Incompressible Euler in Generalized Function Spaces

• The topic presents the use of convex integration to construct weak, oscillatory solutions for incompressible Euler equations within various generalized function spaces.
• It examines the impact of function space properties on well-posedness, energy conservation, and unique phenomena such as nonuniqueness and rigidity across dimensions.
• Insights include the role of geometric and Lagrangian frameworks in bridging analytical results with applications in turbulence and vortex dynamics.

The incompressible Euler equations in generalized function spaces involve the paper of weak, rough, and/or highly oscillatory solutions to the Euler system using a diversity of function space frameworks. These include negative Sobolev spaces, Besov and Triebel–Lizorkin spaces, generalized Campanato and Morrey spaces, spaces with slowly varying regularity weights, and others. The interplay between these function spaces and the structure of the Euler equations reveals a rich array of phenomena—such as nonuniqueness, nonuniform dependence, ill-posedness, or energy conservation breakdown—that are not accessible in the classical smooth or finite-energy frameworks.

1. Convex Integration, Subsolutions, and the h-Principle

The h-principle for the incompressible Euler equations frames weak solution theory as a density statement for subsolutions in generalized function spaces. In this framework, a subsolution is a triple $(v, \mathcal{R}, p)$ solving the Euler–Reynolds system: $$\partial_t v + \operatorname{div}(v \otimes v) + \nabla p = \operatorname{div} \mathcal{R}, \quad \operatorname{div} v = 0,$$ where $\mathcal{R}$ is the Reynolds stress, a symmetric (often trace-free) defect tensor. There exists a sharp energy defect inequality,

$$\frac{e(t) - \langle|v|^2\rangle}{d} \operatorname{Id} - \mathcal{R} > 0,$$

which determines the set of "strict" subsolutions.

The main h-principle asserts that every strict subsolution can be approximated arbitrarily well in the $H^{-1}$-norm by an exact (weak) solution of the incompressible Euler equations. This is achieved by an iterative convex integration scheme: at each step, an oscillatory perturbation is constructed (e.g., using Beltrami flows in 3D or stream functions in 2D) so that the Reynolds stress decreases and the local kinetic energy matches the prescribed profile more closely. The iteration exploits the flexibility of generalized function spaces (in particular, negative Sobolev spaces) to "hide" fast oscillations, enabling the construction of dissipative and wildly oscillatory Hölder continuous solutions for any prescribed, smooth energy profile. The central formulas are: $$v_{1} = v + w, \quad w \approx \sqrt{\rho}\,\sum_k \gamma_k(R_\ell/\rho)\phi_k(v_\ell, \lambda t)b_k(\lambda x),$$ with the building blocks $b_k$ being Beltrami or divergence-free oscillatory modes. The construction ensures the mean $\langle w \otimes w \rangle$ matches the defect Reynolds stress.

2. Stationary Problems, Rigidity, and Dimensional Dependence

While the h-principle applies robustly to both time-dependent and stationary Euler equations, dimensional effects introduce qualitative differences. For stationary solutions, the key tool is the reformulation as a differential inclusion, imposing the pointwise constraint $u = v \otimes v - (r/d)\operatorname{Id}$ for some prescribed energy $r(x)$. Convex integration allows filling the relaxed set (the convex hull of the constraint), but in $d=2$ dimensions, the lamination wave cone is strictly smaller; certain oscillatory directions are forbidden.

This phenomenon implies that the relaxation achievable in generalized function spaces (such as $L^\infty$ or $H^{-1}$) for stationary solutions is more restricted in 2D than in 3D. The lamination–convex hull in 2D does not fill the full convex hull, resulting in a rigidity phenomenon absent in higher dimensions. This has consequences for nonuniqueness and the admissibility criteria for weak solutions in generalized function spaces and may be linked to the observed rigidity of two-dimensional turbulence (Choffrut et al., 2014).

