Weak-strong uniqueness for bi-fluid compressible system with algebraic closure

1. Problem Formulation

The paper investigates a viscous compressible two-fluid system in a bounded three-dimensional domain $\Omega$. The system consists of two barotropic fluids sharing a common velocity field $u$. The key feature of this model is the algebraic closure for the pressure law, where the partial pressures of the two fluids are equal ($p_+ = p_-$), rather than being governed by a differential transport equation for the volume fraction.

The governing equations are:

1. Continuity equations for the partial masses $R = \alpha \varrho_+$ and $Q = (1-\alpha)\varrho_-$:
   $$\partial_t R + \text{div}(Ru) = 0, \quad \partial_t Q + \text{div}(Qu) = 0$$
2. Momentum equation for the mixture:
   $$\partial_t((R+Q)u) + \text{div}((R+Q)u \otimes u) + \nabla(\alpha p_+(\varrho_+) + (1-\alpha)p_-(\varrho_-)) = \text{div} \mathbb{S}(\nabla u)$$
3. Algebraic closure (Pressure equality):
   $$p_+(\varrho_+) = p_-(\varrho_-)$$
   where $p_\pm(\varrho_\pm) = \varrho_\pm^{\gamma_\pm}$ with $\gamma_\pm > 1$.

Context:

• Global Weak Solutions: Existence was established by Novotný and Pokorný (2020) for large initial data.
• Local Strong Solutions: Existence was established by Piasecki and Zatorska (2022) for large initial data using maximal regularity.
• The Gap: While both solution types exist, it was an open problem whether the weak solution is unique as long as a strong solution exists (the weak-strong uniqueness principle). Previous results for similar systems (e.g., differential closure models) relied heavily on the convexity of the free Helmholtz energy with respect to conservative variables, a constraint this paper aims to remove.

2. Methodology

The authors employ the relative entropy method (also known as the relative energy method), a powerful tool for proving weak-strong uniqueness in fluid dynamics.

Key Technical Steps:

1. Derivation of Evolution Equations:
   Unlike standard transport models where the volume fraction $\alpha$ satisfies a pure transport equation ($\partial_t \alpha + u \cdot \nabla \alpha = 0$), the algebraic closure $p_+ = p_-$ implies a more complex evolution for $\alpha$. The authors derive a transport equation with an additional nonlinear source term:
   $$\partial_t \alpha + u \cdot \nabla \alpha + \omega_\alpha \text{div} u = 0$$
   where $\omega_\alpha$ is a bounded coefficient depending on $\alpha$ and the adiabatic exponents. This nonlinear term is the primary source of difficulty, as it generates "troublesome" terms in the relative entropy inequality that do not cancel out naturally.

2. Relative Entropy Functional:
   They define a relative energy functional $\mathcal{E}$ measuring the distance between the weak solution $(\alpha, \varrho_+, \varrho_-, u)$ and a strong solution $(\beta, \tilde{\varrho}_+, \tilde{\varrho}_-, v)$:
   $$\mathcal{E} = \int_\Omega \left[ \frac{1}{2}(R+Q)|u-v|^2 + \text{Internal Energy Terms} + \frac{1}{2}(\alpha-\beta)^2 \right] dx$$
   The inclusion of the $(\alpha-\beta)^2$ term is crucial for controlling the volume fraction difference.

3. Handling the Nonlinear Terms:
   The derivation of the relative entropy inequality produces remainder terms involving the difference in volume fractions and the nonlinear source terms.

   • The authors exploit the specific implicit structure of the pressure law to derive a stability estimate for the volume fractions $\alpha$ and $\beta$.
   • They introduce an auxiliary lemma (Lemma 5.2) to control the $L^2$ difference of the volume fractions, utilizing the boundedness of the coefficients and the regularity of the strong solution.

4. Gronwall Argument:
   By carefully estimating the remainder terms (denoted $J_1$ through $J_8$ in the paper) and coupling the relative energy inequality with the volume fraction stability estimate, they close the estimate using a Gronwall inequality. This shows that if the initial relative energy is zero, it remains zero for the lifespan of the strong solution.

3. Key Contributions

• Proof of Weak-Strong Uniqueness: The paper establishes that any finite-energy weak solution (in the sense of Novotný–Pokorný) coincides with the local strong solution (in the sense of Piasecki–Zatorska) as long as the strong solution exists, provided they share the same initial data.
• Removal of Convexity Constraints: Unlike previous works on differential closure models (e.g., Jin and Novotný [20]), this result does not require a positive lower bound for the volume fraction $\alpha$ away from 0 or 1, nor does it rely on strict convexity assumptions of the free energy with respect to the conservative variables.
• Handling Algebraic Closure: The paper successfully addresses the specific mathematical challenges posed by the algebraic pressure closure ($p_+ = p_-$), specifically the non-cancellation of terms arising from the nonlinear evolution of the volume fraction.
• Structural Insight: The authors demonstrate how the implicit relation between densities and volume fraction can be leveraged to derive stability estimates compatible with the relative energy framework.

4. Main Results

Theorem 2.1 (Weak-Strong Uniqueness):
Let $(\alpha, \varrho_-, \varrho_+, u)$ be a global weak solution and $(\beta, \tilde{\varrho}_-, \tilde{\varrho}_+, v)$ be a strong solution emanating from the same initial data. Then, on the time interval $(0, T)$ where the strong solution exists:
$$(\alpha, \varrho_-, \varrho_+, u)(t, x) = (\beta, \tilde{\varrho}_-, \tilde{\varrho}_+, v)(t, x) \quad \text{a.e. in } (0, T) \times \Omega.$$

Theorem 5.1 (Stability Estimate):
The relative entropy satisfies an inequality of the form:
$$\mathcal{E}(\tau) + \int_0^\tau \int_\Omega \mathbb{S}(\nabla u - \nabla v) : \nabla(u-v) \, dx dt \leq C \mathcal{E}(0)$$
where the constant $C$ depends on the strong solution but not on the regularity of the weak solution.

5. Significance

• Mathematical Rigor: This work resolves a significant open problem in the theory of multi-fluid compressible flows, providing a rigorous justification for the stability of weak solutions in the presence of algebraic closures.
• Singular Limits: The relative entropy inequality derived here provides a robust mechanism for studying singular limits (e.g., low Mach number limits, incompressible limits). It allows for the rigorous justification of formal asymptotic expansions and numerical schemes for these complex systems.
• Generalizability: The techniques developed, particularly the handling of the nonlinear volume fraction transport, offer a pathway to extend weak-strong uniqueness results to other multi-component systems with complex constitutive relations, including those with differential closures or magnetohydrodynamics (MHD) extensions.
• Physical Relevance: By removing the constraint that $\alpha$ must be bounded away from 0 and 1, the results apply to a broader range of physical scenarios, including cases where one fluid phase may vanish or dominate locally.

In summary, this paper bridges the gap between the existence of global weak solutions and local strong solutions for a complex two-fluid model, proving their uniqueness through a sophisticated application of the relative entropy method tailored to the algebraic constraints of the system.