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July  2015, 35(7): 3039-3057. doi: 10.3934/dcds.2015.35.3039

Time-dependent singularities in the Navier-Stokes system

Grzegorz Karch 1, and  Xiaoxin Zheng 2,

1. 

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław
2. 

Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received  April 2014 Revised  October 2014 Published  January 2015

We show that, for a given Hölder continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset \mathbb{R}^3 \times \mathbb{R}^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $\mathbb{R}^3$ which is regular outside this curve and singular on it. This is a solution of the homogeneous system outside the curve, however, as a distributional solution on $\mathbb{R}^3 \times \mathbb{R}^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.
Keywords: time-dependent singularity, Navier-Stokes system, incompressible fluid, Slezkin-Landau solutions..
Mathematics Subject Classification: Primary: 35Q40; Secondary: 76D0.
Citation: Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
References:
[1]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, (1999).   Google Scholar

[2]

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

[3]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, in Handbook of Mathematical Fluid Dynamics. Vol. III (eds. J. Friedlander, (2004), 161.   Google Scholar

[4]

H. J. Choe and H. Kim, Isolated singularity for the stationary Navier-Stokes system,, J. Math. Fluid Mech., 2 (2000), 151.  doi: 10.1007/PL00000951.  Google Scholar

[5]

R. Farwig, G. 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.  doi: 10.2140/pjm.2011.253.367.  Google Scholar

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V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbbR^3$: A view from reaction-diffusion theory,, , ().   Google Scholar

[7]

K. Hirata, Removable singularities of semilinear parabolic equations,, Proc. Amer. Math. Soc., 142 (2014), 157.  doi: 10.1090/S0002-9939-2013-11739-9.  Google Scholar

[8]

S. Y. Hsu, Removable singularites of semilinear parabolic equations,, Adv. Differential Equations, 15 (2010), 137.   Google Scholar

[9]

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

[10]

G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of mild solutions to Navier-Stokes system in $\mathbbR^3$,, , ().   Google Scholar

[11]

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

[12]

H. Kim and H. Kozono, A removable isolated singularity theorem for the stationary Navier-Stokes equations,, J. Differential Equations, 220 (2006), 68.  doi: 10.1016/j.jde.2005.02.002.  Google Scholar

[13]

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.  doi: 10.1016/j.anihpc.2011.01.003.  Google Scholar

[14]

H. Kozono, Removable singularities of weak solutions to the Navier-Stokes equations,, Comm. Partial Differential Equations, 23 (1998), 949.  doi: 10.1080/03605309808821374.  Google Scholar

[15]

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

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (in Russian),, Nauka, (1986).   Google Scholar

[17]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Press, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[18]

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

[19]

F. Lin, A New Proof of the Caffarelli-Kohn-Nirenberg Theorem,, Comm. Pure Appl. Math., 51 (1998), 241.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[20]

Y. Luo and T. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations,, , ().   Google Scholar

[21]

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

[22]

R. O'Neil, Convolution operators and $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[23]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,, Academic Press, (1975).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.  doi: 10.3934/dcds.2010.26.313.  Google Scholar

[29]

N. A. Slezkin, On one integrabile case for the complete differential equations of the motion of a viscous fluid,, Uchen. Zapiski Moskov. Gosud. Universiteta, 11 (1934), 89.   Google Scholar

[30]

H. B. Squire, The round laminar jet,, Quart. J. Mech. Appl. Math., 4 (1951), 321.  doi: 10.1093/qjmam/4.3.321.  Google Scholar

[31]

V. Šverák, On Landau's solutions of the Navier-Stokes equations,, Problems in mathematical analysis, 179 (2011), 208.  doi: 10.1007/s10958-011-0590-5.  Google Scholar

[32]

J. Takahashi and E. Yanagida, Removability of time-dependent singularities in the heat equation,, , ().   Google Scholar

[33]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations,, Topol. Meth. Nonlinear Anal., 11 (1998), 135.   Google Scholar

show all references

References:
[1]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, (1999).   Google Scholar

[2]

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

[3]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, in Handbook of Mathematical Fluid Dynamics. Vol. III (eds. J. Friedlander, (2004), 161.   Google Scholar

[4]

H. J. Choe and H. Kim, Isolated singularity for the stationary Navier-Stokes system,, J. Math. Fluid Mech., 2 (2000), 151.  doi: 10.1007/PL00000951.  Google Scholar

[5]

R. Farwig, G. 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.  doi: 10.2140/pjm.2011.253.367.  Google Scholar

[6]

V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbbR^3$: A view from reaction-diffusion theory,, , ().   Google Scholar

[7]

K. Hirata, Removable singularities of semilinear parabolic equations,, Proc. Amer. Math. Soc., 142 (2014), 157.  doi: 10.1090/S0002-9939-2013-11739-9.  Google Scholar

[8]

S. Y. Hsu, Removable singularites of semilinear parabolic equations,, Adv. Differential Equations, 15 (2010), 137.   Google Scholar

[9]

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

[10]

G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of mild solutions to Navier-Stokes system in $\mathbbR^3$,, , ().   Google Scholar

[11]

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

[12]

H. Kim and H. Kozono, A removable isolated singularity theorem for the stationary Navier-Stokes equations,, J. Differential Equations, 220 (2006), 68.  doi: 10.1016/j.jde.2005.02.002.  Google Scholar

[13]

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.  doi: 10.1016/j.anihpc.2011.01.003.  Google Scholar

[14]

H. Kozono, Removable singularities of weak solutions to the Navier-Stokes equations,, Comm. Partial Differential Equations, 23 (1998), 949.  doi: 10.1080/03605309808821374.  Google Scholar

[15]

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

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (in Russian),, Nauka, (1986).   Google Scholar

[17]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Press, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[18]

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

[19]

F. Lin, A New Proof of the Caffarelli-Kohn-Nirenberg Theorem,, Comm. Pure Appl. Math., 51 (1998), 241.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[20]

Y. Luo and T. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations,, , ().   Google Scholar

[21]

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

[22]

R. O'Neil, Convolution operators and $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[23]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,, Academic Press, (1975).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.  doi: 10.3934/dcds.2010.26.313.  Google Scholar

[29]

N. A. Slezkin, On one integrabile case for the complete differential equations of the motion of a viscous fluid,, Uchen. Zapiski Moskov. Gosud. Universiteta, 11 (1934), 89.   Google Scholar

[30]

H. B. Squire, The round laminar jet,, Quart. J. Mech. Appl. Math., 4 (1951), 321.  doi: 10.1093/qjmam/4.3.321.  Google Scholar

[31]

V. Šverák, On Landau's solutions of the Navier-Stokes equations,, Problems in mathematical analysis, 179 (2011), 208.  doi: 10.1007/s10958-011-0590-5.  Google Scholar

[32]

J. Takahashi and E. Yanagida, Removability of time-dependent singularities in the heat equation,, , ().   Google Scholar

[33]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations,, Topol. Meth. Nonlinear Anal., 11 (1998), 135.   Google Scholar

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