3. Weak Solution Frameworks and Energy Conservation

The paper of energy conservation and anomalous dissipation in generalized function spaces is deeply connected to Onsager’s conjecture. In these settings, weak solutions are often defined using test functions that are divergence-free and adapted to the boundary geometry (e.g., zero normal component), so that the pressure need not appear explicitly: $$(u(t), v(t))_D - (u(0), v(0))_D - \int_0^t (u, \partial_t v)_D \,d\tau - \int_0^t ((u \otimes u): \nabla v)_D\, d\tau = 0.$$ Energy conservation can be established for solutions in $L^3((0, T); L^3)$ satisfying certain bulk and boundary continuity conditions. More generally, sufficient regularity for energy conservation is expressed in precise function spaces, e.g., $C^{0, \alpha}_\lambda$ (Hölog spaces), Besov $B^\alpha_{3,\infty}$, or corresponding local Morrey conditions. The sharp threshold for energy conservation may thus be characterized in various generalized spaces: $u \in L^3((0,T); C^{0, \alpha}_\lambda(\overline{\Omega}))\ \mbox{with}\ \alpha \geq \tfrac{1}{3},\, \lambda>0$ ensures the $L^2$-energy is conserved for weak solutions (Veiga et al., 2019).

4. Well-posedness, Ill-posedness, and Nonuniform Dependence

The behavior of the Cauchy problem for the Euler equations in generalized function spaces varies dramatically with the choice of regularity scale. For spaces slightly "above" the critical scaling (e.g., Besov $B^s_{p,q}$ or Triebel–Lizorkin $F^{s,\psi}_{p,q}$ with $s = d/p + 1$ and a suitable slowly varying $\psi$), local-in-time well-posedness holds. The condition that $\psi$ grows fast enough (subject to certain average integrability) ensures Lipschitz control and permits fixed-point arguments, with precise a priori estimates: $$\|u(t)\|_{X^{s,\psi}} \leq \|u_0\|_{X^{s,\psi}}\exp\left(C\int_0^t \|\nabla u(\tau)\|_{L^\infty}\,d\tau\right),$$ where $X$ is either Besov or Triebel–Lizorkin (Harrison et al., 3 Oct 2025).

In sharp contrast, for the borderline critical spaces (e.g., $B^1_{\infty,1}$, $C^1$, $F^{d+1}_{1,\infty}$), solutions may be ill-posed:

• In $C^1$, small perturbations in the initial data can lead to norm inflation in arbitrarily short time; continuous dependence fails (Misiołek et al., 2014).
• In $B^s_{p,q}$ and $F^s_{1,\infty}$, the data-to-solution map is nowhere uniformly continuous; norm inflation and temporal discontinuity are exhibited explicitly via frequency-localized initial data constructions (Pastrana, 2019, Pak, 2023).
• In $H^{d/2+1}$ (critical Sobolev) and critical Lorentz spaces, instantaneous blow-up may occur due to vortex stretching, even in arbitrarily small time (Jeong et al., 2021).

Extensions to generalized Campanato spaces, Morrey spaces, and slow-scale-modulated Besov/Triebel–Lizorkin spaces further delineate the threshold for well-posedness and continuous dependence (Chae et al., 2019, Cobb et al., 7 Oct 2024).

5. Role of Geometric and Lagrangian Structures

Geometric and Lagrangian frameworks play a crucial role in both well-posedness theory and the analysis of regularity propagation in generalized spaces:

• Geodesic formulations relate the Euler flow to geodesics on the infinite-dimensional manifold $\mathcal{D}^s_\mu(\mathbb{R}^n)$, the group of volume-preserving diffeomorphisms equipped with Sobolev topology. The evolution equation is governed by analytically well-behaved Christoffel maps, guaranteeing local well-posedness as long as $s > n/2 + 1$ (Inci, 2013, Inci, 2013).
• Eulerian–Lagrangian and back-to-labels reformulations provide alternate perspectives for weak solution theory and allow comparison between Eulerian and Lagrangian regularity propagation. Indeed, analytic or Gevrey regularity in Lagrangian variables persists over short times, while Eulerian regularity may degrade instantaneously (Constantin et al., 2015).

