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

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

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.

Citation: Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete and 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.

[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.

[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.

[4]

V. V. ChepyzhovM. 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.

[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.

[6]

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

[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.

[8]

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

[9]

P. ConstantinD. 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.

[10]

P. ConstantinM. 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.

[11]

P. ConstantinA. 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.

[12]

P. ConstantinA. 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.

[13]

P. ConstantinA. 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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

T. DłotkoM.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.

[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.

[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.

[24]

S. FriedlanderN. 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.

[25]

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

[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.

[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.

[28]

C. E. KenigG. 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.

[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.

[30]

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.

[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.

[32]

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

[33]

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

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

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

[36]

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

show all references

References:
[1]

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

[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.

[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.

[4]

V. V. ChepyzhovM. 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.

[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.

[6]

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

[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.

[8]

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

[9]

P. ConstantinD. 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.

[10]

P. ConstantinM. 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.

[11]

P. ConstantinA. 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.

[12]

P. ConstantinA. 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.

[13]

P. ConstantinA. 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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

T. DłotkoM.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.

[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.

[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.

[24]

S. FriedlanderN. 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.

[25]

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

[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.

[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.

[28]

C. E. KenigG. 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.

[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.

[30]

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.

[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.

[32]

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

[33]

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

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

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

[36]

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

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