Convex Integration: Schemes & Applications

• Convex integration schemes are analytical and geometric methods that construct flexible weak solutions by exploiting oscillatory perturbations along constrained wave cones.
• They use iterative corrugation and defect decomposition techniques to progressively reduce errors, achieving convergence in specified regularity classes.
• Applications span fluid dynamics, where nonunique Euler solutions arise, to materials science with multiwell and phase transformation models.

Convex integration schemes are analytic and geometric frameworks for constructing “flexible” solutions to nonlinear partial differential equations and differential inclusions, allowing explicit realization of a plethora of anomalous behaviors such as non-uniqueness and irregularity. At their core, these schemes exploit localized oscillatory perturbations—parametrized via “wave cones” or rank-one directions—within a suitable relaxation of the nonlinear constraint. Modern convex integration has become a foundational tool in the analysis of mathematical fluid dynamics, calculus of variations, geometry, and rigidity/flexibility phenomena in several classes of differential equations.

1. Foundational Principles and Abstract Framework

Convex integration begins with the following abstract structure:

• A domain $\Gamma\subset\mathbb{R}^n$ and an unknown field $z:\Gamma\to\mathbb{R}^m$.
• A first-order linear differential operator $A(z)$, often the row-wise divergence.
• A nonlinear pointwise constraint $z(x)\in K_b$, where to each “parameter” $b(x)$ is assigned a compact set $K_b\subset\mathbb{R}^m$ (the “constitutive set”).

One seeks to solve

$A(z) \leq b(x), \qquad z(x)\in K_{b(x)}.$

Here, $A(z)\leq b(x)$ is interpreted componentwise, encoding systems such as the compressible Euler equations, isometric immersions, or holonomic jet relations (Markfelder, 2023, Massot et al., 2021).

The associated relaxation is given by defining the $\Lambda$–convex hull $U_b := \operatorname{int}(K_b)^\Lambda$, with $\Lambda$ the wave cone associated to the linear operator $A$—that is, the collection of directions $z_0$ such that $A(z) = 0$ admits plane-wave solutions with $z_0\in\Lambda$. The scheme crucially relies on the fact that subsolutions—fields $z$ for which $A(z)\leq b$ and $z(x)\in U_{b(x)}$—form a large set, amenable to iterated perturbation by oscillations directed along $\Lambda$.

If one can produce a single subsolution, then a convex integration theorem yields the existence of infinitely many weak solutions (“wild solutions”) with $z(x)\in K_{b(x)}$ almost everywhere, as $L^\infty$ weak-star limits (Markfelder, 2023).

2. Iterative Methodology: Corrugations and Stage Analysis

At the level of implementation, convex integration proceeds by constructing a sequence of approximate solutions, driving the “defect” or “error” to zero. Taking, for example, the Nash–Kuiper approach to isometric immersions, which appears in all classical convex integration, one iteratively:

• Decomposes the defect $D_q$ (metric error, stress, or residual) as a sum of rank-one or primitive building blocks, i.e., $D_q = \sum a_{i,q}^2 \xi_i\otimes \xi_i$.
• At each step, adds a high-frequency oscillatory “corrugation” in direction $\xi_i$ with amplitude $a_{i,q}$, such that the nonlinear constraint moves closer to being pointwise satisfied up to smaller error.

For higher dimensions or more variables, a “stage” is composed of multiple substeps, each eliminating a portion of the error. The improved decomposition lemma (Su et al., 30 Apr 2025) refines the classical Nash count of $M_0=n(n+1)/2$ steps per stage to a minimal number $\Xi_n$ depending on the Radon–Hurwitz number, enabling convergence in better Hölder regularity.

The typical error iteration is geometric or super-geometric. For instance, in the Lewicka–Pakzad construction—applicable to the Monge–Ampère equation (Codenotti et al., 2018)—the $C^0$ norm of the defect contracts by a factor (e.g., $3/4$) at each full stage, whereas the $C^1$ norm of the solution grows only polynomially in the frequencies, yielding $C^{1,\alpha}$ convergence if frequency growth is chosen rapidly enough.

3. Applications in Fluid Dynamics: Euler Equations and Admissibility

In the context of incompressible and compressible Euler equations, convex integration is used to construct weak solutions exhibiting nonuniqueness, failure of admissibility, and turbulence-like behaviors (Buckmaster et al., 2019, Markfelder, 2023, Markfelder, 2020, Dębiec et al., 2021, Chen et al., 27 Jan 2026, Bruè et al., 2024). The “defect” here is the Reynolds stress $R_q$ in a system of the form: $$\partial_t v_q + \operatorname{div}(v_q\otimes v_q) + \nabla p_q = \operatorname{div} R_q, \quad \operatorname{div} v_q = 0$$ Perturbations are built from Beltrami flows, Mikado flows, or, in recent developments, highly localized spatial building blocks such as Lamb–Chaplygin dipoles (Bruè et al., 2024). The critical parameter hierarchy—frequencies $\lambda_q$, amplitudes $\delta_q$, and mollification scales $\ell_q$—controls the convergence of the scheme and the regularity of the limit. Transport, oscillation, and “Nash” errors are balanced at each stage to ensure that $R_{q+1}\rightarrow0$ while the energy profile or other admissibility criteria are manipulated as desired.

