×

Global regularity for the supercritical active scalars. (English) Zbl 1375.35399

Summary: This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
35Q86 PDEs in connection with geophysics
86A05 Hydrology, hydrography, oceanography
PDFBibTeX XMLCite
Full Text: DOI

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 · doi:10.1137/070682319
[2] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (2011) · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[3] Bergh, J., Löfström, J.: Interpolation Spaces, An Introduction. Springer, Berlin (1976) · Zbl 0344.46071 · doi:10.1007/978-3-642-66451-9
[4] Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171, 1903-1930 (2010) · Zbl 1204.35063 · doi:10.4007/annals.2010.171.1903
[5] Chae, D., Constantin, P., Córdoba, D., Gancedo, F., Wu, J.: Generalized surface quasi-geostrophic equations with singular velocities. Comm. Pure Appl. Math. 65, 1037-1066 (2012) · Zbl 1244.35108 · doi:10.1002/cpa.21390
[6] Chae, D., Constantin, P., Wu, J.: Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations. Arch. Ration. Mech. Anal. 202, 35-62 (2011) · Zbl 1266.76010 · doi:10.1007/s00205-011-0411-5
[7] Chae, D., Constantin, P., Wu, J.: Dissipative models generalizing the 2D Navier-Stokes and surface quasi-geostrophic equations. Indiana Univ. Math. J. 61, 1997-2018 (2012) · Zbl 1288.35416 · doi:10.1512/iumj.2012.61.4756
[8] Chae, D., Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Comm. Math. Phys. 233, 297-311 (2003) · Zbl 1019.86002 · doi:10.1007/s00220-002-0750-z
[9] Chen, Q., Miao, C., Zhang, Z.: A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Comm. Math. Phys. 271, 821-838 (2007) · Zbl 1142.35069 · doi:10.1007/s00220-007-0193-7
[10] Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasigeostrophic equation. Indiana Univ. Math. J. 50(Special Issue), 97-107 (2001). Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000) · Zbl 0989.86004
[11] Constantin, P., Foias, C.: Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988) · Zbl 0687.35071
[12] 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 · doi:10.1512/iumj.2008.57.3629
[13] Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495-1533 (1994) · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[14] Constantin, P., Tarfulea, A., Vicol, V.: Long time dynamics of forced critical SQG. Comm. Math. Phys. 335, 93-141 (2015) · Zbl 1316.35238 · doi:10.1007/s00220-014-2129-3
[15] Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937-948 (1999) · Zbl 0957.76093 · doi:10.1137/S0036141098337333
[16] Constantin, P., Wu, J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1103-1110 (2008) · Zbl 1149.76052 · doi:10.1016/j.anihpc.2007.10.001
[17] Constantin, P., Wu, J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 159-180 (2009) · Zbl 1163.76010 · doi:10.1016/j.anihpc.2007.10.002
[18] Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22, 1289-1321 (2012) · Zbl 1256.35078 · doi:10.1007/s00039-012-0172-9
[19] Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249, 511-528 (2004) · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1
[20] Córdoba, D., Fontelos, M., Mancho, A., Rodrigo, J.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949-5952 (2005) · Zbl 1135.76315 · doi:10.1073/pnas.0501977102
[21] Zelati M.C., Vicol V.: On the global regularity for the supercritical SQG equation, arXiv:1410.3186v1 [math.AP] 13 Oct 2014 · Zbl 1360.35204
[22] Dabkowski, M.: Eventual regularity of the solutions to the supercritical dissipative quasigeostrophic equation. Geom. Funct. Anal. 21, 1-13 (2011) · Zbl 1210.35185 · doi:10.1007/s00039-011-0108-9
[23] Dong, B., Chen, Z.: Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation. Nonlinearity 19, 2919-2928 (2006) · Zbl 1109.76063 · doi:10.1088/0951-7715/19/12/011
