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New Liouville type theorems for the stationary Navier-Stokes equations

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Wenke Tan

Analysis of PDEs · Submitted 7 Jan 2025

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

The authors' main claim From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations.

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arXiv ID2501.03609 (version 1)
Pith IDHYPGWADZ, Pith's permanent record of this paper and its review
CitationWenke Tan. "New Liouville type theorems for the stationary Navier-Stokes equations." arXiv:2501.03609 (2025). https://pith.science/paper/HYPGWADZ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,763 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.54 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$.

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 mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. contribution 0.52 0.84 0.05 asserted 0.76
s1 We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. result 0.60 0.96 0.19 asserted 1.00
s2 From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. result 0.88 0.95 0.40 asserted 1.00
s3 Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$. contribution 0.62 0.95 0.25 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

We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$.

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3 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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