| Title:
|
On the search for singularities in incompressible flows (English) |
| Author:
|
Córdoba, Diego |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
299-320 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities. (English) |
| Keyword:
|
Navier-Stokes equations |
| Keyword:
|
singularities |
| Keyword:
|
incompressible flows |
| MSC:
|
35Q35 |
| MSC:
|
74H35 |
| MSC:
|
76D03 |
| idZBL:
|
Zbl 1164.76320 |
| idMR:
|
MR2291777 |
| DOI:
|
10.1007/s10778-006-0108-x |
| . |
| Date available:
|
2009-09-22T18:26:17Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134643 |
| . |
| Reference:
|
[1] J. T. Beale, T. Kato, and A. Majda: Remarks on the breakdown of smooth solutions for the 3D Euler equations.Commun. Math. Phys. 94 (1984), 61–66. MR 0763762, 10.1007/BF01212349 |
| Reference:
|
[2] A. L. Bertozzi, P. Constantin: Global regularity for vortex patches.Commun. Math. Phys. 152 (1993), 19–28. MR 1207667, 10.1007/BF02097055 |
| Reference:
|
[3] A. L. Bertozzi, A. J. Majda: Vorticity and the Mathematical Theory of Incompresible Fluid Flow. Cambridge Texts in Applied Mathematics No. 27.Cambridge University Press, Cambridge, 2002. MR 1867882 |
| Reference:
|
[4] D. Chae: On the Euler equations in the critical Triebel-Lizorkin spaces.Arch. Ration. Mech. Anal. 170 (2003), 185–210. Zbl 1093.76005, MR 2020259, 10.1007/s00205-003-0271-8 |
| Reference:
|
[5] D. Chae: The quasi-geostrophic equation in the Triebel-Lizorkin spaces.Nonlinearity 16 (2003), 479–495. Zbl 1029.35006, MR 1958612, 10.1088/0951-7715/16/2/307 |
| Reference:
|
[6] D. Chae, J. Lee: Global well-posedness in the super-critical dissipative quasi-geostrophic equations.Commun. Math. Phys. 233 (2003), 297–311. MR 1962043, 10.1007/s00220-002-0750-z |
| Reference:
|
[7] J. Y. Chemin: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels.Ann. Sci. Ec. Norm. Supér. 26 (1993), 517–542. (French) Zbl 0779.76011, MR 1235440, 10.24033/asens.1679 |
| Reference:
|
[8] P. Constantin: Energy spectrum of quasi-geostrophic turbulence.Phys. Rev. Lett. 89 (2002), 1804501–1804504. 10.1103/PhysRevLett.89.184501 |
| Reference:
|
[9] P. Constantin, D. Córdoba, and J. Wu: On the critical dissipative quasi-geostrophic equation.Indiana Univ. Math. J. 50 (2001), 97–107. MR 1855665, 10.1512/iumj.2001.50.2153 |
| Reference:
|
[10] P. Constantin, C. Fefferman, and A. J. Majda: Geometric constraints on potentially singular solutions for the 3-D Euler equations.Commun. Partial Differ. Equation 21 (1996), 559–571. MR 1387460, 10.1080/03605309608821197 |
| Reference:
|
[11] P. Constantin, A. J. Majda, and E. Tabak: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar.Nonlinearity 7 (1994), 1495–1533. MR 1304437, 10.1088/0951-7715/7/6/001 |
| Reference:
|
[12] P. Constantin, Q. Nie, and N. Schorghofer: Nonsingular surface-quasi-geostrophic flow.Phys. Lett. A 241 (1998), 168–172. MR 1613907, 10.1016/S0375-9601(98)00108-X |
| Reference:
|
[13] P. Constantin, J. Wu: Behavior of solutions of 2D quasi-geostrophic equations.SIAM J. Math. Anal. 30 (1999), 937–948. Zbl 0957.76093, MR 1709781, 10.1137/S0036141098337333 |
| Reference:
|
[14] A. Córdoba, D. Córdoba: A pointwise estimate for fractionary derivatives with applications to partial differential equations.Proc. Natl. Acad. Sci. USA 100 (2003), 15316–15317. MR 2032097, 10.1073/pnas.2036515100 |
| Reference:
|
[15] A. Córdoba, D. Córdoba: A maximum principle applied to quasi-geostrophic equations.Commun. Math. Phys. 249 (2004), 511–528. MR 2084005, 10.1007/s00220-004-1055-1 |
| Reference:
|
[16] A. Córdoba, D. Córdoba, C. L. Fefferman, and M. A. Fontelos: A geometrical constraint for capillary jet breakup.Adv. Math. 187 (2004), 228–239. MR 2074177, 10.1016/j.aim.2003.08.009 |
| Reference:
|
[17] D. Córdoba: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation.Ann. Math. 148 (1998), 1135–1152. MR 1670077, 10.2307/121037 |
| Reference:
|
