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July  2017, 22(5): 1857-1873. doi: 10.3934/dcdsb.2017110

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Michele Coti Zelati 1,, and  Piotr Kalita 2,

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742, USA
2. 

Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

* Corresponding author: Michele Coti Zelati

Received  December 2015 Revised  February 2016 Published  March 2017

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

Keywords: Surface quasi-geostrophic equation, critical dissipation, global attractors, Sobolev regularity.
Mathematics Subject Classification: Primary: 35Q35, 35B41; Secondary: 35B4.
Citation: Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland Publishing Co. , Amsterdam, 1992. Google Scholar

[2]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[3]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math.(2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[4]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl.(9), 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence, RI, 2002. Google Scholar

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A. Cheskidov and M. Dai, The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$, preprint, arXiv: 1402.4801. Google Scholar

[8]

A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

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P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97-107.  doi: 10.1512/iumj.2001.50.2153.  Google Scholar

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P. Constantin, M. Coti Zelati and V. Vicol, On the critical dissipative quasi-geostrophic equation, Nonlinearity, 29 (2016), 298-318.  doi: 10.1088/0951-7715/29/2/298.  Google Scholar

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P. Constantin, A. J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.  doi: 10.1088/0951-7715/7/6/001.  Google Scholar

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P. Constantin, A. Tarfulea and V. Vicol, Absence of anomalous dissipation of energy in forced two dimensional fluid equations, Arch. Ration. Mech. Anal., 212 (2014), 875-903.  doi: 10.1007/s00205-013-0708-7.  Google Scholar

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P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93-141.  doi: 10.1007/s00220-014-2129-3.  Google Scholar

[14]

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

[15]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[16]

M. Coti Zelati, Long time behavior of subcritical SQG equations in scale-invariant Sobolev spaces, preprint, arXiv: 1512.00497. Google Scholar

[17]

M. Coti Zelati and V. Vicol, On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535{552, arXiv: 1410.3186. Google Scholar

[18]

M. Coti Zelati, On the theory of global attractors and Lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149.  doi: 10.1007/s11228-012-0215-2.  Google Scholar

[19]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561.  doi: 10.1137/140978995.  Google Scholar

[20]

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

[21]

T. Dłotko, M.B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 231-261.  doi: 10.1016/j.jde.2015.02.022.  Google Scholar

[22]

T. Dłotko and C. Sun, 2D Quasi-Geostrophic equation; sub-critical and critical cases, Nonlinear Anal., 150 (2017), 38-60.  doi: 10.1016/j.na.2016.11.005.  Google Scholar

[23]

H. Dong and D. Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 1095-1101.  doi: 10.3934/dcds.2008.21.1095.  Google Scholar

[24]

S. Friedlander, N. Pavlović and V. Vicol, Nonlinear instability for the critically dissipative quasi-geostrophic equation, Comm. Math. Phys., 292 (2009), 797-810.  doi: 10.1007/s00220-009-0851-z.  Google Scholar

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems American Mathematical Society, Providence, RI, 1988. Google Scholar

[26]

P. Kalita and G. Lukaszewicz, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Anal., 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[27]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[28]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 41 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[29]

A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370 (2009), 58-72.  doi: 10.1007/s10958-010-9842-z.  Google Scholar

[30]

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.  Google Scholar

[31]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[32]

J. Pedlosky, Geophysical Fluid Dynamics Springer, Berlin, 1982. Google Scholar

[33]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations Ph. D thesis, The University of Chicago, 1995. Google Scholar

[34] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[35]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002. Google Scholar

[36]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland Publishing Co. , Amsterdam, 1992. Google Scholar

[2]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[3]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math.(2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[4]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl.(9), 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence, RI, 2002. Google Scholar

[7]

A. Cheskidov and M. Dai, The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$, preprint, arXiv: 1402.4801. Google Scholar

[8]

A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

[9]

P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97-107.  doi: 10.1512/iumj.2001.50.2153.  Google Scholar

[10]

P. Constantin, M. Coti Zelati and V. Vicol, On the critical dissipative quasi-geostrophic equation, Nonlinearity, 29 (2016), 298-318.  doi: 10.1088/0951-7715/29/2/298.  Google Scholar

[11]

P. Constantin, A. J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.  doi: 10.1088/0951-7715/7/6/001.  Google Scholar

[12]

P. Constantin, A. Tarfulea and V. Vicol, Absence of anomalous dissipation of energy in forced two dimensional fluid equations, Arch. Ration. Mech. Anal., 212 (2014), 875-903.  doi: 10.1007/s00205-013-0708-7.  Google Scholar

[13]

P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93-141.  doi: 10.1007/s00220-014-2129-3.  Google Scholar

[14]

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

[15]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[16]

M. Coti Zelati, Long time behavior of subcritical SQG equations in scale-invariant Sobolev spaces, preprint, arXiv: 1512.00497. Google Scholar

[17]

M. Coti Zelati and V. Vicol, On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535{552, arXiv: 1410.3186. Google Scholar

[18]

M. Coti Zelati, On the theory of global attractors and Lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149.  doi: 10.1007/s11228-012-0215-2.  Google Scholar

[19]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561.  doi: 10.1137/140978995.  Google Scholar

[20]

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

[21]

T. Dłotko, M.B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 231-261.  doi: 10.1016/j.jde.2015.02.022.  Google Scholar

[22]

T. Dłotko and C. Sun, 2D Quasi-Geostrophic equation; sub-critical and critical cases, Nonlinear Anal., 150 (2017), 38-60.  doi: 10.1016/j.na.2016.11.005.  Google Scholar

[23]

H. Dong and D. Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 1095-1101.  doi: 10.3934/dcds.2008.21.1095.  Google Scholar

[24]

S. Friedlander, N. Pavlović and V. Vicol, Nonlinear instability for the critically dissipative quasi-geostrophic equation, Comm. Math. Phys., 292 (2009), 797-810.  doi: 10.1007/s00220-009-0851-z.  Google Scholar

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems American Mathematical Society, Providence, RI, 1988. Google Scholar

[26]

P. Kalita and G. Lukaszewicz, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Anal., 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[27]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[28]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 41 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[29]

A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370 (2009), 58-72.  doi: 10.1007/s10958-010-9842-z.  Google Scholar

[30]

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.  Google Scholar

[31]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[32]

J. Pedlosky, Geophysical Fluid Dynamics Springer, Berlin, 1982. Google Scholar

[33]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations Ph. D thesis, The University of Chicago, 1995. Google Scholar

[34] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[35]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002. Google Scholar

[36]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. Google Scholar

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