Asymptotics of Stationary Navier Stokes Equations in Higher Dimensions

Hao Jia; Vladimír Šverák

doi:10.1007/s10114-017-7397-3

Asymptotics of Stationary Navier Stokes Equations in Higher Dimensions

Hao Jia, Vladimír Šverák

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.

Original language	English (US)
Pages (from-to)	598-611
Number of pages	14
Journal	Acta Mathematica Sinica, English Series
Volume	34
Issue number	4
DOIs	https://doi.org/10.1007/s10114-017-7397-3
State	Published - Apr 1 2018

Bibliographical note

Funding Information:
Received August 30, 2017, accepted September 20, 2017 The second author is supported in part by grant DMS (Grant No. 1362467) from the National Science Foundation; the first author is supported in part by DMS (Grant No. 1600779)

Publisher Copyright:
© 2018, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

Navier Stokes equations
asymptotics
steady states

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10.1007/s10114-017-7397-3

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abstract = "We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.",

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AU - Šverák, Vladimír

N1 - Funding Information: Received August 30, 2017, accepted September 20, 2017 The second author is supported in part by grant DMS (Grant No. 1362467) from the National Science Foundation; the first author is supported in part by DMS (Grant No. 1600779) Publisher Copyright: © 2018, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.

AB - We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.

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KW - asymptotics

KW - steady states

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Asymptotics of Stationary Navier Stokes Equations in Higher Dimensions

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