Convex Integration Technique

• Convex Integration is a method that constructs flexible, oscillatory solutions for nonlinear PDEs by iteratively correcting subsolutions.
• It employs subsolution concepts and oscillatory correctors to decompose errors and overcome non-convex constraints in differential systems.
• The technique has broad applications in fluid dynamics, geometry, and material science, challenging conventional notions of uniqueness and regularity.

Convex integration is a fundamental technique in the analysis of nonlinear partial differential equations (PDEs) and differential inclusions, enabling the flexible construction of solutions with intricate oscillatory or "wild" properties. Originating in the study of isometric immersions and subsequently formalized in the works of Nash and Gromov, convex integration revolutionized the understanding of underdetermined systems, has found foundational applications in fluid dynamics, geometry, and the theory of materials, and has undergone extensive developments with algebraic, geometric, and functional-analytic refinements.

1. General Framework and Historical Foundations

Convex integration solves equations or inclusions of the form

$$Du(x) \in K \subset \mathbb{R}^m, \quad x \in \Omega,$$

where $K$ is a (potentially non-convex) set of admissible gradients or jets. The central insight is to construct solutions as limits of "subsolutions"—objects that strictly satisfy a relaxed version of the constraint (typically, that $Du(x)$ lies in the interior of the convex hull of $K$) and admit controllably small "error" terms, which are then eliminated through highly oscillatory perturbations operating in directions consistent with the underlying differential constraints.

Historically, Nash (1954) proved the $C^1$ isometric embedding theorem via iterative addition of oscillatory correctors, while Gromov (1986) provided the h-principle perspective, identifying openness and ampleness of the relation as key topological-geometric criteria for flexibility and the applicability of convex integration (Dong et al., 15 Jan 2026, Massot et al., 2021).

2. Building Blocks and Perturbation Schemes

At the heart of convex integration schemes—be it for differential inclusions, fluid equations, or elasticity—are the recursive correction steps:

• Subsolution Concept: A subsolution is a map $u$ (or tuple) such that $Du$ lies in the relaxed admissible set (e.g., the $\Lambda$-convex or rank-one convex hull), possibly carrying an "error" field (e.g., Reynolds stress $R$ in fluid mechanics) (Sattig et al., 2023, Dong et al., 15 Jan 2026).
• Oscillatory Correctors: At each iteration, a highly oscillatory correction $w_{q+1}$ is constructed using building blocks (e.g., Mikado flows, Beltrami waves, or geometrically-designed corrugations), which are tailored to decrease the error term $R_q$ in a weak topology. The amplitude and frequency of the corrector are finely tuned to ensure convergence and control of the desired norms.
• Geometric Lemma / Decomposition: Defects or errors are decomposed (e.g., as sums of rank-one tensors) so that each correction targets a component of the error, utilizing algebraic or geometric identities such as the rank-one decomposition lemma or the structure of the wave cone (Sattig et al., 2023, Su et al., 30 Apr 2025).

The efficacy of perturbations relies on the understanding of so-called "T$_N$-configurations," wave cones, and the construction of localized plane waves compatible with the PDE's structure.

3. Convex Integration in Fluid Dynamics

Convex integration has led to profound results in hydrodynamics—most notably the construction of Hölder-continuous weak solutions to the Euler and Navier–Stokes equations displaying non-uniqueness and anomalous energy dissipation (violations of the classical Onsager conjecture) (Buckmaster et al., 2019, Bulut et al., 2023, Bruè et al., 2024, Sattig et al., 2023).

Euler and Navier–Stokes

In the incompressible Euler context, the iterative scheme is formulated as: $$\partial_t v_q + v_q \cdot \nabla v_q + \nabla p_q = \operatorname{div} R_q, \quad \operatorname{div} v_q = 0,$$ where $R_q$ is the Reynolds stress. The perturbation $w_{q+1}$ is chosen such that

$$R_{q+1} = R_q - (w_{q+1} \otimes w_{q+1} - \frac{1}{d}|w_{q+1}|^2 \, \operatorname{Id}) + \text{transport/corrector errors}.$$

Building blocks include Mikado flows (highly localized, intermittent vector fields), Beltrami waves, and in recent developments, spatially inhomogeneous dipole flows adapted to achieve new integrability thresholds on vorticity (Sattig et al., 2023, Bruè et al., 2024). The parameter choices---oscillation frequency, concentration, and energy scaling---are critical for achieving specific regularity (e.g., Onsager-critical $C^{1/3-}$) or integrability (e.g., vorticity in $L^p$ with $p > 1$).

