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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 & Continuous Dynamical Systems - A, 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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

show all references

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

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