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September  2021, 29(4): 2719-2739. doi: 10.3934/era.2021010

Local well-posedness of perturbed Navier-Stokes system around Landau solutions

Jingjing Zhang and  Ting Zhang ,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Ting Zhang

Received  December 2020 Published  September 2021 Early access  February 2021

For the incompressible Navier-Stokes system, when initial data are uniformly locally square integral, the local existence of solutions has been obtained. In this paper, we consider perturbed system and show that perturbed solutions of Landau solutions to the Navier-Stokes system exist locally under $ L^q_{\text{uloc}} $-perturbations, $ q\geq 2 $. Furthermore, when $ q\geq 3, $ the solution is well-posed. Precisely, we give the explicit formula of the pressure term.

Keywords: Local well-posedness, Navier-Stokes system, Landau solutions.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 49K40.
Citation: Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29 (4) : 2719-2739. doi: 10.3934/era.2021010
References:
[1]

A. Basson, Solutions Spatialement Homog$\grave{e}$nes Adapt$\acute{e}$es des $\acute{e}$quations de Navier-Stokes, Thesis. University of Evry., 2006. Google Scholar

[2]

Z. Bradshaw and T.-P. Tsai, Self-similar solutions to the Navier-Stokes equations: A survey of recent results, Nonlinear Analysis in Geometry and Applied Mathematics, 2 (2018), 159-181.   Google Scholar

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J. L. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $ L^3_{\rm uloc } (\Bbb{R}^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218.  doi: 10.1007/s00205-013-0643-7.  Google Scholar

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G. Karch and D. Pilarczyk, Asymptotic stability of Landau solutions to Navier-Stokes system, Arch. Ration. Mech. Anal., 202 (2011), 115-131.  doi: 10.1007/s00205-011-0409-z.  Google Scholar

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G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of singular solutions to the Navier-Stokes system of equations in $\mathbb{R}^3$, J. Math. Pures Appl., 108 (2017), 14-40.  doi: 10.1016/j.matpur.2016.10.008.  Google Scholar

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N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2,220, Amer. Math. Soc., Providence, RI, 220 (2007), 141–164. doi: 10.1090/trans2/220/07.  Google Scholar

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H. Koch and D. Tataru, Well posednesss for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar

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L. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.   Google Scholar

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J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[17]

L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅰ. One singularity, Arch. Ration. Mech. Anal., 227 (2018), 1091-1163.  doi: 10.1007/s00205-017-1181-5.  Google Scholar

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L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅱ. Classification of axisymmetric no-swirl solutions, J. Differential Equations, 264 (2018), 6082-6108.  doi: 10.1016/j.jde.2018.01.028.  Google Scholar

[19]

L. Li, Y. Y. Li and X. Yan, Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations, J. Funct. Anal., 277 (2019), 3599-3652.  doi: 10.1016/j.jfa.2019.05.022.  Google Scholar

[20]

L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities, Discrete Contin. Dyn. Syst., 39 (2019), 7163-7211.  doi: 10.3934/dcds.2019300.  Google Scholar

[21]

V. Šverák, On Landau's solutions of the Navier-Stokes equations, Problems in mathematical analysis, No. 61, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.  Google Scholar

[22]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456.  doi: 10.1080/03605309208820892.  Google Scholar

[23]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.  doi: 10.12775/TMNA.1998.008.  Google Scholar

[24]

T.-P. Tsai, Forward discretely self-similar solutions of the Navier-Stokes equations, Comm. Math. Phys., 328 (2014), 29-44.  doi: 10.1007/s00220-014-1984-2.  Google Scholar

[25]

T.-P. Tsai, Lectures on Navier-Stokes Equations, American Mathematical Society, Providence, RI, vol. 192, 2018. doi: 10.1090/gsm/192.  Google Scholar

[26]

J. Zhang and T. Zhang, Global existence of discretely self-similar solutions to the generalized MHD system in Besov space, J. Math. Phys., 60 (2019), 081515, 18 pp. doi: 10.1063/1.5092787.  Google Scholar

[27]

N. Zhao, A Liouville theorem for axially symmetric $D$-solutions to steady Navier-Stokes equations, Nonlinear Anal., 187 (2019), 247-258.  doi: 10.1016/j.na.2019.04.018.  Google Scholar

show all references

References:
[1]

