×

zbMATH — the first resource for mathematics

On the global regularity for the anisotropic dissipative surface quasi-geostrophic equation. (English) Zbl 1429.35206

MSC:
35R11 Fractional partial differential equations
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abidi H and Hmidi T 2008 On the global well-posedness of the critical quasi-geostrophic equation SIAM J. Math. Anal.40 167-85 · Zbl 1157.76054
[2] Bahouri H, Chemin J-Y and Danchin R 2011 Fourier Analysis and Nonlinear Partial Differential Equations(Grundlehren Math. Wiss. vol 343) (Berlin: Springer) · Zbl 1227.35004
[3] Caffarelli L and Vasseur A 2010 Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation Ann. Math.171 1903-30 · Zbl 1204.35063
[4] Cao C, Li J and Titi E 2017 Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity (arXiv:1703.02512v1 [math.AP])
[5] Chae D and Lee J 2003 Global well-posedness in the super-critical dissipative quasi-geostrophic equations Commun. Math. Phys.233 297-311 · Zbl 1019.86002
[6] Chae D, Constantin P, Córdoba D, Gancedo F and Wu J 2012 Generalized surface quasi-geostrophic equations with singular velocities Commun. Pure Appl. Math.65 1037-66 · Zbl 1244.35108
[7] Chae D, Constantin P and Wu J 2011 Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations Arch. Ration Mech. Anal.202 35-62 · Zbl 1266.76010
[8] Chae D, Córdoba A, Córdoba D and Fontelos M 2005 Finite time singularities in a 1D model of the quasi-geostrophic equation Adv. Math.194 203-23 · Zbl 1128.76372
[9] Chen Q, Miao C and Zhang Z 2007 A new Bernstein’s inequality and the 2D dissipative quasigeostrophic equation Commun. Math. Phys.271 821-38 · Zbl 1142.35069
[10] Constantin P, Córdoba D and Wu J 2001 On the critical dissipative quasi-geostrophic equation Indiana Univ. Math. J.50 97-107 · Zbl 0989.86004
[11] Constantin P, Majda A and Tabak E 1994 Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar Nonlinearity7 1495 · Zbl 0809.35057
[12] Constantin P and Vicol V 2012 Nonlinear maximum principles for dissipative linear nonlocal operators and applications Geom. Funct. Anal.22 1289-321 · Zbl 1256.35078
[13] Constantin P and Wu J 1999 Behavior of solutions of 2D quasi-geostrophic equations SIAM J. Math. Anal.30 937-48 · Zbl 0957.76093
[14] Constantin P and Wu J 2008 Regularity of Hölder continuous solutions of the supercritical quasigeostrophic equation Ann. Inst. Henri Poincare Anal. Non Lineaire25 1103-10 · Zbl 1149.76052
[15] Córdoba A and Córdoba D 2004 A maximum principle applied to quasi-geostrophic equations Commun. Math Phys.249 511-28 · Zbl 1309.76026
[16] Coti Zelati M and Vicol V 2016 On the global regularity for the supercritical SQG equation Indiana Univ. Math. J.65 535-52 · Zbl 1360.35204
[17] Dabkowski M 2011 Eventual regularity of the solutions to the supercritical dissipative quasigeostrophic equation Geom. Funct. Anal.21 1-13 · Zbl 1210.35185
[18] Danchin R and Paicu M 2011 Global existence results for the anisotropic Boussinesq system in dimension two Math. Models Methods Appl. Sci.21 421-57 · Zbl 1223.35249
[19] Dong H and Du D 2008 Global well-posedness and a decay estimate for the critical dissipative quasigeostrophic equation in the whole space Discrete Contin. Dyn. Syst.21 1095-101 · Zbl 1141.35436
[20] Dong H and Li D 2008 Spatial analyticity of the solutions to the subcritical dissipative quasigeostrophic equations Arch. Ration. Mech. Anal.189 131-58 · Zbl 1157.35456
[21] Dong H and Li D 2010 On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces J. Differ. Equ.248 2684-702 · Zbl 1193.35151
[22] Dong H and Pavlovic N 2009 A regularity criterion for the dissipation quasi-geostrophic equation Ann. Inst. Henri Poincaré Anal. Non Linéaire26 1607-19 · Zbl 1176.35133
[23] Ju N 2004 Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space Commun. Math. Phys.251 365-76 · Zbl 1106.35061
[24] Kenig C, Ponce G and Vega L 1993 Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle Commun. Pure Appl. Math.46 527-620 · Zbl 0808.35128
[25] Kiselev A 2011 Nonlocal maximum principles for active scalars Adv. Math.227 1806-26 · Zbl 1244.35022
[26] Kiselev A and Nazarov F 2010 A variation on a theme of Caffarelli and Vasseur Zap. Nauchn. Sem. POMI370 58-72 · Zbl 1288.35393
[27] Kiselev A, Nazarov F and Volberg A 2007 Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Inventiones Math.167 445-53 · Zbl 1121.35115
[28] Kozono H, Ogawa T and Taniuchi Y 2002 The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations Math. Z.242 251-78 · Zbl 1055.35087
[29] Li J and Titi E 2016 Global well-posedness of the 2D Boussinesq equations with vertical dissipation Arch. Ration. Mech. Anal.220 983-1001 · Zbl 1336.35297
[30] Majda A and Bertozzi A 2001 Vorticity and Incompressible Flow (Cambridge: Cambridge University Press) · Zbl 0983.76001
[31] Miao C and Xue L 2012 Global wellposedness for a modified critical dissipative quasi-geostrophic equation J. Differ. Equ.252 792-818 · Zbl 1382.35233
[32] Pedlosky J 1987 Geophysical Fluid Dynamics (New York: Springer) · Zbl 0713.76005
[33] Resnick S 1995 Dynamical problems in nonlinear advective partial differential equations PhD Thesis The University of Chicago, Pro-Quest LLC, Ann Arbor, MI
[34] Schonbek M and Schonbek T 2003 Asymptotic behavior to dissipative quasi-geostrophic flows SIAM J. Math. Anal.35 357-75 · Zbl 1126.76386
[35] Silvestre L 2010 Eventual regularization for the slightly supercritical quasi-geostrophic equation Ann. Inst. Henri Poincare Anal Non Lineaire27 693-704 · Zbl 1187.35186
[36] Silvestre L, Vicol V and Zlatoš A 2013 On the loss of continuity for the super-critical drift-diffusion equations Arch. Ration. Mech. Anal.27 845-77 · Zbl 1264.35077
[37] Wang H and Zhang Z 2011 A frequency localized maximum principle applied to the 2D quasi-geostrophic equation Commun. Math. Phys.301 105-29 · Zbl 1248.35211
[38] Wu J 2002 The quasi-geostrophic equation and its two regularizations Commun. PDE27 1161-81 · Zbl 1012.35067
[39] Wu J, Xu X and Ye Z 2015 Global smooth solutions to the n-dimensional damped models of incompressible fluid mechanics with small initial datum J. Nonlinear Sci.25 157-92 · Zbl 1311.35236
[40] Wu J, Xu X and Ye Z 2017 Global regularity for several incompressible fluid models with partial dissipation J. Math. Fluid Mech.19 423-44 · Zbl 1379.35255
[41] Wu J, Xu X and Ye Z 2018 The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusion J. Math. Pures Appl.115 187-217 · Zbl 1392.35240
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.