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July  2019, 39(7): 3749-3765. doi: 10.3934/dcds.2019152

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

Received  January 2018 Published  April 2019

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.
Citation: Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
References:
[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.  Google Scholar

[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.  Google Scholar

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Q. ChenC. 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.  Google Scholar

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P. ConstantinG. 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.  Google Scholar

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P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[21]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.  Google Scholar

[22]

A. Kiselev, Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.  doi: 10.1016/j.aim.2011.03.019.  Google Scholar

[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.  Google Scholar

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A. KiselevF. 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.  Google Scholar

[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.  Google Scholar

[26]

J. Leray, Sur le mouvement dun liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

K. Yamazaki, A remark on the global well-posedness of a modified critical quasi-geostrophic equation, arXiv: 1006.0253 [math.AP]. Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Q. ChenC. 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.  Google Scholar

[7]

P. ConstantinG. 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.  Google Scholar

[8]

P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.  Google Scholar

[22]

A. Kiselev, Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.  doi: 10.1016/j.aim.2011.03.019.  Google Scholar

[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.  Google Scholar

[24]

A. KiselevF. 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.  Google Scholar

[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.  Google Scholar

[26]

J. Leray, Sur le mouvement dun liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

K. Yamazaki, A remark on the global well-posedness of a modified critical quasi-geostrophic equation, arXiv: 1006.0253 [math.AP]. Google Scholar

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