A. Basson, Solutions Spatialement Homog$\grave{e}$nes Adapt$\acute{e}$es des $\acute{e}$quations de Navier-Stokes, Thesis. University of Evry., 2006. Google Scholar

[2]

Z. Bradshaw and T.-P. Tsai, Self-similar solutions to the Navier-Stokes equations: A survey of recent results, Nonlinear Analysis in Geometry and Applied Mathematics, 2 (2018), 159-181.   Google Scholar

[3]

Y. Giga and T. Miyakawa, Navier-Stokes flow in ${\bf R}^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618.  doi: 10.1080/03605308908820621.  Google Scholar

[4]

G. H. Hardy, Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.  doi: 10.1007/BF01199965.  Google Scholar

[5]

G. H. Hardy, An inequality between integrals, Messenger of Mathematics, 54 (1925), 150-156.   Google Scholar

[6]

J. L. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $ L^3_{\rm uloc } (\Bbb{R}^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218.  doi: 10.1007/s00205-013-0643-7.  Google Scholar

[7]

G. Karch and D. Pilarczyk, Asymptotic stability of Landau solutions to Navier-Stokes system, Arch. Ration. Mech. Anal., 202 (2011), 115-131.  doi: 10.1007/s00205-011-0409-z.  Google Scholar

[8]

G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of singular solutions to the Navier-Stokes system of equations in $\mathbb{R}^3$, J. Math. Pures Appl., 108 (2017), 14-40.  doi: 10.1016/j.matpur.2016.10.008.  Google Scholar

[9]

T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in R$^{m}$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.  Google Scholar

[10]

N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2,220, Amer. Math. Soc., Providence, RI, 220 (2007), 141–164. doi: 10.1090/trans2/220/07.  Google Scholar

[11]

H. Koch and D. Tataru, Well posednesss for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar

[12]

H. Kwon and T.-P. Tsai, Global Navier-Stokes flows for non-decaying initial data with slowly decaying oscillation, Comm. Math. Phys., 375 (2020), 1665-1715.  doi: 10.1007/s00220-020-03695-3.  Google Scholar

[13]

L. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.   Google Scholar

[14]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, 2002. doi: 10.1201/9781420035674.  Google Scholar

[15] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.  Google Scholar
[16]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[17]

L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅰ. One singularity, Arch. Ration. Mech. Anal., 227 (2018), 1091-1163.  doi: 10.1007/s00205-017-1181-5.  Google Scholar

[18]

L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅱ. Classification of axisymmetric no-swirl solutions, J. Differential Equations, 264 (2018), 6082-6108.  doi: 10.1016/j.jde.2018.01.028.  Google Scholar

[19]

L. Li, Y. Y. Li and X. Yan, Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations, J. Funct. Anal., 277 (2019), 3599-3652.  doi: 10.1016/j.jfa.2019.05.022.  Google Scholar

[20]

L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities, Discrete Contin. Dyn. Syst., 39 (2019), 7163-7211.  doi: 10.3934/dcds.2019300.  Google Scholar

[21]

V. Šverák, On Landau's solutions of the Navier-Stokes equations, Problems in mathematical analysis, No. 61, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.  Google Scholar

[22]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456.  doi: 10.1080/03605309208820892.  Google Scholar

[23]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.  doi: 10.12775/TMNA.1998.008.  Google Scholar

[24]

T.-P. Tsai, Forward discretely self-similar solutions of the Navier-Stokes equations, Comm. Math. Phys., 328 (2014), 29-44.  doi: 10.1007/s00220-014-1984-2.  Google Scholar

[25]

T.-P. Tsai, Lectures on Navier-Stokes Equations, American Mathematical Society, Providence, RI, vol. 192, 2018. doi: 10.1090/gsm/192.  Google Scholar

[26]

J. Zhang and T. Zhang, Global existence of discretely self-similar solutions to the generalized MHD system in Besov space, J. Math. Phys., 60 (2019), 081515, 18 pp. doi: 10.1063/1.5092787.  Google Scholar

[27]

N. Zhao, A Liouville theorem for axially symmetric $D$-solutions to steady Navier-Stokes equations, Nonlinear Anal., 187 (2019), 247-258.  doi: 10.1016/j.na.2019.04.018.  Google Scholar

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