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May  2021, 41(5): 2301-2340. doi: 10.3934/dcds.2020366

On the asymptotic properties for stationary solutions to the Navier-Stokes equations

Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA

* Corresponding author: Oleg Imanuvilov

Received  August 2019 Revised  July 2020 Published  May 2021 Early access  November 2020

Fund Project: The author is supported by NSF grant DMS 1312900

In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution $ v $ and its derivatives decay like $ O(|x|^{-k}) $ for sufficiently large $ k $, depending on the velocity field, as $ |x|\to \infty $, then $ v $ is zero on that exterior domain. Constructive estimate for $ k $ is given. In the case where velocity field is only bounded at infinity, we show that the infimum of $ L^2 $ norm of a velocity field on a unit ball located at distance $ t $ from an origin is bounded from below as $ Ce^{-\beta t^\frac 43\ln(t)}. $ The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.

Citation: Oleg Imanuvilov. On the asymptotic properties for stationary solutions to the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2301-2340. doi: 10.3934/dcds.2020366
References:
[1]

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L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Fourier Integral Operators, Springer-Verlag, Berin, 1985.  Google Scholar

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L. Hörmander, Linear Partial Differential Operators, Spring-Verlag, Berlin, 1963.  Google Scholar

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C. E. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

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C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273–1290. doi: 10.3934/dcds.2010.28.1273.  Google Scholar

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C.-L. Lin, G. Uhlmann and J.-N. Wang, Asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domain, Indiana Univ. Math. J., 60 (2011), 2093–2106. doi: 10.1512/iumj.2011.60.4490.  Google Scholar

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C.-L. Lin and J.-N. Wang, Quantitative uniqueness estimates for the general second order elliptic equations, J. Func. Anal., 266 (2014), 5108–5125. doi: 10.1016/j.jfa.2014.02.016.  Google Scholar

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R. Regbaoui, Strong unique continuation for Stokes equation, Comm. Partial Differential Equations, 24 (1999), 1891–1902. doi: 10.1080/03605309908821486.  Google Scholar

show all references

References:
[1]

J. Bourgain and C. E. Kenig, On localization in the Andersen-Bernoulli model in higher dimensions, Invent. Math., 161, (2005), 389–426. doi: 10.1007/s00222-004-0435-7.  Google Scholar

[2]

A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Am. J. Math., 80 (1958), 16–36. doi: 10.2307/2372819.  Google Scholar

[3]

T. Carleman, Sur ur problème d'unicité pur les systémes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 9 pp.  Google Scholar

[4]

R. H. Dyer and D. E. Edmunds, Asymptotic behavior of solutions of the stationary Navier-Stokes equations, J. London Math. Soc., 44 (1969), 340-346.  doi: 10.1112/jlms/s1-44.1.340.  Google Scholar

[5]

R. Finn, Stationary solutions of the Navier-Stokes equations, Proc. Symp. Appl. Math. Amer. Math. Soc., 17 (1965), 121–153. Google Scholar

[6]

X. Fu, Q. Lü and X. Zhang, Carleman Estimates for Second Order Partial Differential Operators and Applications, A unified approach, Springer, 2019. doi: 10.1007/978-3-030-29530-1.  Google Scholar

[7]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-differential Operators, Springer-Verlag, Berin, 1985.  Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Fourier Integral Operators, Springer-Verlag, Berin, 1985.  Google Scholar

[9]

L. Hörmander, Linear Partial Differential Operators, Spring-Verlag, Berlin, 1963.  Google Scholar

[10]

C. E. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[11]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273–1290. doi: 10.3934/dcds.2010.28.1273.  Google Scholar

[12]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domain, Indiana Univ. Math. J., 60 (2011), 2093–2106. doi: 10.1512/iumj.2011.60.4490.  Google Scholar

[13]

C.-L. Lin and J.-N. Wang, Quantitative uniqueness estimates for the general second order elliptic equations, J. Func. Anal., 266 (2014), 5108–5125. doi: 10.1016/j.jfa.2014.02.016.  Google Scholar

[14]

R. Regbaoui, Strong unique continuation for Stokes equation, Comm. Partial Differential Equations, 24 (1999), 1891–1902. doi: 10.1080/03605309908821486.  Google Scholar

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