Finite time singularities for a class of generalized surface quasi-geostrophic equations

Dong, Hongjie; Li, Dong

doi:10.1090/S0002-9939-08-09328-3

Finite time singularities for a class of generalized surface quasi-geostrophic equations
HTML articles powered by AMS MathViewer

by Hongjie Dong and Dong Li

Proc. Amer. Math. Soc. 136 (2008), 2555-2563

DOI: https://doi.org/10.1090/S0002-9939-08-09328-3

Published electronically: February 21, 2008

PDF | Request permission

Abstract:

We propose and study a class of generalized surface quasi-geo- strophic equations. We show that in the inviscid case certain radial solutions develop gradient blow-up in finite time. In the critical dissipative case, the equations are globally well-posed with arbitrary $H^1$ initial data.

References

Balodis, P., Córdoba, A., Inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math. 214 (2007) no. 1, 1–39.
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762
Caffarelli, L., Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, preprint.
Carrilo, J.A., Ferreira, L.C.F., Asymptotic behavior for the sub-critical dissipative quasi-geostrophic equations, preprint.
Dongho Chae, Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math. 194 (2005), no. 1, 203–223. MR 2141858, DOI 10.1016/j.aim.2004.06.004
Peter Constantin, Qing Nie, and Norbert Schörghofer, Nonsingular surface quasi-geostrophic flow, Phys. Lett. A 241 (1998), no. 3, 168–172. MR 1613907, DOI 10.1016/S0375-9601(98)00108-X
Peter Constantin, Diego Cordoba, and Jiahong Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 97–107. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855665, DOI 10.1512/iumj.2008.57.3629
P. Constantin, P. D. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724. MR 812343, DOI 10.1002/cpa.3160380605
Diego Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math. (2) 148 (1998), no. 3, 1135–1152. MR 1670077, DOI 10.2307/121037
Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389. MR 2179734, DOI 10.4007/annals.2005.162.1377
Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl. (9) 86 (2006), no. 6, 529–540 (English, with English and French summaries). MR 2281451, DOI 10.1016/j.matpur.2006.08.002
Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
Dong, H., Higher regularity for the critical and super-critical dissipative quasi-geostrophic equations, arXiv:math.AP/0701826.
Dong, H., On the well-posedness for a transport equation with nonlocal velocity, preprint.
Dong, H., Du, D., Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, to appear in Discrete Contin. Dyn. Syst., arXiv:math.AP/0701828.
Dong, H., Li, D., Spatial analyticity of the solutions to the sub-critical dissipative quasi-geostrophic equations, submitted.
Salvatore De Gregorio, A partial differential equation arising in a $1$D model for the $3$D vorticity equation, Math. Methods Appl. Sci. 19 (1996), no. 15, 1233–1255. MR 1410208, DOI 10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.3.CO;2-N
Ju, N., Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions, Indiana Univ. Math. J. (2006), in press.
A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260, DOI 10.1007/s00222-006-0020-3
Andrew J. Majda and Esteban G. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow, Phys. D 98 (1996), no. 2-4, 515–522. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). MR 1422288, DOI 10.1016/0167-2789(96)00114-5
Koji Ohkitani and Michio Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids 9 (1997), no. 4, 876–882. MR 1437554, DOI 10.1063/1.869184
Pedlosky, J., Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.
Takashi Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity 16 (2003), no. 4, 1319–1328. MR 1986297, DOI 10.1088/0951-7715/16/4/307
Steven Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math. 39 (1986), no. 4, 531–537. MR 840339, DOI 10.1002/cpa.3160390404
Yahan Yang, Behavior of solutions of model equations for incompressible fluid flow, J. Differential Equations 125 (1996), no. 1, 133–153. MR 1376063, DOI 10.1006/jdeq.1996.0027