Breakthroughs and Extensions

• Alternating Schemes and Above-Onsager Exponents: Alternating the perturbation between two flows allows for improved regularity thresholds. For forced Euler, convex integration can construct non-unique solutions with regularity $C^{1/2-}$, surpassing Onsager's $1/3$ (Bulut et al., 2023).
• Baire Category Argument: The convex integration output is often a residual set in a suitable function space, identified as points of continuity of an energy- or defect-gap functional in a complete metric topology, ensuring genericity of wild solutions (Sattig et al., 2023).
• Intermittency and Non-Periodic Perturbations: To bypass spectral concentration or integrability barriers, strategies such as time-averaged error cancellation and non-periodic building blocks (e.g., using the Lamb–Chaplygin dipole) are leveraged for 2D Euler (Bruè et al., 2024).

4. Convex Integration in Non-Elliptic and Nonlocal PDEs

Convex integration also applies to a range of geometric and analytical PDEs beyond fluid mechanics:

• Monge–Ampère Equation and Scalar Curvature: Convex integration can construct dense sets of $C^{1,\alpha}$ solutions to the 2D Monge–Ampère equation for all $\alpha < 1/7$, and even provides solutions to prescribed scalar curvature problems via mixed (multi-scale) corrugation schemes (Lewicka et al., 2015, Aliouane et al., 13 May 2025). Rigidity arises beyond sharp regularity thresholds (e.g., $\alpha > 2/3$ for Monge–Ampère).
• Diffusion, Perona–Malik, and Forward-Backward PDEs: The method is impactful for nonmonotone diffusion and gradient-flow equations, yielding infinite families of Lipschitz solutions under rank-one convexity and structural hypotheses on the nonlinearity (Yan, 2018, Kim et al., 2015). The explicit lamination convex hull and T$_N$-configurations underpin the construction.
• Active Scalar Equations: For non-dissipative systems including the magnetostrophic and porous media models, convex integration produces $L^\infty$ weak solutions, using frequency-localized and Fourier-constrained oscillatory increments adapted to nonlocal multipliers (Shvydkoy, 2010).

5. Algebraic and Geometric Advances: Decomposition and Regularity

A significant recent advance is the reduction in the number of rank-one building blocks needed to decompose the defect in the Nash–Kuiper–Conti–De Lellis–Székelyhidi framework. This is achieved using tools from algebraic topology, classical algebraic geometry, and representation theory (Su et al., 30 Apr 2025):

• Decomposition Lemma: Any symmetric matrix-valued error can be written as a sum of as few as $\Xi_n = n(n+1)/2 - \rho(n/2)$ rank-one projectors, where $\rho(n)$ is the Radon–Hurwitz number (Bott periodicity), with the remainder handled by solving a linear elliptic system for an auxiliary field. This reduction yields improved Hölder exponents for flexible $C^{1,\alpha}$ solutions, e.g., $\alpha < (n^2+1)^{-1}$ for $n=2,4,8,16$.

Dimension $n$	Classical bound	New bound via (Su et al., 30 Apr 2025)
2, 4, 8, 16	$(n^2+n+1)^{-1}$	$(n^2+1)^{-1}$
other	$(n^2+n+1)^{-1}$	$(n^2+n-2\rho(\frac{n}{2})-1)^{-1}$

The analytic aspect employs an elliptic method, solving away the non-rank-one component without any loss of differentiability, thus integrating advances from classical PDE regularity theory with convex integration.

