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Failure of the chain rule for the divergence of bounded vector fields

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Gianluca Crippa, Nikolay Gusev, Stefano Spirito, Emil Wiedemann

Analysis of PDEs · Submitted 8 Dec 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions.

how it is stated · asserted 1.00 read the record →

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arXiv ID1412.2476 (version 1)
Pith IDRC66JVHZ, Pith's permanent record of this paper and its review
CitationEmil Wiedemann, Gianluca Crippa, Nikolay Gusev, Stefano Spirito. "Failure of the chain rule for the divergence of bounded vector fields." arXiv:1412.2476 (2014). https://pith.science/paper/RC66JVHZ
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,907 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. contribution 0.58 0.95 0.90 asserted 1.00
s1 Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. method 0.93 0.93 0.16 asserted 0.91
s2 For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures. result 0.84 0.93 0.10 asserted 0.76

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.

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