Pith. Sign in

RECORD

Instabilities and oscillations in coagulation equations with kernels of homogeneity one

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Michael Herrmann, Barbara Niethammer, Juan J.L. Velázquez

Analysis of PDEs · Submitted 30 Jun 2016

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

The authors' main claim However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data.

how it is stated · asserted 0.99 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1606.09405 (version 1)
Pith IDHO74YPMN, Pith's permanent record of this paper and its review
CitationBarbara Niethammer, Juan J.L. Vel\'azquez, Michael Herrmann. "Instabilities and oscillations in coagulation equations with kernels of homogeneity one." arXiv:1606.09405 (2016). https://pith.science/paper/HO74YPMN
Claim paper

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.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,930 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.52 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.

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 discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. contribution 0.79 0.88 0.05 asserted 0.83
s1 The case of what we call diagonally dominant kernels is particularly interesting. background 0.47 0.17 0.03 asserted 0.89
s2 Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. background 0.58 0.17 0.08 hedged possibility 0.71
s3 However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. result 1.00 0.86 0.40 asserted 0.99
s4 On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. result 1.00 0.78 0.18 asserted 1.00
s5 This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front. interpretation 0.98 0.68 0.19 asserted 0.96

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from simulation or computation 0.64.

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
levels run
  • 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 discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.