Acta mathematica scientia, Series B >
GLOBAL REGULARITY FOR MODIFIED CRITICAL DISSIPATIVE QUASI-GEOSTROPHIC EQUATIONS
Received date: 2013-09-22
Revised date: 2014-05-18
Online published: 2014-11-20
Supported by
Jiu was supported by Project of Beijing Chang Cheng Xue Zhe (11228102), and was also supported by NSF of China (11171229, 11231006).
We consider the n-dimensional modified quasi-geostrophic (SQG) equations
∂tθ + u · ∇θ + K∧αθ = 0,
u = ∧α−1R±θ
with K> 0, α∈ (0, 1] and θ0 ∈ W1,∞(Rn). In this paper, we establish a different proof for the global regularity of this system. The original proof was given by Constantin, Iyer, and Wu[5], who employed the approach of Besov space techniques to study the global existence and regularity of strong solutions to modified critical SQG equations for two dimensional case. The proof provided in this paper is based on the nonlinear maximum principle as well as the
approach in Constantin and Vicol [2].
YANG Wan-Rong , JIU Quan-Sen . GLOBAL REGULARITY FOR MODIFIED CRITICAL DISSIPATIVE QUASI-GEOSTROPHIC EQUATIONS[J]. Acta mathematica scientia, Series B, 2014 , 34(6) : 1741 -1748 . DOI: 10.1016/S0252-9602(14)60119-3
[1] Beale J, Kato T, Majda A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm Math Phys, 1984, 94: 61–66
[2] Constantin P, Vicol V. Nonlinear maximum principles for dissipative linear nonlocal operators and applications.
Geom Funct Anal, 2012, 22(5): 1289–1321
[3] Constantin P, Wu J. H¨older continuity solutions of the supercritical disspative hydrodynamic transport equations. Ann Inst H Poincar´e Anal Non Lin´eaire, 2009, 26: 159–180
[4] Constantin P,Wu J. Regularity of H¨older continuous solutions of the supercritical quasi-geostrophic eqution. Ann Inst H Poincar´e Anal Non Lin´eaire, 2008, 25: 1103–1110
[5] Constantin P, Iyer G, Wu J. Global regularity or a modified critical dissipative quasi-geostrophic equation. Indiana Univ Math J, 2008, 57: 2681–2692
[6] Caffarelli L, Vasseur A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math, 2010, 171(3): 1903–1930
[7] C¨orfoba A, C¨ordoba D. Amaximum principle applied to quasi-geostrophic equations. Math Phys, 2004, 249: 511–528
[8] Jiu Q, Miao C,Wu J, Zhang Z. The 2D incompressible Boussinesq equations with general critical dissipation. 2012, arXiv: 1212.3227v1
[9] Kiselev A, Nazarov F, Volberg A. Global well-posedness foe the critical 2D dissipative quasi-geostrophic equation. Invent Math, 2007, 167: 445–453
[10] Majda A, Bertozzi A. Vorticity and Incompressible Flow. Cambridge, England: Cambridge University Press, 2002
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