Previous |  Up |  Next

Article

Keywords:
Navier-Stokes equations; singularities; incompressible flows
Summary:
In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.
References:
[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. DOI 10.1007/BF01212349 | MR 0763762
[2] A. L.  Bertozzi, P.  Constantin: Global regularity for vortex patches. Commun. Math. Phys. 152 (1993), 19–28. DOI 10.1007/BF02097055 | MR 1207667
[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
[4] D.  Chae: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal. 170 (2003), 185–210. DOI 10.1007/s00205-003-0271-8 | MR 2020259 | Zbl 1093.76005
[5] D.  Chae: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16 (2003), 479–495. DOI 10.1088/0951-7715/16/2/307 | MR 1958612 | Zbl 1029.35006
[6] D.  Chae, J.  Lee: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233 (2003), 297–311. DOI 10.1007/s00220-002-0750-z | MR 1962043
[7] J. Y.  Chemin: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. Ec. Norm. Supér. 26 (1993), 517–542. (French) DOI 10.24033/asens.1679 | MR 1235440 | Zbl 0779.76011
[8] P.  Constantin: Energy spectrum of quasi-geostrophic turbulence. Phys. Rev. Lett. 89 (2002), 1804501–1804504. DOI 10.1103/PhysRevLett.89.184501
[9] P. Constantin, D. Córdoba, and J. Wu: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50 (2001), 97–107. DOI 10.1512/iumj.2001.50.2153 | MR 1855665
[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. DOI 10.1080/03605309608821197 | MR 1387460
[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. DOI 10.1088/0951-7715/7/6/001 | MR 1304437
[12] P. Constantin, Q.  Nie, and N. Schorghofer: Nonsingular surface-quasi-geostrophic flow. Phys. Lett.  A 241 (1998), 168–172. DOI 10.1016/S0375-9601(98)00108-X | MR 1613907
[13] P. Constantin, J. Wu: Behavior of solutions of 2D  quasi-geostrophic equations. SIAM J.  Math. Anal. 30 (1999), 937–948. DOI 10.1137/S0036141098337333 | MR 1709781 | Zbl 0957.76093
[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. DOI 10.1073/pnas.2036515100 | MR 2032097
[15] A.  Córdoba, D.  Córdoba: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249 (2004), 511–528. DOI 10.1007/s00220-004-1055-1 | MR 2084005
[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. DOI 10.1016/j.aim.2003.08.009 | MR 2074177
[17] D.  Córdoba: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math. 148 (1998), 1135–1152. DOI 10.2307/121037 | MR 1670077
[18] D.  Córdoba, C.  Fefferman: On the collapse of tubes carried by 3D  incompressible flows. Commun. Math. Phys. 222 (2001), 293–298. DOI 10.1007/s002200100502 | MR 1859600
[19] D. Córdoba, C. Fefferman, and R. de la  Llave: On squirt singularities in hydrodynamics. SIAM J.  Math. Anal. 36 (2004), 204–213. DOI 10.1137/S0036141003424095 | MR 2083858
[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. DOI 10.1073/pnas.0308154101 | MR 2036970
[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. DOI 10.1073/pnas.0501977102 | MR 2141918
[22] R. J.  Diperna, P. L. Lions: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547. DOI 10.1007/BF01393835 | MR 1022305
[23] C. R.  Doering, J. D.  Gibbon: Applied analysis of the Navier Stokes equations. Cambridge University Press, Cambridge, 1995. MR 1325465
[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. DOI 10.1080/03605308108820180 | MR 0607552
[25] I. M. Held, R. Pierrehumbert, and S. T.  Garner: Surface quasi-geostrophic dynamics. J.  Fluid Mech. 282 (1995), 1–20. DOI 10.1017/S0022112095000012 | MR 1312238
[26] N. Ju: The maximum principle and the global attractor for the dissipative 2D  quasi-geostrophic equations. Commun. Math. Phys. 255 (2005), 161–181. DOI 10.1007/s00220-004-1256-7 | MR 2123380 | Zbl 1088.37049
[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. DOI 10.1007/s002200000267 | MR 1794270
[28] T. A.  Kowalewski: On the separation of droplets from a liquid jet. Fluid Dyn. Res. 17 (1996), 121–145.
[29] K. Ohkitani, M. Yamada: Inviscid and inviscid-limit behavior of a surface quasi-geostrophic flow. Phys. Fluids 9 (1997), 876–882. DOI 10.1063/1.869184 | MR 1437554
[30] J. Pedlosky: Geophysical Fluid Dynamics. Springer-Verlag, New York, 1987. Zbl 0713.76005
[31] M. T.  Plateau: Smithsonian Report 250. 1863.
[32] Rayleigh, Lord (J. W.  Strutt): On the instability of jets. Proc. L.  M.  S. 10 (1879), 4–13.
[33] S. Resnick: Dynamical problem in nonlinear advective partial differential equations. PhD.  Thesis, University of Chicago, 1995.
[34] J. L.  Rodrigo: On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Math. 58 (2005), 821–866. DOI 10.1002/cpa.20059 | MR 2142632 | Zbl 1073.35006
[35] R.  Salmon: Lectures on Geophysical Fluid Dynamics. Oxford University Press, New York, 1998. MR 1718369
[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)
[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. DOI 10.1126/science.265.5169.219 | MR 1282463
[38] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970. MR 0290095 | Zbl 0207.13501
[39] E. M. Stein: Harmonic Analysis. Princeton University Press, Princeton, 1993. MR 1232192 | Zbl 0821.42001
[40] M.  Sussman, P.  Smereka: Axisymmetric free boundary problems. J.  Fluid Mech. 341 (1997), 269–294. DOI 10.1017/S0022112097005570 | MR 1457712
[41] L. Tartar: Topics in Nonlinear Analysis. Publications Mat. D’Orsay, No.  7813. Univ. de Paris-Sud, Orsay, 1978. MR 0532371
Partner of
EuDML logo