zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Infinite energy solutions of the surface quasi-geostrophic equation. (English) Zbl 1205.35219
The authors study the formation of singularities of a 1D nonlinear and non-local (surface quasi-geostrophic, SQG) equation, which is also a model for 3D vorticity Euler equations. One shows that this equation provides solutions of the surface quasi-geostrophic equation with infinite energy. The existence of self-similar solutions and the blow-up for classical solutions are also shown.

35Q35PDEs in connection with fluid mechanics
35Q31Euler equations
76B47Vortex flows
76E30Nonlinear effects (fluid mechanics)
35B40Asymptotic behavior of solutions of PDE
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI
[1] Baker, G. R.; Li, X.; Morlet, A. C.: Analytic structure of two 1D-transport equations with nonlocal fluxes, Phys. D 91, 349-375 (1996) · Zbl 0899.76104 · doi:10.1016/0167-2789(95)00271-5
[2] Castro, A.; Córdoba, D.: Self-similar solutions for a transport equation with non-local flux, Chinese ann. Math. 30, 505-512 (2009) · Zbl 1186.35154 · doi:10.1007/s11401-009-0180-8
[3] Chae, D.: Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. math. Phys. 273, 203-215 (2007) · Zbl 1157.35079 · doi:10.1007/s00220-007-0249-8
[4] Chae, D.: On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations, Adv. math. 221, 1678-1702 (2009) · Zbl 1173.35094 · doi:10.1016/j.aim.2009.02.015
[5] Childress, S.; Ierley, G. R.; Spiegel, E. A.; Young, W. R.: Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. fluid mech. 203, 1-22 (1989) · Zbl 0674.76013 · doi:10.1017/S0022112089001357
[6] Constantin, P.; Lax, P.; Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equation, Comm. pure appl. Math. 38, No. 6, 715-724 (1985) · Zbl 0615.76029 · doi:10.1002/cpa.3160380605
[7] Constantin, P.; Majda, A.; Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7, 1495-1533 (1994) · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[8] Constantin, P.; Nie, Q.; Schörghofer, N.: Nonsingular surface quasi-geostrophic flow, Phys. lett. A 241, 168-172 (1998) · Zbl 0974.76512 · doi:10.1016/S0375-9601(98)00108-X
[9] Córdoba, A.; Córdoba, D.; Fontelos, M. A.: Formation of singularities for a transport equation with nonlocal velocity, Ann. of math. 162, 1377-1389 (2005) · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377
[10] Córdoba, A.; Córdoba, D.; Fontelos, M. A.: Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. math. Pures appl. (9) 86, 529-540 (2006) · Zbl 1106.35059 · doi:10.1016/j.matpur.2006.08.002
[11] Córdoba, D.: Nonexistence of simple hyperbolic blow up for the quasi-geostrophic equation, Ann. of math. 148, 1135-1152 (1998) · Zbl 0920.35109 · doi:10.2307/121037 · http://www.math.princeton.edu/~annals/issues/1998/148_3.html
[12] Córdoba, D.; Fefferman, C.: Growth of solutions for QG and 2D Euler equations, J. amer. Math. soc. 15, 665-670 (2002) · Zbl 1013.76011 · doi:10.1090/S0894-0347-02-00394-6
[13] De Gregorio, S.: On a one-dimensional model for the three dimensional vorticity equation, J. stat. Phys. 59, 1251 (1990) · Zbl 0712.76027 · doi:10.1007/BF01334750
[14] De Gregorio, S.: A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. methods appl. Sci. 19, 1233 (1996) · Zbl 0860.35101 · doi:10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.0.CO;2-W
[15] Deng, J.; Hou, T. Y.; Li, R.; Yu, X.: Level sets dynamics and the non-blowup of the 2D quasi-geostrophic equation, Methods appl. Anal. 13, No. 2, 157-180 (2006) · Zbl 1173.76006 · euclid:maa/1200694870
[16] Friedlander, S.; Shvydkoy, R.: The unstable spectrum of the surface quasi-geostrophic equation, J. math. Fluid mech. 7, No. suppl. 1, S81-S93 (2005) · Zbl 1064.76046 · doi:10.1007/s00021-004-0129-3
[17] Getoor, R. K.: First passage times for symmetric stable processes in space, Trans. amer. Math. soc. 101, 75-90 (1961) · Zbl 0104.11203 · doi:10.2307/1993412
[18] A. Kiselev, F. Nazarov, A simple energy pump for periodic 2D QGE, preprint.
[19] Morlet, A. C.: Further properties of a continuum of model equations with globally defined flux, J. math. Anal. appl. 221, 132-160 (1998) · Zbl 0916.35049 · doi:10.1006/jmaa.1997.5801
[20] Ohkitani, K.; Yamada, M.: Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. fluids 9, No. 4, 876-882 (1997) · Zbl 1185.76841 · doi:10.1063/1.869184
[21] Okamoto, H.; Ohkitani, K.: On the role of the convection term in the equations of motion of incompressible fluid, J. phys. Soc. Japan 74, 2737-2742 (2005) · Zbl 1083.76007 · doi:10.1143/JPSJ.74.2737
[22] Okamoto, H.; Sakajo, T.; Wunsch, M.: On a generalization of the constantin-Lax-majda equation, Nonlinearity 21, No. 10, 2447-2461 (2008) · Zbl 1221.35300 · doi:10.1088/0951-7715/21/10/013
[23] Sato, H.; Sakajo, T.: Numerical study of de gregorio’s model for the 3D vorticity equation, Trans. Japan SIAM 16, 221 (2006)