Sufficient conditions on Liouville type theorems for the 3D steady Navier–Stokes equations

Seregin, G.; Wang, W.

doi:10.1090/spmj/1603

Sufficient conditions on Liouville type theorems for the 3D steady Navier–Stokes equations
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by G. Seregin and W. Wang

St. Petersburg Math. J. 31 (2020), 387-393

DOI: https://doi.org/10.1090/spmj/1603

Published electronically: February 4, 2020

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Abstract:

Our aim is to prove Liouville type theorems for the three dimensional steady-state Navier–Stokes equations provided the velocity field belongs to some Lorentz spaces. The corresponding statement contains several known results as a particular case.

References

Dongho Chae, Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations, Comm. Math. Phys. 326 (2014), no. 1, 37–48. MR 3162482, DOI 10.1007/s00220-013-1868-x
Dongho Chae and Tsuyoshi Yoneda, On the Liouville theorem for the stationary Navier-Stokes equations in a critical space, J. Math. Anal. Appl. 405 (2013), no. 2, 706–710. MR 3061045, DOI 10.1016/j.jmaa.2013.04.040
Dongho Chae and Jörg Wolf, On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb {R}^3$, J. Differential Equations 261 (2016), no. 10, 5541–5560. MR 3548261, DOI 10.1016/j.jde.2016.08.014
Dongho Chae and Shangkun Weng, Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5267–5285. MR 3543547, DOI 10.3934/dcds.2016031
G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162, DOI 10.1007/978-0-387-09620-9
Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), no. 1, 83–105. MR 2545826, DOI 10.1007/s11511-009-0039-6
Hideo Kozono, Yutaka Terasawa, and Yuta Wakasugi, A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions, J. Funct. Anal. 272 (2017), no. 2, 804–818. MR 3571910, DOI 10.1016/j.jfa.2016.06.019
G. Seregin, Liouville type theorem for stationary Navier-Stokes equations, Nonlinearity 29 (2016), no. 8, 2191–2195. MR 3538409, DOI 10.1088/0951-7715/29/8/2191
—, A liouville type theorem for steady-state Navier–Stokes equations, J. Équations dérivées partielles (2016), Exp. IX, 1–5; arXiv: 1611.01563
G. Seregin, Remarks on Liouville type theorems for steady-state Navier-Stokes equations, Algebra i Analiz 30 (2018), no. 2, 238–248; English transl., St. Petersburg Math. J. 30 (2019), no. 2, 321–328. MR 3790738, DOI 10.1090/spmj/1544