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  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 - A, doi: 10.3934/dcds.2020366
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

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

[1]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[2]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[3]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[4]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[5]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[6]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[7]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[8]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[9]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[10]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

[11]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[12]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[13]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[14]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[15]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[16]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[17]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[18]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[19]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[20]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (19)
  • HTML views (45)
  • Cited by (0)

Other articles
by authors

[Back to Top]