[24] Dong, H.: Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness. Discrete Contin. Dyn. Syst. 26, 1197-1211 (2010) · Zbl 1186.35158 · doi:10.3934/dcds.2010.26.1197
[25] Dong, H., Pavlović, N.: A regularity criterion for the dissipative quasi-geostrophic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1607-1619 (2009) · Zbl 1176.35133 · doi:10.1016/j.anihpc.2008.08.001
[26] Dong, H., Pavlović, N.: Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces. Comm. Math. Phys. 290, 801-812 (2009) · Zbl 1185.35187 · doi:10.1007/s00220-009-0756-x
[27] Hmidi, T., Keraani, S.: Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces. Adv. Math. 214, 618-638 (2007) · Zbl 1119.76070 · doi:10.1016/j.aim.2007.02.013
[28] Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys. 255, 161-181 (2005) · Zbl 1088.37049 · doi:10.1007/s00220-004-1256-7
[29] Ju, N.: Dissipative 2D quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J. 56, 187-206 (2007) · Zbl 1129.35062 · doi:10.1512/iumj.2007.56.2851
[30] Kiselev, A.: Regularity and blow up for active scalars. Math. Model. Nat. Phenom. 5, 225-255 (2010) · Zbl 1194.35490 · doi:10.1051/mmnp/20105410
[31] Kiselev, A., Nazarov, F.: A variation on a theme of Caffarelli and Vasseur. Zap. Nauchn. Sem. S. -Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370, 58-72, 220 (2009) · Zbl 1288.35393
[32] 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 · doi:10.1007/s00222-006-0020-3
[33] Li, D., Rodrigo, J.: Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Comm. Math. Phys. 286, 111-124 (2009) · Zbl 1172.86301 · doi:10.1007/s00220-008-0585-3
[34] Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002) · Zbl 0983.76001
[35] May, R.: Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space \[H^1\] H1. J. Differ. Equ. 250, 320-339 (2011) · Zbl 1210.35262 · doi:10.1016/j.jde.2010.09.021
[36] Miao, C., Wu, J., Zhang, Z.: Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics. Science Press, Beijing (2012). (in Chinese)
[37] Miao, C., Xue, L.: Global well-posedness for a modified critical dissipative quasi-geostrophic equation. J. Differ. Equ. 252, 792-818 (2012) · Zbl 1382.35233 · doi:10.1016/j.jde.2011.08.018
[38] Pedlosky, J.: Geophysical fluid dynamics. Springer, New York (1987) · Zbl 0713.76005 · doi:10.1007/978-1-4612-4650-3
[39] Resnick, S.: Dynamical problems in nonlinear partial differential equations, Ph.D. Thesis, University of Chicago, (1995) · Zbl 1316.35238
[40] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin, New York (1996) · Zbl 0873.35001 · doi:10.1515/9783110812411
[41] Silvestre, L.: Eventual regularization for the slightly supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 693-704 (2010) · Zbl 1187.35186 · doi:10.1016/j.anihpc.2009.11.006
[42] Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence (2006) · Zbl 1106.35001
[43] Triebel, H.: Theory of Function Spaces II. Birkhauser, Basel (1992) · Zbl 0763.46025 · doi:10.1007/978-3-0346-0419-2
[44] Wang, H., Zhang, Z.: A frequency localized maximum principle applied to the 2D quasi-geostrophic equation. Comm. Math. Phys. 301, 105-129 (2011) · Zbl 1248.35211 · doi:10.1007/s00220-010-1144-2
[45] Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal., 36, 1014-1030, (2004/05) · Zbl 1083.76064
[46] Wu, J.: The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18, 139-154 (2005) · Zbl 1067.35002 · doi:10.1088/0951-7715/18/1/008
[47] Wu, J.: Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Anal. 67, 3013-3036 (2007) · Zbl 1122.76014 · doi:10.1016/j.na.2006.09.050
[48] Yamazaki, K.: On the regularity criteria of a surface quasi-geostrophic equation. Nonlinear Anal. 75, 4950-4956 (2012) · Zbl 1242.35075 · doi:10.1016/j.na.2012.04.010
[49] 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 · doi:10.1016/j.jmaa.2007.06.064
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.