Time-dependent singularities in the Navier-Stokes system

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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

Abstract
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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 and Continuous Dynamical Systems, 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, Cambridge, 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]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics. Vol. III (eds. J. Friedlander, D. Serre), North-Holland, Amsterdam, 2004, 161-244.
[4]	H. J. Choe and H. Kim, Isolated singularity for the stationary Navier-Stokes system, J. Math. Fluid Mech., 2 (2000), 151-184. doi: 10.1007/PL00000951.
[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), 367-382. doi: 10.2140/pjm.2011.253.367.
[6]	V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbb{R}^3$: A view from reaction-diffusion theory, arXiv:0901.4286.
[7]	K. Hirata, Removable singularities of semilinear parabolic equations, Proc. Amer. Math. Soc., 142 (2014), 157-171. doi: 10.1090/S0002-9939-2013-11739-9.
[8]	S. Y. Hsu, Removable singularites of semilinear parabolic equations, Adv. Differential Equations, 15 (2010), 137-158.
[9]	G. Karch and D. Pilarczyk, Asymptotic stability of Landau solutions to Navier-Stokes system, Arch. Rational Mech. Anal., 202 (2011), 115-131. doi: 10.1007/s00205-011-0409-z.
[10]	G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of mild solutions to Navier-Stokes system in $\mathbb{R}^3$, arXiv:1308.6667v1.
[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), 1717-1753. doi: 10.1080/03605302.2012.708082.
[12]	H. Kim and H. Kozono, A removable isolated singularity theorem for the stationary Navier-Stokes equations, J. Differential Equations, 220 (2006), 68-84. doi: 10.1016/j.jde.2005.02.002.
[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), 303-313. doi: 10.1016/j.anihpc.2011.01.003.
[14]	H. Kozono, Removable singularities of weak solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 23 (1998), 949-966. doi: 10.1080/03605309808821374.
[15]	L. D. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.
[16]	L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (in Russian), Nauka, Moscow, 1986; English translation by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.
[17]	P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Press, Boca Raton, 2002. doi: 10.1201/9781420035674.
[18]	J. Leray, Sur le mouvement d'un liquide visqeux emplissant l'space, Acta. Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[19]	F. Lin, A New Proof of the Caffarelli-Kohn-Nirenberg Theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.
[20]	Y. Luo and T. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations, arXiv:1310.8307v2.
[21]	H. Miura and T. Tsai, Point singularities of 3D stationary Navier-Stokes flows, J. Math. Fluid Mech., 14 (2012), 33-41. doi: 10.1007/s00021-010-0046-6.
[22]	R. O'Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.
[23]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-London, 1975.
[24]	S. Sato and E. Yanagida, Solutions with moving singularities for a semlinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: 10.1016/j.jde.2008.09.004.
[25]	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.
[26]	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.
[27]	S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906. doi: 10.3934/dcdss.2011.4.897.
[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), 313-331. doi: 10.3934/dcds.2010.26.313.
[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), 89-90.
[30]	H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329. doi: 10.1093/qjmam/4.3.321.
[31]	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.
[32]	J. Takahashi and E. Yanagida, Removability of time-dependent singularities in the heat equation, arXiv:1304.5147v2.
[33]	G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Meth. Nonlinear Anal., 11 (1998), 135-145.

show all references

References:

[1]	G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 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]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics. Vol. III (eds. J. Friedlander, D. Serre), North-Holland, Amsterdam, 2004, 161-244.
[4]	H. J. Choe and H. Kim, Isolated singularity for the stationary Navier-Stokes system, J. Math. Fluid Mech., 2 (2000), 151-184. doi: 10.1007/PL00000951.
[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), 367-382. doi: 10.2140/pjm.2011.253.367.
[6]	V. A. Galaktionov, On blow-up "twistors" for the Navier-Stokes equations in $\mathbb{R}^3$: A view from reaction-diffusion theory, arXiv:0901.4286.
[7]	K. Hirata, Removable singularities of semilinear parabolic equations, Proc. Amer. Math. Soc., 142 (2014), 157-171. doi: 10.1090/S0002-9939-2013-11739-9.
[8]	S. Y. Hsu, Removable singularites of semilinear parabolic equations, Adv. Differential Equations, 15 (2010), 137-158.
[9]	G. Karch and D. Pilarczyk, Asymptotic stability of Landau solutions to Navier-Stokes system, Arch. Rational Mech. Anal., 202 (2011), 115-131. doi: 10.1007/s00205-011-0409-z.
[10]	G. Karch, D. Pilarczyk and M. E. Schonbek, $L^2$-asymptotic stability of mild solutions to Navier-Stokes system in $\mathbb{R}^3$, arXiv:1308.6667v1.
[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), 1717-1753. doi: 10.1080/03605302.2012.708082.
[12]	H. Kim and H. Kozono, A removable isolated singularity theorem for the stationary Navier-Stokes equations, J. Differential Equations, 220 (2006), 68-84. doi: 10.1016/j.jde.2005.02.002.
[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), 303-313. doi: 10.1016/j.anihpc.2011.01.003.
[14]	H. Kozono, Removable singularities of weak solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 23 (1998), 949-966. doi: 10.1080/03605309808821374.
[15]	L. D. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.
[16]	L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (in Russian), Nauka, Moscow, 1986; English translation by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.
[17]	P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Press, Boca Raton, 2002. doi: 10.1201/9781420035674.
[18]	J. Leray, Sur le mouvement d'un liquide visqeux emplissant l'space, Acta. Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[19]	F. Lin, A New Proof of the Caffarelli-Kohn-Nirenberg Theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.
[20]	Y. Luo and T. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations, arXiv:1310.8307v2.
[21]	H. Miura and T. Tsai, Point singularities of 3D stationary Navier-Stokes flows, J. Math. Fluid Mech., 14 (2012), 33-41. doi: 10.1007/s00021-010-0046-6.
[22]	R. O'Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.
[23]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-London, 1975.
[24]	S. Sato and E. Yanagida, Solutions with moving singularities for a semlinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: 10.1016/j.jde.2008.09.004.
[25]	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.
[26]	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.
[27]	S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906. doi: 10.3934/dcdss.2011.4.897.
[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), 313-331. doi: 10.3934/dcds.2010.26.313.
[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), 89-90.
[30]	H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329. doi: 10.1093/qjmam/4.3.321.
[31]	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.
[32]	J. Takahashi and E. Yanagida, Removability of time-dependent singularities in the heat equation, arXiv:1304.5147v2.
[33]	G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Meth. Nonlinear Anal., 11 (1998), 135-145.

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