×

On blow-up space jets for the Navier-Stokes equations in \(\mathbb R^{3}\) with convergence to Euler equations. (English) Zbl 1159.81322

Summary: Formation of blow-up singularities for the Navier-Stokes equations \[ \mathbf u_t+(\mathbf u\cdot\nabla)\mathbf u=-\nabla p+\Delta \mathbf u\quad\text{and}\quad \text{div}\,\mathbf u=0\;\text{in}\;\mathbb R^3\times(0,T) \] is studied. In cylindrical polar coordinates \(\{r,\varphi,z\}\) in \(\mathbb R^3\), their restriction to the linear subspace \(W_2=\text{Span}\{1,z\}\) is shown to be consistent. Using links with blow-up theory for nonlinear reaction-diffusion partial differential equations, the following questions are under scrutiny: (ii) introducing a self-similar blow-up “swirl mechanism” with the angular swirl divergences \(\varphi(t)=-\sigma \ln(T-t)\) and \(\varphi'(t)=(\sigma/(T-t))\to\infty\) as \(t\to T-\), where \(\sigma\in\mathbb R\) is a parameter; (ii) existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional/global blow-up of the \(z\) component of the velocity; (iii) as an intrinsic part of the construction, convergence of the above rescaled patterns as \(t\to T-\) to smooth blow-up self-similar solutions of the corresponding three-dimensional Euler equations \(\mathbf u_t+(\mathbf u\cdot\nabla)\mathbf u=-\nabla p\) and \(\text{div}\,\mathbf u=0\) in \(\mathbb R^3\times(0,T)\), which (iv) are also shown to admit single point blow-up in the similarity variables.
Editorial remark: No review copy delivered.

MSC:

