Diffeological Tangent Spaces and Distributional Linearization for Lifted Euler--Reynolds Limits

Technical Summary: Diffeological Tangent Spaces and Distributional Linearization for Lifted Euler–Reynolds Limits

Problem Statement
The incompressible Euler equations occupy a central position in fluid mechanics and the modern theory of convex integration. A key insight in this field is that exact solutions are effectively studied through the larger class of Euler–Reynolds subsolutions, triples $(v, p, R)$ satisfying $\partial_t v + \text{div}(v \otimes v) + \nabla p = -\text{div} R$ and $\text{div} v = 0$, where $R$ is the Reynolds stress tensor representing the defect. Convex integration schemes generate sequences of such subsolutions where the Reynolds stress converges to zero while velocity fields develop fine oscillations. However, the nonlinear fluxes do not converge to the flux of the limit ($v_\alpha \otimes v_\alpha \not\to v \otimes v$), leading to measure-valued descriptions of weak limits.

A fundamental geometric challenge arises in these spaces: unlike classical configuration spaces, the space of Euler–Reynolds subsolutions does not form a smooth manifold. The defining constraints are distributional, and weak limits often violate the nonlinear constitutive relation $Q = v \otimes v$. Consequently, there is no canonical notion of a tangent vector, making the basic question of what it means to infinitesimally perturb an Euler–Reynolds state non-trivial. Existing frameworks (Young measures, convex integration) focus on existence and relaxation but lack a unified first-order geometric theory for the spaces in which these limits reside.

Methodology
The authors address this by developing a diffeological framework, a generalization of smooth manifolds that replaces coordinate charts with families of compatible smooth parametrizations (plots). This approach allows for the definition of intrinsic internal tangent spaces without requiring an underlying manifold structure, making it suitable for infinite-dimensional spaces defined by weak analytical constraints.

The methodology proceeds in five stages:

1. Lifted Formulation: The authors separate linear and nonlinear structures by treating the velocity $v$, the quadratic flux $Q$, and the Reynolds stress $R$ as independent variables in an ambient space $E$. The nonlinear constraint $Q = v \otimes v$ is recovered only as a specific locus (the "Euler locus") within this lifted space.
2. Diffeological Construction: They define the lifted limit space $X(v_0, A)$ as the weak closure of smooth strict subsolutions within the ambient convenient vector space $E$. This space is equipped with a subspace diffeology.
3. Internal Tangent Spaces: Using the theory of internal tangent spaces for diffeological spaces, they construct tangent vectors intrinsically. They establish an "ambient realization" map that associates these intrinsic vectors with concrete distributional perturbations in $E$.
4. Distributional Linearization: They prove that these ambient realizations satisfy the linearized Euler–Reynolds equations in the sense of distributions.
5. Finite-Mixture Models: To illustrate the abstract theory, they introduce a finite-mixture model of lifted states, analyzing how differentials of mixture maps realize specific tangent directions and relate Reynolds stress to infinitesimal phase splitting.

Key Contributions and Results

• Intrinsic First-Order Geometry: The paper endows lifted Euler–Reynolds limit spaces with an intrinsic first-order geometry. It proves that every internal tangent vector, when realized in the ambient space, satisfies the linearized Euler–Reynolds equations distributionally (Theorem 3.7, Propositions 4.11–4.12). This establishes a natural distributional linearization for weak lifted limit spaces.
• Tangent Directions to the Euler Locus: The authors distinguish between tangent directions that preserve the exact Euler state (where $Q = v \otimes v$ and $R=0$) and those that generate Reynolds stress. They show that for smooth subsolutions, the ambient realization of the internal tangent space coincides with the full solution space of the linearized Euler–Reynolds system (Proposition 4.5).
• Observable Hierarchy and Kernels: A significant contribution is the characterization of "observable" versus "hidden" infinitesimal directions. The authors define kernels for velocity, flux, and full-state observables:
  • Velocity Kernel: Tangents invisible to velocity measurements ($w=0$).
  • Flux Kernel: Tangents invisible to both velocity and flux measurements ($w=0, H=0$).
  • Stress-Gauge Directions: Tangents in the flux kernel that are non-zero only in the stress component ($S \neq 0$). These represent genuine first-order deformations of the Reynolds stress that are undetectable by velocity or flux probes. The authors show these directions satisfy $\text{div} S = \nabla \pi$ (Theorem 4.20) and construct explicit smooth representatives (Proposition 4.21).
• Finite-Mixture Realization: The paper develops a finite-dimensional model where Reynolds stress appears as a covariance defect generated by infinitesimal phase splitting. Under genericity assumptions, every deviatoric stress tensor is realized by such a first-order mixture defect (Theorem 5.13). This yields phase-counting bounds and a minimality result for the observable hierarchy (Theorem 5.15), determining the minimal order of moments required to separate weight directions in a mixture.
• Energy Constraints: The theory is extended to energy-constrained strata, identifying tangent directions compatible with both linearized dynamics and prescribed kinetic energy profiles (Theorem 3.9).

Significance and Claims
The paper claims to provide a unified framework for distributional linearization, observable tangent structures, and finite-dimensional mixture realizations of Euler–Reynolds limits. By shifting the emphasis from the analysis of weak limits to the differential geometry of the spaces in which they reside, the work offers a rigorous geometric interpretation of which infinitesimal variations are physically visible and which remain concealed within the lifted formulation.

The authors emphasize that their approach is compatible with classical measure-valued descriptions (Young measures) but provides a distinct geometric perspective. The results distinguish observable perturbations from hidden stress-gauge directions, clarifying the geometric flexibility inherent in the Euler–Reynolds system that is absent in the exact Euler equations. The finite-mixture model serves as a concrete realization of these abstract tangent directions, linking the algebraic structure of phase splitting to the generation of Reynolds stress.

The paper does not propose new convex integration schemes or experimental applications but rather establishes the foundational geometric language necessary to study the infinitesimal structure of weak solutions and their limits.