• Previous Article
    A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors
  • DCDS Home
  • This Issue
  • Next Article
    Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models
January  2020, 40(1): 189-206. doi: 10.3934/dcds.2020008

Singularities of certain finite energy solutions to the Navier-Stokes system

1. 

Instytut Matematyczny, Uniwersytet Wroclawski, pl.Gruwaldzki 2/4 Wroclaw, Poland

2. 

University of California, Department of Mathematics, Santa Cruz, CA 95064, USA

3. 

Florida Atlantic University, Department of Mathematical Sciences, Boca Raton, FL 33431, USA

* Corresponding author: Tomas P. Schonbek

Received  December 2018 Revised  June 2019 Published  October 2019

We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in $ \big(\gamma(t), t\big)\in \mathbb{R}^3\times [0, \infty) $ and singular on it. We revisit some of the existence results as well as some of the asymptotic estimates obtained in that work in order prove that those solutions belongs to the space $ C\big([0, \infty), L^2( \mathbb{R}^3)^3\big) $.

Citation: Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
References:
[1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, paperback ed., 1999. 
[2]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system?, J. Differential Equations, 197 (2004), 247-274.  doi: 10.1016/j.jde.2003.10.003.

[3]

A. Decaster and D. Iftimie, On the asymptotic behaviour of solutions of the stationary Navier-Stokes equations in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 277-291.  doi: 10.1016/j.anihpc.2015.12.002.

[4]

R. FarwigG. P. Galdi and M. Kyed, Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body, Pacific J. Math., 253 (2011), 367-382.  doi: 10.2140/pjm.2011.253.367.

[5]

V. A. Galaktionov, On blow-up "twistors" for the Navier–Stokes equations in $\mathbb{R}^3$: A view from reaction-diffusion theory, preprint, arXiv: 0901.4286.

[6]

K. KangH. Miura and T.-P. Tsai, Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data, Comm. Partial Differential Equations, 37 (2012), 1717-1753.  doi: 10.1080/03605302.2012.708082.

[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.

[8]

G. KarchD. Pilarczyk and M. E. Schonbek, L2-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.

[9]

G. Karch and X. Zheng, Time-dependent singularities in the Navier-Stokes system, Discrete Contin. Dyn. Syst., 35 (2015), 3039-3057.  doi: 10.3934/dcds.2015.35.3039.

[10]

A. Korolev and V. Šverák, On the large-distance asymptotics of steady state solutions of the Navier-Stokes equations in 3D exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 303-313.  doi: 10.1016/j.anihpc.2011.01.003.

[11]

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

[12] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[13]

H. Miura and T.-P. Tsai, Point singularities of 3D stationary Navier-Stokes flows, J. Math. Fluid Mech., 14 (2012), 33-41.  doi: 10.1007/s00021-010-0046-6.

[14]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748.  doi: 10.1016/j.jde.2008.09.004.

[15]

S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 313-331.  doi: 10.3934/dcds.2010.26.313.

[16]

S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906.  doi: 10.3934/dcdss.2011.4.897.

[17]

S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation, Commun. Pure Appl. Anal., 11 (2012), 387-405.  doi: 10.3934/cpaa.2012.11.387.

[18]

S. Sato and E. Yanagida, Asymptotic behavior of singular solutions for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 32 (2012), 4027-4043.  doi: 10.3934/dcds.2012.32.4027.

[19]

N. A. Slëzkin, On an integrability case of full differential equations of the motion of a viscous fluid, Moskov. Gos. Univ. Uč. Zap., 2 (1934), 89-90. 

[20]

V. Šverák, On Landau's solutions of the Navier-Stokes equations, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.

[21]

J. Takahashi and E. Yanagida, Time-dependent singularities in the heat equation, Commun. Pure Appl. Anal., 14 (2015), 969-979.  doi: 10.3934/cpaa.2015.14.969.

[22]

——, Time-dependent singularities in a semilinear parabolic equation with absorption, Commun. Contemp. Math., 18 (2016), 1550077, 27pp.

show all references

References:
[1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, paperback ed., 1999. 
[2]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system?, J. Differential Equations, 197 (2004), 247-274.  doi: 10.1016/j.jde.2003.10.003.

