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Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$
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 |
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.
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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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