Weak-strong uniqueness and low Mach number limit for a viscous compressible fluid around a rotating body

Technical Summary: Weak-Strong Uniqueness and Low Mach Number Limit for a Viscous Compressible Fluid Around a Rotating Body

Problem Formulation
The paper investigates the mathematical behavior of an isothermal compressible Newtonian fluid flowing around a rigid body $S$ that undergoes a time-independent rigid motion (rotation with fixed angular velocity $\omega$) in a three-dimensional unbounded domain $\Omega = \mathbb{R}^3 \setminus S$. The fluid dynamics are governed by the compressible Navier–Stokes equations in a frame attached to the rotating body. The system includes mass conservation, momentum balance with Coriolis and centrifugal forces, no-slip boundary conditions on the body, and specific asymptotic conditions at spatial infinity where the density approaches a constant $\rho_\infty$ and the velocity approaches a constant $a_\infty$ (parallel to $\omega$).

The study focuses on two primary analytical challenges:

1. Weak-Strong Uniqueness: Establishing that a finite-energy weak solution to the compressible system coincides with a strong solution (if one exists) sharing the same initial data.
2. Low Mach Number Limit: Proving that as the Mach number tends to zero (formally represented by a scaling parameter $\varepsilon \to 0$), the solutions to the compressible system converge to the solution of the incompressible Navier–Stokes equations for a rotating flow.

Methodology
The central tool employed throughout the paper is the Relative Energy Inequality (REI). The authors derive a relative energy functional, $E(\rho, u|\sigma, V)$, which measures the "distance" between a weak solution $(\rho, u)$ and a sufficiently smooth comparison function $(\sigma, V)$. This functional combines the kinetic energy difference and the relative internal energy (based on the convexity of the pressure law).

The methodology proceeds as follows:

• Derivation of REI: The authors first establish a relative energy inequality for finite-energy weak solutions to the exterior domain problem with a rotating body. This involves testing the weak formulation of the momentum and continuity equations with the comparison functions and carefully handling the unbounded domain and the rotational terms.
• Approximation Arguments: To handle the unbounded domain and the specific boundary conditions at infinity, the authors utilize cut-off functions and the Bogovski\u{i} operator to construct divergence-free test functions that match the asymptotic behavior of the flow.
• Weak-Strong Uniqueness: By setting the comparison function $(\sigma, V)$ to be a strong solution, the REI is used to bound the relative energy. Using Korn-type inequalities and the coercivity of the relative energy, the authors show that the relative energy must vanish, implying the weak and strong solutions are identical.
• Low Mach Number Limit: The authors introduce a rescaled pressure term ($\varepsilon^{-2}$) to model the low Mach number regime. They apply the scaled REI with the incompressible solution as the comparison function. By deriving uniform bounds on the relative energy and utilizing compactness arguments (specifically the convergence of the "essential" and "residual" parts of the density), they demonstrate the convergence of the compressible solutions to the incompressible limit.

Key Contributions and Results

1. Weak-Strong Uniqueness Principle:
   The paper proves that for the compressible Navier–Stokes equations around a rotating body in an exterior domain, any finite-energy weak solution coincides with a strong solution emanating from the same initial data, provided the strong solution exists. This result holds under the assumption that the adiabatic exponent satisfies $\gamma > 6/5$, which covers a broader range of physically relevant gases (including monoatomic gases where $\gamma = 5/3$) than previous existence results for weak solutions (which often require $\gamma > 3/2$).

2. Low Mach Number Limit:
   The authors establish the convergence of solutions to the compressible system (with rescaled pressure) to the solution of the incompressible Navier–Stokes equations for a rotating flow as the Mach number parameter $\varepsilon \to 0$.

   • Qualitative Convergence: They prove that if the initial data are "well-prepared" (i.e., the initial relative energy with respect to the incompressible limit vanishes as $\varepsilon \to 0$), then the relative energy of the solution remains small uniformly in time.
   • Quantitative Estimates: Under additional regularity assumptions on the pressure of the incompressible limit, the paper provides explicit convergence rates. The error in the relative energy is shown to be bounded by terms of order $O(\varepsilon + \varepsilon^{2/\gamma})$.

3. Handling of Rotating Exterior Domains:
   The work extends the theory of relative energy inequalities to the specific case of an exterior domain with a rotating boundary. This requires careful treatment of the Coriolis force terms and the construction of appropriate test functions that respect the rotation and the behavior at infinity.

Significance and Scope
The paper claims significance in providing a rigorous mathematical framework for analyzing compressible flows around rotating bodies, a scenario relevant to various engineering applications but mathematically challenging due to the unbounded domain and the coupling of rotation with compressibility.

• Uniqueness: It resolves the uniqueness issue in the "weak-strong" sense for this specific configuration, a property that remains an open question for general weak solutions in the absence of a strong solution.
• Limit Passage: It justifies the use of incompressible models for low-speed rotating flows by proving the convergence of the compressible model, including quantitative error estimates.
• Generality: The results are derived for a general pressure law satisfying standard physical conditions and for a range of adiabatic exponents $\gamma > 6/5$, making the results applicable to a wide class of physical fluids.

The authors explicitly note that their analysis relies on "well-prepared" initial data to avoid the generation of acoustic waves, which distinguishes their approach from studies involving general initial data or additional limits (such as low Froude number limits). The work builds upon and generalizes existing results for bounded domains and stationary bodies, adapting the relative energy method to the complexities of rotating exterior domains.