[18] D. Córdoba, C. Fefferman: On the collapse of tubes carried by 3D incompressible flows.Commun. Math. Phys. 222 (2001), 293–298. MR 1859600, 10.1007/s002200100502 |
| Reference:
|
[19] D. Córdoba, C. Fefferman, and R. de la Llave: On squirt singularities in hydrodynamics.SIAM J. Math. Anal. 36 (2004), 204–213. MR 2083858, 10.1137/S0036141003424095 |
| Reference:
|
[20] D. Córdoba, C. Fefferman, and J. L. Rodrigo: Almost sharp fronts for the surface quasi-geostrophic equations.Proc. Natl. Acad. Sci. USA 101 (2004), 2687–2691. MR 2036970, 10.1073/pnas.0308154101 |
| Reference:
|
[21] D. Córdoba, M. Fontelos, A. Mancho, and J. L. Rodrigo: Evidence of singularities for a family of contour dynamics equations.Proc. Natl. Acad. Sci. USA 102 (2005), 5949–5952. MR 2141918, 10.1073/pnas.0501977102 |
| Reference:
|
[22] R. J. Diperna, P. L. Lions: Ordinary differential equations, transport theory and Sobolev spaces.Invent. Math. 98 (1989), 511–547. MR 1022305, 10.1007/BF01393835 |
| Reference:
|
[23] C. R. Doering, J. D. Gibbon: Applied analysis of the Navier Stokes equations.Cambridge University Press, Cambridge, 1995. MR 1325465 |
| Reference:
|
[24] C. Foias, C. Guillopé, and R. Témam: New a priori estimates for Navier-Stokes equations in dimension 3.Commun. Partial Differ. Equations 6 (1981), 329–359. MR 0607552, 10.1080/03605308108820180 |
| Reference:
|
[25] I. M. Held, R. Pierrehumbert, and S. T. Garner: Surface quasi-geostrophic dynamics.J. Fluid Mech. 282 (1995), 1–20. MR 1312238, 10.1017/S0022112095000012 |
| Reference:
|
[26] N. Ju: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations.Commun. Math. Phys. 255 (2005), 161–181. Zbl 1088.37049, MR 2123380, 10.1007/s00220-004-1256-7 |
| Reference:
|
[27] H. Kozono, Y. Taniuchi: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations.Commun. Math. Phys. 214 (2000), 191–200. MR 1794270, 10.1007/s002200000267 |
| Reference:
|
[28] T. A. Kowalewski: On the separation of droplets from a liquid jet.Fluid Dyn. Res. 17 (1996), 121–145. |
| Reference:
|
[29] K. Ohkitani, M. Yamada: Inviscid and inviscid-limit behavior of a surface quasi-geostrophic flow.Phys. Fluids 9 (1997), 876–882. MR 1437554, 10.1063/1.869184 |
| Reference:
|
[30] J. Pedlosky: Geophysical Fluid Dynamics.Springer-Verlag, New York, 1987. Zbl 0713.76005 |
| Reference:
|
[31] M. T. Plateau: Smithsonian Report 250.1863. |
| Reference:
|
[32] Rayleigh, Lord (J. W. Strutt): On the instability of jets.Proc. L. M. S. 10 (1879), 4–13. |
| Reference:
|
[33] S. Resnick: Dynamical problem in nonlinear advective partial differential equations.PhD. Thesis, University of Chicago, 1995. |
| Reference:
|
[34] J. L. Rodrigo: On the evolution of sharp fronts for the quasi-geostrophic equation.Commun. Pure Appl. Math. 58 (2005), 821–866. Zbl 1073.35006, MR 2142632, 10.1002/cpa.20059 |
| Reference:
|
[35] R. Salmon: Lectures on Geophysical Fluid Dynamics.Oxford University Press, New York, 1998. MR 1718369 |
| Reference:
|
[36] F. Savart: Mémoire sur la Constitution des veines liquides lancées par des orifices circulaires en mince paroi.Ann. Chim. Phys. 53 (1833), 337–386. (French) |
| Reference:
|
[37] X. D. Shi, M. P. Brenner, and S. R. Nagel: A cascade of structure in a drop falling from a faucet.Science 265 (1994), 219–222. MR 1282463, 10.1126/science.265.5169.219 |
| Reference:
|
[38] E. M. Stein: Singular Integrals and Differentiability Properties of Functions.Princeton University Press, Princeton, 1970. Zbl 0207.13501, MR 0290095 |
| Reference:
|
[39] E. M. Stein: Harmonic Analysis.Princeton University Press, Princeton, 1993. Zbl 0821.42001, MR 1232192 |
| Reference:
|
[40] M. Sussman, P. Smereka: Axisymmetric free boundary problems.J. Fluid Mech. 341 (1997), 269–294. MR 1457712, 10.1017/S0022112097005570 |
| Reference:
|
[41] L. Tartar: Topics in Nonlinear Analysis. Publications Mat. D’Orsay, No. 7813.Univ. de Paris-Sud, Orsay, 1978. MR 0532371 |
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