Weak-Strong Uniqueness for a Rigid Body Immersed in an Inviscid Compressible Fluid

1. Problem Statement

The paper addresses the mathematical analysis of the fluid-structure interaction (FSI) between a free rigid body and an inviscid, compressible, isentropic fluid in a fixed 3D domain $\Omega$.

• Governing Equations:
  • Fluid: The compressible Euler equations (mass and momentum conservation) in the fluid domain $F_t = \Omega \setminus B_t$.
  • Solid: The rigid body $B_t$ moves with translational velocity $V(t)$ and angular velocity $\omega(t)$. Its motion is governed by Newton-Euler equations driven by the fluid pressure forces on the boundary $\partial B_t$.
  • Boundary Conditions: Slip conditions are imposed. The fluid velocity normal to the boundary matches the solid's normal velocity ($(u_F - u_B) \cdot n = 0$), and there is no penetration at the outer domain boundary ($u_F \cdot n = 0$).
• The Challenge:
  • Inviscid Limit: Proving the existence of solutions for the inviscid system directly is difficult due to the lack of compactness in the vanishing viscosity limit ($\epsilon \to 0$). Standard weak solutions to the Euler equations often fail to exist globally or are not unique.
  • Moving Domain: The domain of the fluid changes over time as the rigid body moves. Crucially, the position of the body depends on the fluid velocity, which in the viscous approximation depends on the viscosity parameter $\epsilon$. This makes the space of admissible test functions dependent on $\epsilon$, complicating the passage to the limit.
  • Uniqueness: Establishing uniqueness for weak solutions to the Euler equations is a major open problem. The paper tackles this via the Weak-Strong Uniqueness principle: if a strong (smooth) solution exists, any dissipative measure-valued solution with the same initial data must coincide with it.

2. Methodology

The authors employ a vanishing viscosity limit approach combined with measure-valued solution theory and relative energy analysis.

A. Existence via Vanishing Viscosity Limit

1. Approximation: The authors start with the compressible Navier-Stokes equations coupled with the rigid body, subject to Navier slip boundary conditions (which allow for slip proportional to the shear stress).
2. Convergence: They utilize known existence results for the viscous system (for $\gamma > 3/2$) and pass to the limit as viscosity $\epsilon \to 0$.
3. Young Measures: Due to the lack of compactness, the sequence of viscous solutions does not converge to a standard function but to a Young measure (a probability measure describing the oscillations and concentrations of the sequence).
4. Dissipative Measure-Valued Solutions: The limit is characterized as a "dissipative Young measure-valued solution," which includes a defect measure (energy dissipation) accounting for the loss of energy in the limit.

B. Key Technical Innovation: Test Function Approximation

A central novelty of the paper is the construction of a suitable approximation for test functions in the weak formulation.

• The Problem: In the weak formulation of the viscous system, test functions must satisfy boundary conditions on the moving boundary $\partial B^\epsilon_t$. In the inviscid limit, the boundary is $\partial B_t$. Since $B^\epsilon_t$ depends on $\epsilon$, a fixed test function $\phi$ for the inviscid system is generally not admissible for the viscous system.
• The Solution: The authors construct a sequence of approximating test functions $\phi^\epsilon$ that:
  1. Belong to the admissible space for the viscous problem (satisfying slip conditions on $\partial B^\epsilon_t$).
  2. Converge to the fixed test function $\phi$ of the inviscid problem.
  3. Crucial Regularity: They impose a $C^1$ regularity requirement on the normal component of the test functions across the fluid-solid interface. This is stronger than the $C^0$ requirement in the viscous case and is necessary to control the convergence of nonlinear terms (like $\rho u \otimes u$) near the boundary layer where the test function is modified.

C. Weak-Strong Uniqueness via Relative Energy

To prove uniqueness, the authors use the Relative Energy Method (also known as the relative entropy method):

1. Coordinate Transformation: Since the strong solution and the measure-valued solution may define different moving domains ($B_{2,t}$ vs $B_{1,t}$), the authors introduce a coordinate transformation (flow map) to map the strong solution onto the domain of the measure-valued solution.
2. Relative Energy Functional: They define a functional $E_{rel}(t)$ measuring the distance between the measure-valued solution and the strong solution. This functional includes kinetic energy differences, internal energy differences, and a defect measure term.
3. Grönwall Inequality: By deriving a differential inequality for $E_{rel}(t)$, they show that if the initial data is identical, the relative energy remains zero for all time, provided the strong solution is sufficiently regular. This implies the measure-valued solution coincides with the strong solution.

3. Key Contributions

1. First Result on Compressible Inviscid FSI: This is the first mathematical result establishing the existence and weak-strong uniqueness for a compressible inviscid fluid interacting with a rigid body. Previous results were limited to incompressible flows or viscous fluids.
2. Novel Test Function Construction: The paper resolves a critical technical gap overlooked in previous incompressible studies (e.g., [3]). It rigorously constructs test functions that bridge the gap between the $\epsilon$-dependent viscous domain and the inviscid domain, handling the nonlinearity and boundary layer effects correctly.
3. Dissipative Measure-Valued Framework: The authors successfully formulate the problem within the framework of dissipative Young measure-valued solutions, which is robust enough to handle the lack of compactness inherent in the inviscid limit of compressible flows.
4. Weak-Strong Uniqueness Proof: They prove that if a strong solution exists, the dissipative measure-valued solution is unique and coincides with it. This provides a selection criterion for physically relevant solutions in the absence of global weak uniqueness.

4. Main Results

• Theorem 1.1 (Existence): Under the assumption that the adiabatic exponent $\gamma > 3/2$ (required for the existence of the viscous approximations) and regular initial data, the system admits a local-in-time dissipative Young measure-valued solution.
• Theorem 1.2 (Weak-Strong Uniqueness): If a dissipative measure-valued solution and a strong solution (with sufficient regularity, e.g., $C^1$ for the solid motion and density/velocity gradients) exist with the same initial data, they coincide. Specifically:
  • The rigid body domains are identical ($B_{1,t} = B_{2,t}$).
  • The measure-valued solution reduces to a Dirac mass concentrated on the strong solution values.
  • The energy dissipation defect measure $D(t)$ is zero.
  • Note: The uniqueness result holds for any $\gamma > 1$, even though existence requires $\gamma > 3/2$ due to the approximation method.

5. Significance

• Theoretical Advancement: This work bridges a significant gap in fluid dynamics, extending the theory of fluid-structure interaction to the challenging regime of compressible, inviscid flows.
• Methodological Rigor: The detailed analysis of the test function approximation provides a blueprint for handling moving boundary problems in the vanishing viscosity limit, particularly where the boundary geometry depends on the solution itself.
• Physical Relevance: While ideal fluids (inviscid) are an approximation, understanding their interaction with structures is crucial for high-speed aerodynamics and astrophysical flows. The result validates the use of measure-valued solutions as a mathematically sound concept for these systems when classical solutions break down.
• Foundation for Future Work: By establishing the weak-strong uniqueness property, the paper sets the stage for further investigations into the stability of such systems and the potential development of numerical schemes that converge to these measure-valued solutions.

In summary, the paper provides a rigorous mathematical foundation for modeling compressible inviscid fluid-structure interactions, overcoming the dual challenges of moving boundaries and the lack of compactness in the inviscid limit through innovative test function constructions and relative energy analysis.