zbMATH — the first resource for mathematics

Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces. (English) Zbl 1185.35187
Summary: We study regularity criteria for weak solutions of the initial value problem for the dissipative quasi-geostrophic equation
\[ \begin{cases} \theta_t+u\cdot\nabla\theta+ (-\delta)^{\gamma/2}\theta=0, &x\in\mathbb R^2,\;t\in (0,\infty),\\ \theta(0,x)= \theta_0(x), \end{cases} \]
where \(\gamma\in(0,2]\) is a fixed parameter and \(u=(u_1,u_2)\) is the velocity. We show in this paper that if \({\theta \in C((0, T); C^{1-\gamma})}\), or \({\theta \in L^{r}((0, T); C^\alpha)}\) with \({\alpha = 1 - \gamma + \frac{\gamma}{r}}\) is a weak solution of the 2D quasi-geostrophic equation, then \(\theta \) is a classical solution in \({(0, T] \times {\mathbb R}^2}\). This result improves our previous result in [the authors, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 5, 1607–1619 (2009; Zbl 1176.35133)].

35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
86A05 Hydrology, hydrography, oceanography
76E20 Stability and instability of geophysical and astrophysical flows
35D30 Weak solutions to PDEs
Zbl 1176.35133
Full Text: DOI
[1] Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (to appear) · Zbl 1204.35063
[2] Chae D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16(2), 479–495 (2003) · Zbl 1029.35006
[3] Chae D.: On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J. Math. Anal. 37(5), 1649–1656 (2006) · Zbl 1141.76010
[4] Chae D., Lee J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003) · Zbl 1019.86002
[5] Chen Q., Miao C., Zhang Z.: A new Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation. Commun. Math. Phys. 271(3), 821–838 (2007) · Zbl 1142.35069
[6] Cheskidov, A., Shvydkoy, R.: On the regularity of weak solutions of the 3D Navier-Stokes equations in \({B^{-1}_{\infty,\infty}}\) . http://arXiv.org/abs/math.AP/0708.3067v2[math.AP] , 2007 · Zbl 1186.35137
[7] Cheskidov A., Constantin P., Friedlander S., Shvydkoy R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21(6), 1233–1252 (2008) · Zbl 1138.76020
[8] Constantin P., Cordoba D., Wu J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) · Zbl 0989.86004
[9] Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) · Zbl 0809.35057
[10] Constantin P., Wu J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999) · Zbl 0957.76093
[11] Constantin P., Wu J.: Hölder continuity of solutions of super-critical dissipative hydrodynamic transport equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 159–180 (2009) · Zbl 1163.76010
[12] Constantin P., Wu J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(6), 1103–1110 (2008) · Zbl 1149.76052
[13] Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004) · Zbl 1309.76026
[14] Dong B.-Q., Chen Z.-M.: A remark on regularity criterion for the dissipative quasi-geostrophic equations. J. Math. Anal, Appl. 329(2), 1212–1217 (2007) · Zbl 1154.76339
[15] Dong, H.: Higher regularity for the critical and super-critical dissipative quasi-geostrophic equations. http://arXiv.org/abs/math/0701826v1[math.AP] , 2007
[16] Dong, H., Li, D.: On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces, Preprint 2007 · Zbl 1193.35151
[17] 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
[18] Dong, H., Pavlović, N.: A regularity criterion for the dissipative quasi-geostrophic equations, To appear in Annales de l’Institut Henri Poincare - Non Linear Analysis, doi: 10.1016/j.anihpe.2008.08.001 , 2008
[19] Hmidi T., Keraani S.: Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces. Adv. Math. 214(2), 618–638 (2007) · Zbl 1119.76070
[20] Kiselev A., Nazarov F., Volberg A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445–453 (2007) · Zbl 1121.35115
[21] Ju N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun. Math. Phys. 255(1), 161–181 (2005) · Zbl 1088.37049
[22] Ju N.: Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J. 56(1), 187–206 (2007) · Zbl 1129.35062
[23] Miura H.: Dissipative quasi-geostrophic equation for large initial data in the critical sobolev space. Commun. Math. Phys. 267(1), 141–157 (2006) · Zbl 1113.76029
[24] Pedlosky J.: Geophysical fluid dynamics. Springer, New York (1987) · Zbl 0713.76005
[25] Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago, 1995
[26] Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equations in Besov spaces. SIAM J. Math. Anal. 36(3), 1014–1030 (2004/05) (electronic)
[27] Wu J.: Solutions of the 2D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal. 62(4), 579–594 (2005) · Zbl 1116.35348
[28] Wu J.: Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Commun. Math. Phys. 263(3), 803–831 (2006) · Zbl 1104.35037
[29] Wu J.: Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Analysis 67, 3013–3036 (2007) · Zbl 1122.76014
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.