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

Michele Coti Zelati; Piotr Kalita

doi:10.3934/dcdsb.2017110

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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
[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

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

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