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Time-dependent singularities in the Navier-Stokes system
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 |
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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