6. Baire Category, Selection Mechanisms, and Limitations

Convex integration constructions typically proceed by establishing density of subsolutions and employing genericity tools (e.g., the Baire category theorem): "wild" or flexible solutions form a residual set in an appropriate function space. However, additional constraints (e.g., surface energy terms in variational models for microstructures) can serve as selection mechanisms, imposing upper bounds on the attainable regularity, thus restoring rigidity or compactness in regimes where wild convex-integration solutions are otherwise permitted (Rüland et al., 2018).

There are also intrinsic limitations: convex integration cannot produce non-uniqueness if energy dissipation or structural damping prevails (e.g., in certain nonlinear heat equations or in the stochastic $\Phi^4$ model without sufficiently rough noise) (Dong et al., 15 Jan 2026).

7. Contemporary Extensions and Open Directions

Current research explores convex integration for singular stochastic PDEs (in combination with rough paths, paracontrolled calculus, and regularity structures), hybrid methods integrating high-frequency oscillation with probabilistic estimates, and the sharpness of non-uniqueness thresholds for continuity and transport equations beyond Sobolev embedding limits (Colombo et al., 4 Apr 2025, Dong et al., 15 Jan 2026).

The general framework is applicable across an array of systems:

• Isometric immersions and holonomic approximation (Massot et al., 2021)
• Compressible and incompressible fluid flows, multi-phase models
• Nonlinear elasticity, shape-memory alloys, and microstructure generation
• Higher-order semilinear and fully nonlinear geometric PDEs

8. Bibliographic and Comparative Table

Application Area	Regularity Threshold/Property	Key References
Incompressible Euler (3D)	$C^{\theta}$ for $\theta<1/3$ (Onsager)	(Buckmaster et al., 2019, Sattig et al., 2023)
Forced Euler	$C^{1/2-}$ above Onsager	(Bulut et al., 2023)
Navier–Stokes (Leray)	Nonuniqueness in $C^0_tL^{2+}_x \cap C^0_tW^{1,1+}_x$	(Buckmaster et al., 2019)
Monge–Ampère (2D)	Flexible for $\alpha<1/7$, rigid for $\alpha>2/3$	(Lewicka et al., 2015)
Shape-memory microstructure	Non-uniqueness vs. regularity via surface energy	(Rüland et al., 2018)
Diffusion/Perona–Malik	Infinitely many Lipschitz solutions	(Yan, 2018, Kim et al., 2015)
Scalar curvature prescription	Mixed (multi-scale) convex integration	(Aliouane et al., 13 May 2025)
Active scalar equations	Bounded nonunique weak solutions	(Shvydkoy, 2010)

• (Sattig et al., 2023) The Baire category method for intermittent convex integration
• (Su et al., 30 Apr 2025) A decomposition lemma in convex integration via classical algebraic geometry
• (Rüland et al., 2018) Convex integration arising in the modelling of shape-memory alloys
• (Lewicka et al., 2015) Convex integration for the Monge-Ampère equation in two dimensions
• (Dębiec et al., 2021) A general convex integration scheme for the isentropic compressible Euler equations
• (Markfelder, 2023) A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion
• (Aliouane et al., 13 May 2025) Almost prescribing scalar curvature by mixed convex integration
• (Bruè et al., 2024) Flexibility of two-dimensional Euler flows with integrable vorticity
• (Bulut et al., 2023) Convex integration above the Onsager exponent for the forced Euler equations
• (Buckmaster et al., 2019) Convex integration and phenomenologies in turbulence
• (Massot et al., 2021) Holonomic approximation through convex integration
• (Shvydkoy, 2010) Convex integration for a class of active scalar equations
• (Kim et al., 2015) Convex integration and infinitely many weak solutions to the Perona-Malik equation
• (Yan, 2018) Convex integration for diffusion equations
• (Dong et al., 15 Jan 2026) Remarks on the convex integration technique applied to singular stochastic partial differential equations
• (Colombo et al., 4 Apr 2025) A convex integration scheme for the continuity equation past the Sobolev embedding threshold

Convex integration, both in philosophy and technical machinery, now stands as a universal scheme to realize flexibility in PDE and geometric constraint systems, systematically producing a spectrum of solutions beyond the scope of classical compactness and regularity theory. Its ongoing development continues to challenge foundational notions of well-posedness, regularity, and selection in nonlinear analysis.