Importantly, new frameworks are able to encode not just momentum conservation but also energy and energy-flux directly at the level of the convex integration system. The scheme described in (Markfelder, 2023) constructs admissible solutions for the barotropic compressible Euler system: $$\partial_t\rho + \nabla\cdot(\rho u) = 0,\quad \partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla p(\rho) = 0$$ supplemented by an energy inequality. Perturbations are performed simultaneously in $(m, U, q, F)$ to “build” the energy and energy-flux, increasing flexibility and allowing the construction of subsolutions not accessible by previous approaches.

4. Key Advances and Extensions

Several recent advances have broadened the reach of convex integration:

• Improved Regularity: The new decomposition lemma (Su et al., 30 Apr 2025) leverages topological results (Adams–Lax–Phillips) to reduce the number of primitive steps required in the rank-one decomposition of the defect. For dimensions $n=2,4,8,16$ one achieves $C^{1,\alpha}$ convergence for $\alpha<1/(n^2+1)$, exceeding the classical threshold $1/(n^2+n+1)$.
• Handling Constraints: Algebraic and topological structures (projective duality, invertibility under doubling the diagonal) allow the correct orthogonality and positivity in the decomposition even after elliptic elimination, thus avoiding Nash–Moser smoothing losses and simplifying iteration.
• Extensions to Multiwell and Phase Transformation Models: The flexibility of convex integration has been demonstrated in the modeling of microstructures in shape-memory alloys, with explicitly implementable algorithms for multiwell energy landscapes and careful control of oscillation scales for regularity versus physical selection by surface energies (Rüland et al., 2018).
• Nonuniqueness Beyond the Onsager Criticality: By alternating which variable carries the Reynolds stress, the regularity barrier of the Onsager exponent ($1/3$ for Euler) can be exceeded for forced systems ($\beta<1/2$) (Bulut et al., 2023).

5. Relaxation, Subsolutions, and Baire-Category Arguments

The systematization of convex integration proceeds via:

• Subsolution Framework: Identification of the $\Lambda$-convex hull $K^\Lambda$ of the constitutive set $K$, characterized directly in terms of inequalities on the energy, stress, or other physical quantities (Markfelder, 2023, Dębiec et al., 2021). Subsolutions are constructed to lie strictly inside this hull.
• Perturbation Property: In each small space–time region, a localized corrector is constructed—often a plane wave along a direction in the wave cone—reducing the local defect. This is achieved through potential theory or explicit explicit oscillatory profiles.
• Residuality via Baire-Category: By constructing a complete metric space of approximate subsolutions and a lower semicontinuous defect functional (e.g., $I_{K}(z) = \int \operatorname{dist}(z(x),K)\,dx$) that can be strictly improved at every iteration, the set of exact solutions is shown to be residual by the Baire-category theorem (Markfelder, 2023).

6. Impact, Optimality, and Directions

Convex integration has revealed a fundamental flexibility considered impossible for many decades: wild solutions violating classical rigidity and uniqueness, density/flexibility theorems for Monge–Ampère-type systems (Codenotti et al., 2018, Lewicka, 2022), and efficient holonomic approximations in jet-space geometry (Massot et al., 2021). These constructions definitively separate the notions of well-posedness, regularity, and physical admissibility, prompting reconsideration of selection criteria and “maximal dissipation” conjectures (e.g., failure of the local maximal-dissipation criterion for compressible shocks (Markfelder, 2023)).

Recent schemes have focused on:

• Achieving higher regularity (supersaturation of classical exponent bounds (Su et al., 30 Apr 2025)).
• Incorporation of physical constraints such as monotonicity, energy inequalities, and boundary conditions.
• Application to SPDEs, though strong nonuniqueness via convex integration has so far been blocked for certain stochastic models (e.g., $\Phi^4$) by underlying deterministic uniqueness (Dong et al., 15 Jan 2026).
• Numerical implementations in multiwell landscape and visualization in geometric PDE (Rüland et al., 2018, Codenotti et al., 2018).

Convex integration thus stands as a cornerstone methodology for creating wild, flexible, and sometimes nonphysical solutions in nonlinear PDE, geometry, and applied analysis, subject to both geometric insight and intricate analytic bookkeeping.