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A connection between symmetry breaking for Sobolev minimizers and stationary Navier-Stokes flows past a circular obstacle

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Filippo Gazzola, Gianmarco Sperone, Tobias Weth

Analysis of PDEs · Submitted 28 Dec 2021

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

The authors' main claim We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

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arXiv ID2112.14122 (version 1)
Pith IDLBDV3XEM, Pith's permanent record of this paper and its review
CitationFilippo Gazzola, Gianmarco Sperone, Tobias Weth. "A connection between symmetry breaking for Sobolev minimizers and stationary Navier-Stokes flows past a circular obstacle." arXiv:2112.14122 (2021). https://pith.science/paper/LBDV3XEM
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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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,318 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.76 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

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 Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. background 1.00 0.05 0.04 hedged possibility 1.00
s1 We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. contribution 0.84 0.86 0.04 asserted 0.95
s2 In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. background 0.75 0.18 0.11 asserted 1.00
s3 Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. contribution 0.70 0.94 0.47 asserted 1.00
s4 We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity. result 0.99 0.95 0.30 asserted 1.00

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

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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

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

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