×

zbMATH — the first resource for mathematics

Global well-posedness for a modified critical dissipative quasi-geostrophic equation. (English) Zbl 1382.35233
Summary: We consider the following modified quasi-geostrophic equation \[ \partial _t \delta + u \cdot \nabla \theta + \nu |D|^\alpha \theta = 0, \quad u = |D|^{\alpha -1}\mathcal R^\perp \theta, \quad x \in \mathbb R^2 \] with \(\nu >0\) and \(\alpha \in ]0,1[\cup ]1,2[\). When \(\alpha \in ]0,1[\), the equation was firstly introduced by P. Constantin et al. [Indiana Univ. Math. J. 57, No. 6, 2681–2692 (2008; Zbl 1159.35059)]. Here, by using the modulus of continuity method, we prove the global well-posedness of the system. As a byproduct, we also show that for every \(\alpha \in ]0,2[\), the Lipschitz norm of the solution has a uniform exponential upper bound.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76U05 General theory of rotating fluids
Citations:
Zbl 1159.35059
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abidi, H.; Hmidi, T., On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. math. anal., 40, 167-185, (2008) · Zbl 1157.76054
[2] Caffarelli, L.; Vasseur, V., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations, Ann. of math., 171, 3, 1903-1930, (2010) · Zbl 1204.35063
[3] D. Chae, P. Constantin, J. Wu, Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., doi:10.1007/s00205-011-0411-5, in press. · Zbl 1266.76010
[4] Chae, D.; Constantin, P.; Wu, J., Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations · Zbl 1288.35416
[5] Chae, D.; Constantin, P.; Córdoba, D.; Gancedo, F.; Wu, J., Generalized surface quasi-geostrophic equations with singular velocities · Zbl 1244.35108
[6] Chemin, J.-Y., Perfect incompressible fluids, (1998), Clarendon Press Oxford
[7] Chen, Q.; Miao, C.; Zhang, Z., A new bernsteinʼs inequality and the 2D dissipative quasi-geostrophic equation, Comm. math. phys., 271, 821-838, (2007) · Zbl 1142.35069
[8] Constantin, P.; Majda, A.J.; Tabak, E., Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7, 1495-1533, (1994) · Zbl 0809.35057
[9] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. math. anal., 30, 937-948, (1999) · Zbl 0957.76093
[10] Constantin, P.; Cordoba, D.; Wu, J., On the critical dissipative quasi-geostrophic equation, J. indiana univ. math., 50, 97-107, (2001) · Zbl 0989.86004
[11] Constantin, P.; Iyer, G.; Wu, J., Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana univ. math. J., 57, 2681-2692, (2008) · Zbl 1159.35059
[12] Córdoba, A.; Córdoba, D., A maximum principle applied to the quasi-geostrophic equations, Comm. math. phys., 249, 511-528, (2004) · Zbl 1309.76026
[13] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two · Zbl 1223.35249
[14] Dong, H.; Du, D., Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete contin. dyn. syst., 21, 4, 1095-1101, (2008) · Zbl 1141.35436
[15] Dong, H., Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, Discrete contin. dyn. syst., 26, 4, 1197-1211, (2010) · Zbl 1186.35158
[16] Duoandikoetxea, J., Fourier analysis, Grad. stud. math., vol. 29, (2001), Amer. Math. Soc. Providence, RI, translated and revised by D. Cruz-Uribe
[17] Friedlander, S.; Vicol, V., Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics, Ann. inst. H. Poincaré anal. non linéaire, 28, 2, 283-301, (2011) · Zbl 1277.35291
[18] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. math., 167, 445-453, (2007) · Zbl 1121.35115
[19] Kiselev, A., Regularity and blow up for active scalars, Math. model. nat. phenom., 5, 225-255, (2010) · Zbl 1194.35490
[20] Kiselev, A., Nonlocal maximum principle for active scalars, Adv. math., 227, 5, 1806-1826, (2011) · Zbl 1244.35022
[21] May, R., Global well-posedness for a modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space \(H^1\), J. differential equations, 250, 1, 320-339, (2011) · Zbl 1210.35262
[22] Miao, C.; Xue, L., On the regularity of a class of generalized quasi-geostrophic equations, J. differential equations, 251, 10, 2789-2821, (2011) · Zbl 1292.76071
[23] Wu, J., Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. math. anal., 36, 1014-1030, (2004) · Zbl 1083.76064
[24] Yu, X., Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation, J. math. anal. appl., 339, 359-371, (2008) · Zbl 1128.35006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.