On strong solutions of Navier-Stokes equations with two velocity components

Dandan Ma; Maria Alessandra Ragusa; Fan Wu; Dandan Ma; Maria Alessandra Ragusa; Fan Wu

doi:10.3934/math.2026386

AIMS Mathematics

2026, Volume 11, Issue 4: 9334-9346. doi: 10.3934/math.2026386

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On strong solutions of Navier-Stokes equations with two velocity components

1.
College of Science & Key Laboratory of Engineering Mathematics and Advanced Computing, Jiangxi University of Water Resources and Electric Power, Nanchang, Jiangxi 330099, China
2.
Corresponding author. Department of Mathematics and Computer Science, University of Catania, Viale Andrea Doria No. 6, Catania 95125, Italy
3.
Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
4.
College of Science, Jiangxi University of Water Resources and Electric Power, Nanchang, Jiangxi 330099, China

Received: 09 January 2026 Revised: 18 March 2026 Accepted: 24 March 2026 Published: 07 April 2026
MSC : 35B65, 35Q35, 76D05

In this paper, we established a continuation criterion for local strong solutions to the three-dimensional incompressible Navier-Stokes equations based on partial velocity components. More precisely, we showed that a unique local strong solution $ u $ does not blow up at time $ T $ provided that the two horizontal velocity components $ u_h $ belong to the Banach spaces $ \dot{V}_{p, q, \theta}^s $ and $ \dot{U}_{p, \beta, \sigma}^s $. These functional spaces strictly contained the homogeneous Besov space $ \dot{B}_{p, q}^s $, and thus allowed a wider admissible class than those considered in earlier works. Our results can therefore be viewed as an extension and refinement of previously known criteria.
- Navier-Stokes equations,
- extension criteria,
- two velocity components
Citation: Dandan Ma, Maria Alessandra Ragusa, Fan Wu. On strong solutions of Navier-Stokes equations with two velocity components[J]. AIMS Mathematics, 2026, 11(4): 9334-9346. doi: 10.3934/math.2026386

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Abstract

In this paper, we established a continuation criterion for local strong solutions to the three-dimensional incompressible Navier-Stokes equations based on partial velocity components. More precisely, we showed that a unique local strong solution $ u $ does not blow up at time $ T $ provided that the two horizontal velocity components $ u_h $ belong to the Banach spaces $ \dot{V}_{p, q, \theta}^s $ and $ \dot{U}_{p, \beta, \sigma}^s $. These functional spaces strictly contained the homogeneous Besov space $ \dot{B}_{p, q}^s $, and thus allowed a wider admissible class than those considered in earlier works. Our results can therefore be viewed as an extension and refinement of previously known criteria.

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