35Q30 Navier-Stokes equations
35B44 Blow-up in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ames W. F., Symmetries, Exact Solutions and Conservation Laws 1, in: CRC Handbook of Lie Group Analysis of Differential Equations (1994)
[2] DOI: 10.1007/978-94-017-0745-9 · doi:10.1007/978-94-017-0745-9
[3] Bakirova M. I., Sov. Phys. Dokl. 33 pp 187– (1988)
[4] Berker, R.Handbook der Physik(Springer, Berlin, 1963), Vol. 8, pp. 1–384.
[5] Birman M. S., Spectral Theory of Self-Adjoint Operators in Hilbert Space (1987) · Zbl 0744.47017
[6] Bitev V. O., Prikl. Mekh. Tekh. Fiz. 6 pp 56– (1972)
[7] DOI: 10.1098/rspa.1998.0263 · Zbl 1023.35053 · doi:10.1098/rspa.1998.0263
[8] DOI: 10.1137/S003613990241552X · Zbl 1112.35095 · doi:10.1137/S003613990241552X
[9] Burgers J. M., Adv. Appl. Math. 1 pp 171– (1948)
[10] DOI: 10.1002/cpa.3160350604 · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[11] Dold J. W., Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 24 pp 663– (1998)
[12] Eidelman S. D., Parabolic Systems (1969)
[13] DOI: 10.1017/S0956792507006900 · Zbl 1221.35296 · doi:10.1017/S0956792507006900
[14] DOI: 10.1137/S0036141004440289 · Zbl 1110.35023 · doi:10.1137/S0036141004440289
[15] Fefferman C., Existence and Smoothness of the Navier–Stokes Equation · Zbl 1194.35002
[16] DOI: 10.1098/rspa.2000.0648 · Zbl 0988.35032 · doi:10.1098/rspa.2000.0648
[17] Friedman A., Partial Differential Equations (1983)
[18] DOI: 10.1098/rspa.2000.0733 · Zbl 0995.35009 · doi:10.1098/rspa.2000.0733
[19] DOI: 10.1002/mma.568 · Zbl 1065.35157 · doi:10.1002/mma.568
[20] DOI: 10.1201/9780203998069 · doi:10.1201/9780203998069
[21] DOI: 10.1016/S0022-0396(02)00151-1 · Zbl 1019.35042 · doi:10.1016/S0022-0396(02)00151-1
[22] Galaktionov V. A., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (2007) · Zbl 1153.35001
[23] DOI: 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R · doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R
[24] DOI: 10.1007/978-1-4612-2050-3 · doi:10.1007/978-1-4612-2050-3
[25] Galaktionov V. A., Adv. Differ. Equ. 4 pp 297– (1999)
[26] DOI: 10.1098/rsta.2002.1068 · Zbl 1048.35055 · doi:10.1098/rsta.2002.1068
[27] DOI: 10.1007/s002050200200 · Zbl 1042.37058 · doi:10.1007/s002050200200
[28] DOI: 10.1016/S0167-2789(99)00067-6 · Zbl 0956.76018 · doi:10.1016/S0167-2789(99)00067-6
[29] DOI: 10.1088/0951-7715/16/5/315 · Zbl 1040.35069 · doi:10.1088/0951-7715/16/5/315
[30] DOI: 10.1002/cpa.3160380304 · Zbl 0585.35051 · doi:10.1002/cpa.3160380304
[31] Herrero M. A., C. R. Acad. Sci., Ser. I: Math. 319 pp 141– (1994)
[32] DOI: 10.3934/dcds.2007.18.637 · Zbl 1194.35307 · doi:10.3934/dcds.2007.18.637
[33] Il’in A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (1992)
[34] DOI: 10.1002/zamm.19210010611 · doi:10.1002/zamm.19210010611
[35] DOI: 10.1002/cpa.3160140335 · Zbl 0107.20403 · doi:10.1002/cpa.3160140335
[36] DOI: 10.1017/S0022112005004179 · Zbl 1073.76018 · doi:10.1017/S0022112005004179
[37] Ladyzhenskaya O. A., Sov. Phys. Dokl. 123 pp 1128– (1958)
[38] Ladyzhenskaya O. A., Mathematical Problems of the Dynamics of Viscous Incompressible Flow (1961)
[39] Mathematical Problems of the Dynamics of Viscous Incompressible Flow, 2. ed. (1970)
[40] Mathematical Problems of the Dynamics of Viscous Incompressible Flow, 2. ed. (1969)
[41] Ladyzhenskaya O. A., Sem. Math. V.A. Steklov Math. Inst. Leningrad 7 pp 70– (1968)
[42] Ladyzhenskaya O. A., Linear and Quasilinear Equations of Parabolic Type (1967) · Zbl 0182.43204
[43] Landau L. D., Dokl. Akad. Nauk SSSR 43 pp 286– (1944)
[44] Landau L. D., Fluid Mechanics, 2. ed. (1987)
[45] Leray J., J. Math. Pures Appl. 12 pp 1– (1933)
[46] Leray J., Acad. Sci., Paris, C. R. 196 pp 527– (1933)
[47] DOI: 10.1007/BF02547354 · JFM 60.0726.05 · doi:10.1007/BF02547354
[48] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995) · Zbl 1261.35001 · doi:10.1007/978-3-0348-0557-5
[49] DOI: 10.1007/BF00381234 · Zbl 0708.76044 · doi:10.1007/BF00381234
[50] Majda A. J., Vosticity and Incompressible Flow (2002)
[51] DOI: 10.1002/cpa.20044 · Zbl 1112.35098 · doi:10.1002/cpa.20044
[52] DOI: 10.1016/j.jde.2004.03.001 · Zbl 1072.35043 · doi:10.1016/j.jde.2004.03.001
[53] DOI: 10.1016/j.jde.2006.04.002 · Zbl 1122.35038 · doi:10.1016/j.jde.2006.04.002
[54] DOI: 10.1017/S002211209900782X · Zbl 0962.76027 · doi:10.1017/S002211209900782X
[55] DOI: 10.1007/s00021-003-0087-1 · Zbl 1082.76026 · doi:10.1007/s00021-003-0087-1
[56] DOI: 10.1007/BF02551584 · Zbl 0884.35115 · doi:10.1007/BF02551584
[57] DOI: 10.2977/prims/1145475447 · Zbl 1070.76012 · doi:10.2977/prims/1145475447
[58] DOI: 10.1063/1.2347898 · Zbl 1144.81396 · doi:10.1063/1.2347898
[59] Oseen C. W., Ark. Mat., Astron. Fys. 6 pp 29– (1910)
[60] Oseen C. W., Ark. Mat., Astron. Fys. 20 pp 1– (1927)
[61] Pai S. I., Viscous Flow Theory: I–Laminar Flow (1965) · Zbl 0074.41603
[62] Panton R. L., Incompressible Flow (1984)
[63] DOI: 10.1007/978-1-4684-0392-3 · doi:10.1007/978-1-4684-0392-3
[64] DOI: 10.1016/j.physd.2005.03.010 · Zbl 1115.76010 · doi:10.1016/j.physd.2005.03.010
[65] Pukhnachev V. V., Uspehi Mehaniki 4 pp 6– (2006)
[66] DOI: 10.1515/9783110889864 · doi:10.1515/9783110889864
[67] DOI: 10.1137/060669838 · Zbl 1169.35366 · doi:10.1137/060669838
[68] Slezkin N. A., Uchen. Zapiski Moskov. Gosud. Universiteta, Gosud. Tehniko–Teoret. Izdat., Moskva/Leningrad pp 89– (1934)
[69] Slezkin N. A., Prikl. Mat. Mekh. 18 pp 764– (1954)
[70] DOI: 10.1007/s002050050099 · Zbl 0916.35084 · doi:10.1007/s002050050099
[71] DOI: 10.1016/0021-8928(68)90147-0 · doi:10.1016/0021-8928(68)90147-0
[72] DOI: 10.1512/iumj.1993.42.42021 · Zbl 0802.35073 · doi:10.1512/iumj.1993.42.42021
[73] DOI: 10.1146/annurev.fluid.23.1.159 · doi:10.1146/annurev.fluid.23.1.159
[74] DOI: 10.1002/mma.775 · Zbl 1184.35254 · doi:10.1002/mma.775
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.