial velocity $u(r)$. For the impermeable wall case, the velocity vanishes identically, reducing the problem to a single nonlinear second-order ODE for density. For inflow/outflow cases, the velocity is expressed in terms of density via mass conservation, leading to a coupled integro-differential equation.
• Linearization and Fixed Point Arguments: The existence proofs rely on perturbation theory. The nonlinear equations are linearized around the far-field state $(\rho_+, 0)$. The linearized operators are identified as spherically symmetric Helmholtz-type equations.
• Green's Function Construction: The authors construct Green's functions for the linearized operators using modified Bessel functions of the first and second kinds ($I_\nu$ and $K_\nu$). Explicit pointwise bounds on these Green functions and their derivatives are derived to handle the non-homogeneous terms.
• Banach Space Framework: The existence of solutions is established using the contraction mapping principle (Banach fixed-point theorem).
  • For the impermeable wall, the analysis is conducted in a weighted Banach space $X_\sigma$ with an exponential weight $e^{\sigma r}$ to capture exponential decay.
  • For inflow/outflow, the analysis uses a Banach space $Y$ with algebraic weights ($r^{2(n-1)}$ and $r^{2n-1}$) to accommodate the slower, algebraic decay induced by the source terms.
• Vanishing Capillarity Limit: The asymptotic behavior as $\kappa \to 0$ is analyzed using energy methods (multiplying the equation by the perturbation and integrating). Two cases are considered: fixed boundary data and singularly scaled boundary data ($\rho_b \sim \kappa^{-1/2}$).

Key Contributions and Results

1. Existence and Uniqueness for Impermeable Wall:
   The paper proves that for sufficiently small boundary data $|\rho_b|$, there exists a unique smooth stationary solution for the impermeable wall problem. Unlike the standard NS equations where the stationary solution is trivial (constant density), the NSK system yields a non-trivial density profile due to the capillarity term. The solution decays exponentially to the far-field state $\rho_+$.

2. Existence and Uniqueness for Inflow/Outflow:
   For small boundary data $|\rho_b|$ and $|u_-|$, unique smooth stationary solutions exist for both inflow and outflow problems. The density perturbation and velocity exhibit algebraic decay rates:

   • Density deviation: $O(r^{-2(n-1)})$.
   • Velocity: $O(r^{-(n-1)})$.
     The results hold even for the inviscid limit ($\mu=0$), covering the spherically symmetric Euler–Korteweg system.

3. Vanishing Capillarity Limit:
   The paper establishes the convergence of the stationary solution $\tilde{\rho}_\kappa$ as $\kappa \to 0$:

   • Fixed Boundary Condition: If $\rho_b$ is fixed, the solution converges to the constant far-field state $\rho_+$. The convergence rates are quantified in $L^2_r$, $L^2_r$-derivative, and $L^\infty$ norms as $O(\kappa^{3/4})$, $O(\kappa^{1/4})$, and $O(\kappa^{1/2})$ respectively.
   • Singularly Scaled Boundary Condition: If $\rho_b = \rho_b^0 / \sqrt{\kappa}$, the solution converges to a non-trivial profile $\bar{\rho}(y)$ (where $y = (r-1)/\sqrt{\kappa}$) satisfying a second-order boundary value problem. The convergence rates in the rescaled variable are similarly established.

4. Numerical Validation:
   Numerical experiments are presented to validate the theoretical convergence rates. The results confirm that the derived rates ($O(\kappa^{3/4})$ and $O(\kappa^{1/2})$) are optimal.

Significance and Claims
The author claims this work represents the first step in extending the theory of stationary solutions from the standard Navier–Stokes equations to the compressible fluid with capillarity effects in an exterior domain. The paper highlights that the inclusion of the capillarity coefficient $\kappa$ fundamentally alters the nature of the stationary solution for the impermeable wall problem, rendering it non-trivial even when the velocity is zero.

The significance lies in:

• Filling a gap in the literature regarding spherically symmetric NSK solutions, which had not been previously considered.
• Providing rigorous decay estimates that distinguish between the exponential decay of the impermeable wall case and the algebraic decay of inflow/outflow cases.
• Establishing the optimal convergence rates for the vanishing capillarity limit, demonstrating how the capillarity term regularizes the solution structure near the boundary.

The paper maintains a modest tone, noting that while it extends known NS results to the NSK system, the analysis requires delicate handling of the non-local capillarity terms and the specific geometry of the exterior domain. The results are presented as foundational for understanding the stability and asymptotic behavior of fluids with capillary effects.