The use of these geometric structures clarifies the influence of function space properties on well-posedness, the loss of regularity, and the nature of the weak solution space.

6. Applications to Turbulence, Vortex Theory, and Dissipation

The emergent phenomena in generalized function spaces directly connect to central themes in turbulence and vortex dynamics:

• Wild (nonunique, possibly dissipative) weak solutions, constructed by convex integration, reflect the microscopic mixing and anomalous dissipation mechanisms postulated in turbulence models, as originally conjectured by Onsager.
• The construction of concentrated helical vortices and their singular limits as vortex filaments evolving by binormal curvature flow illustrates the flexibility of generalized function spaces in modeling singular structures in three-dimensional flows (Qin et al., 14 Dec 2024).
• Port-Hamiltonian and energy-based formulations in Sobolev or differential form settings allow for the rigorous handling of free-surface problems and interconnection with control and energy transfer theories (Cheng et al., 2023).
• The paper of unbounded Yudovich-type solutions in Morrey spaces captures physically relevant, nondecaying flows that evade the constraints of finite energy and ensures Galilean invariance (Cobb et al., 7 Oct 2024).

7. Summary Table: Principal Approaches and Consequences

Generalized Space	Phenomenon Observed	References
$H^{-1}$, $L^\infty$,	Convex integration; h-principle; wild nonuniqueness	(Choffrut, 2012, Choffrut et al., 2014)
$B^{s,\psi}_{p,q}$, $F^{s,\psi}_{p,q}$	Well-posedness (if $\psi$ grows sufficiently); BKM-type global-in-time results in 2D	(Harrison et al., 3 Oct 2025)
$B^1_{\infty,1}$, $C^1$	Norm inflation; ill-posedness; failure of continuous dependence	(Misiołek et al., 2014, Pastrana, 2019)
$H^{d/2+1}$, critical Lorentz	Instantaneous blow-up; nonexistence or ill-posedness	(Jeong et al., 2021)
Triebel–Lizorkin $F^{d+1}_{1,\infty}$	Temporal discontinuity—even with continuous dependence in weaker spaces	(Pak, 2023)
$C^{0, \alpha}_\lambda$, local Morrey	Energy conservation, even for unbounded solutions	(Veiga et al., 2019, Cobb et al., 7 Oct 2024)
Sobolev conormal	Boundary layer regularity, local well-posedness	(Aydın et al., 25 Jul 2024)

• "h-Principles for the incompressible Euler equations" (Choffrut, 2012)
• "Weak solutions to the stationary incompressible Euler equations" (Choffrut et al., 2014)
• "The Euler equations in a critical case of the generalized Campanato space" (Chae et al., 2019)
• "Regularity results for rough solutions of the incompressible Euler equations via interpolation methods" (Colombo et al., 2019)
• "Ill-posedness of the incompressible Euler equations in the $C^1$ space" (Misiołek et al., 2014)
• "A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces" (Jeong et al., 2021)
• "Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space" (Pak, 2023)
• "Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness" (Harrison et al., 3 Oct 2025)
• "On concentrated vortices of 3D incompressible Euler equations under helical symmetry: with swirl" (Qin et al., 14 Dec 2024)
• "Unbounded Yudovich Solutions of the Euler Equations" (Cobb et al., 7 Oct 2024)
• "Euler Equations in Sobolev conormal spaces" (Aydın et al., 25 Jul 2024)
• "Onsager's Conjecture for the Incompressible Euler Equations in the Hölog Spaces $C^{0,α}_λ(\barΩ)$" (Veiga et al., 2019)
• "Energy conservation for the Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$ for weak solutions defined without reference to the pressure" (Robinson et al., 2018)

This synthesis encapsulates the landscape of the incompressible Euler equations in generalized function spaces, emphasizing how the interplay of functional analytic properties, convex integration, Lagrangian geometry, and boundary/energy effects shapes the existence, uniqueness, stability, and physical relevance of weak solutions.