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A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A generalized L^r convex integration framework constructs weak solutions in L^∞ for geophysical fluid models.

desk verdict The paper gives a reusable L^r convex integration scheme that recovers known Euler results and delivers a first admissible-solution construction for hydrostatic Euler via a new convex-hull calculation. read the letter →

arxiv 2606.12192 v1 pith:Z3EM57AN submitted 2026-06-10 math.AP

Daniel W. Boutros , Simon Markfelder , Edriss S. Titi This is my paper · ORCID
classification math.AP
keywords convexintegrationweaksolutionsEulerequationshydrostaticgeophysicalfluidmodelsenergyinequalityLinfinityprimitive
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces a general framework for convex integration designed to build bounded weak solutions to initial value problems for various partial differential equations. The solutions are in L^∞ and weakly continuous in time in the weak L^r topology for r between 1 and infinity, while also satisfying an energy inequality. The framework recovers existing results for the incompressible and barotropic compressible Euler equations and establishes new global existence results for the incompressible Euler equations. It further provides the first convex integration constructions of admissible solutions with natural energy for the hydrostatic Euler equations and yields initial existence results for wild data in the compressible inviscid primitive equations and the inviscid quasi-geostrophic equations.

What carries the argument

The L^r convex integration framework for constructing energy-inequality satisfying weak solutions in L^∞ that are weakly continuous in time.

What would settle it

An explicit computation demonstrating that no point in the computed convex hull subset satisfies the required energy bound for the hydrostatic Euler system would disprove the existence claim for admissible solutions in that model.

Watch

Extended reading notes

Core claim

The central discovery is a generalized L^r convex integration framework that produces weak solutions belonging to L^∞((0,T) × Ω) for bounded domains Ω. These solutions are weakly continuous in time with respect to the weak topology of L^r(Ω) for r in (1,∞) and satisfy an energy inequality. When applied to geophysical models, this yields global existence of weak solutions for all initial data in the incompressible Euler equations, admissible solutions with natural energy for the hydrostatic Euler equations via explicit convex hull computation, and first wild data results for the compressible primitive equations and quasi-geostrophic equations.

Load-bearing premise

Computing a large enough subset of the convex hull for the hydrostatic Euler equations is possible and allows construction of solutions obeying the natural energy inequality.

Editorial extensions

If this is right

  • The framework establishes global existence of L^∞ weak solutions to the incompressible Euler Cauchy problem.
  • It enables the first convex integration construction of admissible solutions satisfying the natural energy inequality for the hydrostatic Euler equations.
  • Existence of infinitely many energy-inequality solutions is shown for some initial data in the considered models.
  • New existence results for weak solutions are obtained for the compressible inviscid primitive equations and inviscid quasi-geostrophic equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the convex hull computation generalizes, similar constructions could apply to other related fluid systems.
  • The results indicate potential non-uniqueness of solutions in these geophysical models beyond the cases explicitly treated.
  • Extensions might incorporate additional constraints or different function spaces in future applications of the framework.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper develops a generalized framework for L^r convex integration, extending De Lellis-Szekelyhidi and Markfelder, to construct weak solutions in L^∞((0,T)×Ω) that are weakly continuous in L^r(Ω) and satisfy energy inequalities. It recovers known results for incompressible and barotropic compressible Euler, proves a new global existence result for incompressible Euler in L^∞, and applies the framework to the hydrostatic Euler equations (first convex integration construction of admissible solutions with natural energy, relying on computation of a large convex hull subset), as well as to compressible inviscid primitive equations and inviscid quasi-geostrophic equations to obtain existence and non-uniqueness results for wild data.

Significance. If the generalized framework is sound and the convex hull computations are accurate, the work provides a unified method for existence and non-uniqueness of weak solutions in geophysical fluid models. The new L^∞ global existence for incompressible Euler and the first admissible convex integration result for hydrostatic Euler with natural energy are notable extensions of the literature. The explicit convex hull subset computation for hydrostatic Euler is credited as a key technical step enabling the energy inequality.

major comments (2)
  1. [§5 (Hydrostatic Euler equations)] §5 (Hydrostatic Euler equations): The claim that the computed large subset of the convex hull enables construction of admissible solutions satisfying the natural energy inequality is load-bearing for the main new result. The manuscript must verify explicitly that this subset contains a sufficiently rich open set in the state space (allowing absorption of arbitrary small Reynolds stress defects while preserving hydrostatic balance, L^∞ bounds, and the energy inequality at each iteration step); without such verification the application-specific result does not follow from the general framework.
  2. [§3 (General Framework)] §3 (General Framework), the iterative scheme: The proof that the L^r weak continuity and energy inequality are preserved under the generalized convex integration procedure relies on estimates that must be checked for compatibility with the specific convex hull subsets used in each geophysical application; the reduction from the general case to the hydrostatic Euler case appears to require additional parameter control not detailed in the framework statement.
minor comments (2)
  1. The notation distinguishing the state variables (velocity, pressure, etc.) across the different models could be standardized with a summary table in the framework section to improve readability.
  2. [Introduction] Theorem statements for the new incompressible Euler L^∞ result and the hydrostatic Euler admissible solutions should include explicit references to the convex hull subset used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will incorporate clarifications and explicit verifications into a revised version of the manuscript.

read point-by-point responses

  1. Referee: [§5 (Hydrostatic Euler equations)] §5 (Hydrostatic Euler equations): The claim that the computed large subset of the convex hull enables construction of admissible solutions satisfying the natural energy inequality is load-bearing for the main new result. The manuscript must verify explicitly that this subset contains a sufficiently rich open set in the state space (allowing absorption of arbitrary small Reynolds stress defects while preserving hydrostatic balance, L^∞ bounds, and the energy inequality at each iteration step); without such verification the application-specific result does not follow from the general framework.

