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 & 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*
[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

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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