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

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

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.

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

[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. KarchD. 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. LiY. 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. LiY. 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. LiY. 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. LiY. 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. KarchD. 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. LiY. 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. LiY. 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. LiY. 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. LiY. 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

[1]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[2]

Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158

[3]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[4]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517

[5]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143

[6]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[7]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[8]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005

[9]

Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3545-3565. doi: 10.3934/cpaa.2021121

[10]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148

[11]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[12]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845

[13]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[14]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[15]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

[16]

Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021147

[17]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[18]

Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021146

[19]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019

[20]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

2020 Impact Factor: 1.833

Metrics

  • PDF downloads (92)
  • HTML views (306)
  • Cited by (0)

Other articles
by authors

[Back to Top]