    Authors: We agree that an explicit verification of the openness and richness of the computed convex-hull subset is necessary to close the argument. In the revised manuscript we will add a dedicated lemma (or subsection) in §5 that confirms the subset contains a nonempty open set in the state space, that this open set is compatible with the hydrostatic constraint, and that the quantitative estimates from the general framework (smallness of the Reynolds stress, preservation of L^∞ bounds and the energy inequality) remain valid at each iterative step. This will make the reduction from the abstract framework to the hydrostatic Euler equations fully rigorous. revision: yes

  2. Referee: [§3 (General Framework)] §3 (General Framework), the iterative scheme: The proof that the L^r weak continuity and energy inequality are preserved under the generalized convex integration procedure relies on estimates that must be checked for compatibility with the specific convex hull subsets used in each geophysical application; the reduction from the general case to the hydrostatic Euler case appears to require additional parameter control not detailed in the framework statement.

    Authors: The estimates in §3 are stated with explicit dependence on the diameter of the convex-hull subset and on the admissible range of the iteration parameters (δ, λ, etc.). For the hydrostatic Euler application these parameters are chosen inside the open intervals guaranteed by the general framework; the specific numerical ranges appear in the proof of the main theorem in §5. In the revision we will insert a short paragraph (or remark) immediately after the statement of the general iterative scheme that records the precise parameter restrictions needed for each application, thereby making the compatibility check transparent and uniform across all models treated in the paper. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation to prior framework; new convex-hull computation for hydrostatic Euler is independent

full rationale

The paper develops a generalized L^r convex integration framework explicitly building on De Lellis-Szekelyhidi (2010) and Markfelder (2024), then applies it to recover known results for Euler equations and to obtain new existence statements for incompressible Euler in L^∞ and admissible solutions for hydrostatic Euler. The abstract identifies the computation of a large subset of the convex hull as the crucial new ingredient for the hydrostatic case; this step is performed within the paper and does not reduce by definition or by self-citation chain to the cited prior works. No self-definitional relations, fitted parameters renamed as predictions, or ansatz smuggling appear. The self-citation is therefore not load-bearing for the central claims.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The framework relies on standard convex integration techniques and computations of convex hulls in function spaces. No free parameters or invented entities are apparent from the abstract.

assumptions (2)
  • standard math Standard properties of convex hulls in appropriate function spaces hold for the systems considered.
    The framework relies on convex integration which uses convex hull computations.
  • domain assumption The PDEs satisfy the necessary structural conditions for the framework to apply, such as being in the family of equations amenable to L^r convex integration.
    Abstract states the framework applies to a large family of PDEs.

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Pith. "Pith review of A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models." pith.science (2026). https://pith.science/paper/Z3EM57AN

@misc{pith2026260612192,
  author       = {Pith},
  title        = {Pith review of: A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z3EM57AN}},
  note         = {Machine review of arXiv:2606.12192}
}
abstract

In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Sz\'ekelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\infty ((0,T) \times \Omega)$ (for a bounded domain $\Omega)$ which are weakly continuous in time with respect to the weak topology of $L^r (\Omega)$ for some $r \in (1,\infty)$. This allows us to construct solutions which obey an energy inequality. In the second part of the paper we apply the framework to several inviscid models appearing in the field of geophysical fluid mechanics in order to show existence of weak solutions for all initial data, and to prove that there exist initial data for which there are infinitely many solutions which satisfy an energy inequality. We first consider the incompressible and the barotropic compressible Euler equations to recover the corresponding results from the literature. In addition, the framework allows us to prove a new result for the incompressible Euler equations, namely the global existence for the Cauchy problem in $L^\infty$. Moreover, we use the framework in the context of the hydrostatic Euler equations (also known as the incompressible inviscid primitive equations), which leads to the first convex integration approach which is able to construct admissible solutions with the natural energy for this system. A crucial ingredient in the proof of this result is the computation of a large subset of the convex hull. Finally, we apply the framework to the compressible inviscid primitive equations and to the inviscid quasi-geostrophic equations to obtain the first results on existence of wild data for these two geophysical models.

Figures

Figures reproduced from arXiv: 2606.12192 by the authors.

Figure 2.1
Figure 2.1. Illustration of Defn. 2.27. While examples (a) and (b) are not suitable, (c) is. [PITH_FULL_IMAGE:figures/full_fig_p036_2_1.png] view at source ↗

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