Universality in the Low Mach number limit via a convex integration framework

Technical Summary: Universality in the Low Mach Number Limit via a Convex Integration Framework

Problem Statement
The paper investigates the low Mach number limit of the compressible Euler equations, a singular limit where the Mach number $\delta$ tends to zero. Physically, this regime describes fluid flows where velocities are small compared to the speed of sound, leading to negligible density variations and incompressible behavior. Mathematically, the goal is to justify the transition from the compressible Euler system (1.1) to the incompressible Euler system (1.2).

While previous works (e.g., Ebin; Klainerman–Majda) established this limit for smooth, well-prepared initial data, and later extensions addressed ill-prepared data and weak solutions using compactness or relative entropy methods, these approaches typically filter out highly oscillatory or turbulent behaviors. A central question remains: Is the incompressible system a "universal attractor" in the context of weak solutions? Specifically, can every incompressible weak solution be realized as the strong limit of a sequence of weak solutions to the compressible system, even when the compressible solutions exhibit the non-uniqueness and turbulence permitted by convex integration?

Methodology
The authors employ a refined convex integration framework, building upon their prior work [5] which established non-uniqueness for the incompressible and compressible isentropic Euler systems. The strategy proceeds in three conceptual stages:

1. Subsolution Construction: Starting from an arbitrary weak solution $u$ of the incompressible Euler equations, the authors construct a smooth incompressible subsolution $(u_\varepsilon, \pi_\varepsilon, R_\varepsilon)$ to a relaxed system, where $R_\varepsilon$ is a positive definite Reynolds stress tensor. This is then lifted to a corresponding compressible subsolution $(\rho_\delta, V_\delta, R_\delta)$ for the rescaled compressible equations (1.3). This lifting ensures that the density $\rho_\delta$ remains close to 1 and the Reynolds stress remains positive definite.

2. Refined Convex Integration: The core innovation lies in modifying the standard convex integration scheme. The authors introduce a new $L^2$-type constraint on the set of admissible perturbations. Specifically, the perturbations $\tilde{V}$ must satisfy:
   $$\left| \int V_\delta \cdot \tilde{V} \, \frac{dx dt}{\rho_\delta} \right| \leq \frac{1}{8} \int \frac{|\tilde{V}|^2}{\rho_\delta} \, dx dt$$
   This constraint is absent in previous $L^\infty$ convex integration theories (which rely on the Baire category theorem) but is naturally aligned with $C^\alpha$ theories.

3. Uniform Estimates and Convergence: By imposing this constraint, the authors derive a uniform energy estimate (Proposition 3.1) showing that the kinetic energy of the perturbations is controlled by the trace of the Reynolds stress tensor:
   $$\int \frac{|\tilde{V}|^2}{\rho_\delta} \, dx dt \leq C \int \text{tr}(R_\delta) \, dx dt$$
   Since the construction of the subsolution ensures $\text{tr}(R_\delta) = O(\delta)$, this estimate tames the turbulent oscillations uniformly with respect to the Mach number $\delta$. This control is crucial for proving strong convergence as $\delta \to 0$.

Key Contributions

• Universal Attractor Result: The paper proves that the incompressible system acts as a universal attractor for the compressible system within the convex integration framework. For any given $L^2$ weak solution of the incompressible Euler equations, there exists a family of initial data for the compressible system such that the resulting weak solutions converge strongly to the given incompressible solution as the Mach number vanishes.
• New Constraint for $L^\infty$ Theory: The introduction of the $L^2$-type constraint (1.4) into the non-constructive $L^\infty$ convex integration scheme is a significant technical advancement. It bridges the gap between the control of turbulent perturbations by the Reynolds tensor (previously seen in $C^\alpha$ contexts) and the Baire category approach used for $L^\infty$ solutions.
• Generality in Adiabatic Exponent: Unlike previous convex integration results for compressible flows that required constraints on the adiabatic exponent $\gamma$ (e.g., $1 < \gamma \leq 1 + 2/n$) to satisfy energy inequalities, this result holds for the full range $\gamma > 1$ because the energy inequality is not imposed as an admissibility criterion.

Main Results
The main theorem (Theorem 1.1) states that for dimensions $n=2$ or $3$ and any $\gamma > 1$, given any weak solution $u$ to the incompressible Euler equations with initial data $u_0 \in L^2$, there exists a family of initial data $(\rho_\delta^0, m_\delta^0)$ for the rescaled compressible Euler equations. For any Mach number $\delta$ sufficiently small, there exist infinitely many weak solutions to the compressible system emanating from this data. As $\delta \to 0$:

• The density $\rho_\delta$ converges to 1 in $C^0$.
• The momentum $\rho_\delta v_\delta$ converges strongly to $u$ in $L^2(0, T; L^2)$.
• The initial momentum $m_\delta^0$ converges weakly to $u_0$ in $L^2$.

Significance
The paper claims to provide a rigorous mathematical justification for the robustness of the incompressible approximation. It demonstrates that any large-scale, divergence-free flow field can emerge as the effective, averaged description of a compressible fluid in the low Mach number regime, regardless of the fine-scale, non-unique oscillatory dynamics modeled by convex integration. This establishes a new form of universality for singular limits, shifting the analytical paradigm from investigating whether a specific family of compressible solutions converges to constructively proving that any incompressible weak solution can be targeted and approximated by compressible weak solutions.