[3]

A. Decaster and D. Iftimie, On the asymptotic behaviour of solutions of the stationary Navier-Stokes equations in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 277-291.  doi: 10.1016/j.anihpc.2015.12.002.

[4]

R. FarwigG. P. Galdi and M. Kyed, Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body, Pacific J. Math., 253 (2011), 367-382.  doi: 10.2140/pjm.2011.253.367.

[5]

V. A. Galaktionov, On blow-up "twistors" for the Navier–Stokes equations in $\mathbb{R}^3$: A view from reaction-diffusion theory, preprint, arXiv: 0901.4286.

[6]

K. KangH. Miura and T.-P. Tsai, Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data, Comm. Partial Differential Equations, 37 (2012), 1717-1753.  doi: 10.1080/03605302.2012.708082.

[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.

[8]

G. KarchD. Pilarczyk and M. E. Schonbek, L2-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.

[9]

G. Karch and X. Zheng, Time-dependent singularities in the Navier-Stokes system, Discrete Contin. Dyn. Syst., 35 (2015), 3039-3057.  doi: 10.3934/dcds.2015.35.3039.

[10]

A. Korolev and V. Šverák, On the large-distance asymptotics of steady state solutions of the Navier-Stokes equations in 3D exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 303-313.  doi: 10.1016/j.anihpc.2011.01.003.

[11]

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

[12] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[13]

H. Miura and T.-P. Tsai, Point singularities of 3D stationary Navier-Stokes flows, J. Math. Fluid Mech., 14 (2012), 33-41.  doi: 10.1007/s00021-010-0046-6.

[14]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748.  doi: 10.1016/j.jde.2008.09.004.

[15]

S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 313-331.  doi: 10.3934/dcds.2010.26.313.

[16]

S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906.  doi: 10.3934/dcdss.2011.4.897.

[17]

S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation, Commun. Pure Appl. Anal., 11 (2012), 387-405.  doi: 10.3934/cpaa.2012.11.387.

[18]

S. Sato and E. Yanagida, Asymptotic behavior of singular solutions for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 32 (2012), 4027-4043.  doi: 10.3934/dcds.2012.32.4027.

[19]

N. A. Slëzkin, On an integrability case of full differential equations of the motion of a viscous fluid, Moskov. Gos. Univ. Uč. Zap., 2 (1934), 89-90. 

[20]

V. Šverák, On Landau's solutions of the Navier-Stokes equations, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.

[21]

J. Takahashi and E. Yanagida, Time-dependent singularities in the heat equation, Commun. Pure Appl. Anal., 14 (2015), 969-979.  doi: 10.3934/cpaa.2015.14.969.

[22]

——, Time-dependent singularities in a semilinear parabolic equation with absorption, Commun. Contemp. Math., 18 (2016), 1550077, 27pp.

[1]

Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039

[2]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222

[3]

Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065

[4]

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

[5]

Wei Wang, Jianliang Zhai, Tusheng Zhang. Large deviations for stochastic $ 2D $ Navier-Stokes equations on time-dependent domains. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3479-3498. doi: 10.3934/cpaa.2022111

[6]

Young-Pil Choi, Jinwook Jung. On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum. Kinetic and Related Models, 2022, 15 (5) : 843-891. doi: 10.3934/krm.2022016

[7]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044

[8]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062

[9]

Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312

[10]

Wei Shi, Xiaona Cui, Xuezhi Li, Xin-Guang Yang. Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5503-5534. doi: 10.3934/dcdsb.2021284

[11]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009

[12]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783

[13]

Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic and Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017

[14]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215

[15]

Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397

[16]

Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4347-4386. doi: 10.3934/dcdsb.2021231

[17]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

[18]

Eduard Marušić-Paloka, Igor Pažanin. Reaction of the fluid flow on time-dependent boundary perturbation. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1227-1246. doi: 10.3934/cpaa.2019059

[19]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239

[20]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (313)
  • HTML views (112)
  • Cited by (1)

[Back to Top]