Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

Wen Tan; Bo-Qing Dong; Zhi-Min Chen

doi:10.3934/dcds.2019152

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2019, Volume 39, Issue 7: 3749-3765. Doi: 10.3934/dcds.2019152

This issue Previous Article Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains Next Article Classification of linear skew-products of the complex plane and an affine route to fractalization

Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518052, China

^* Corresponding author. Zhi-Min Chen
^* Corresponding author. Zhi-Min Chen

Received: January 2018

Published: April 2019

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Abstract

This paper is devoted to the study of the modified quasi-geostrophic equation

$ \partial_t\theta+u\cdot\nabla\theta+\nu\Lambda^\alpha\theta = 0 \ \ \mbox{ with } \ \ u = \Lambda^\beta\mathcal{R}^\perp\theta $

in $ \mathbb{R}^2 $. By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case $ \beta = \alpha-1 $ and the existence of regular solutions for large time $ t>T $ with respect to the supercritical case $ \beta >\alpha -1 $ in Besov spaces. Earlier results for the equation in Hilbert spaces $ H^s $ spaces are improved.

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Mathematics Subject Classification: 35Q35, 76D03.

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[1]	H. Bae, Global well-posedness of dissipative quasi-geostrophic equations in critical spaces, Proc. Am. Math. Soc., 136 (2008), 257-261. doi: 10.1090/S0002-9939-07-09060-0.
[2]	H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften 343, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.
[3]	L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[4]	D. Chae and J. Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys., 233 (2003), 297-311. doi: 10.1007/s00220-002-0750-z.
[5]	Z. M. Chen and Z. Xin, Homogeneity Criterion for the Navier-Stokes equations in the whole spaces, J. Math. Fluid Mech., 3 (2001), 152-182. doi: 10.1007/PL00000967.
[6]	Q. Chen, C. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838. doi: 10.1007/s00220-007-0193-7.
[7]	P. Constantin, G. Iyer and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 57 (2008), 2681-2692. doi: 10.1512/iumj.2008.57.3629.
[8]	P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.
[9]	P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9.
[10]	P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333.
[11]	P. Constantin and J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1103-1110. doi: 10.1016/j.anihpc.2007.10.001.
[12]	P. Constantin and J. Wu, Hölder Continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 159-180. doi: 10.1016/j.anihpc.2007.10.002.
[13]	A. Córdoba and D. Córdoba, A maximum principle applied to the quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.
[14]	M. Dabkowski, Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal., 21 (2011), 1-13. doi: 10.1007/s00039-011-0108-9.
[15]	R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differ. Equ., 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.
[16]	H. Dong, Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, Discrete Contin. Dyn. Syst., 26 (2010), 1197-1211. doi: 10.3934/dcds.2010.26.1197.
[17]	H. Dong and D. Li, On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces, J. Differential Equations, 248 (2010), 2684-2702. doi: 10.1016/j.jde.2010.02.015.
[18]	H. Dong and N. Pavlović, Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces, Comm. Math. Phys., 290 (2009), 801-812. doi: 10.1007/s00220-009-0756-x.
[19]	H. Dong and N. Pavlović, A regularity criterion for the dissipative quasi-geostrophic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1607-1619. doi: 10.1016/j.anihpc.2008.08.001.
[20]	N. Ju, Dissipative quasi-geostrophic equation: Local well-posedness, global regularity and similarity solutions, Indiana Univ. Math. J., 56 (2007), 187-206. doi: 10.1512/iumj.2007.56.2851.
[21]	A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255. doi: 10.1051/mmnp/20105410.
[22]	A. Kiselev, Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826. doi: 10.1016/j.aim.2011.03.019.
[23]	A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov., 370 (2009), 58-72. doi: 10.1007/s10958-010-9842-z.
[24]	A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.
[25]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[26]	J. Leray, Sur le mouvement dun liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[27]	R. May, Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space $H^1$, J. Differential Equations, 250 (2011), 320-339. doi: 10.1016/j.jde.2010.09.021.
[28]	C. Miao and L. Xue, On the regularity of a class of generalized quasi-geostrophic equations, J. Differential Equations, 251 (2011), 2789-2821. doi: 10.1016/j.jde.2011.04.018.
[29]	C. Miao and L. Xue, Global well-posedness for a modified critical dissipative quasi-geostrophic equation, J. Differential Equations, 252 (2012), 792-818. doi: 10.1016/j.jde.2011.08.018.
[30]	H. Miura, Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space, Comm. Math. Phys., 267 (2006), 141-157. doi: 10.1007/s00220-006-0023-3.
[31]	S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995.
[32]	L. Silvestre, Eventual regularization for the slightly supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 693-704. doi: 10.1016/j.anihpc.2009.11.006.
[33]	J. Wu, Quasi-geostrophic-type equations with initial data in Morrey spaces, Nonlinearity, 10 (1997), 1409-1420. doi: 10.1088/0951-7715/10/6/002.
[34]	J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263 (2006), 803-831. doi: 10.1007/s00220-005-1483-6.
[35]	K. Yamazaki, A remark on the global well-posedness of a modified critical quasi-geostrophic equation, arXiv: 1006.0253 [math.AP].