doi: 10.3934/dcdsb.2021093

Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation

1. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

* Corresponding author

Received  November 2020 Revised  February 2021 Early access  March 2021

Fund Project: This work was supported by the National Natural Science Foundation of China under grant 41875084, and by FEDER and Ministerio de Ciencia, Innovación y Universidades of Spain (Grant PGC2018-096540-B-I00), and Junta de Andalucía, Spain (Grant US-1254251)

In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in $ H^{2\alpha+s}(\mathbb{T}^2) $ with $ \alpha>\frac{1}{2} $ and $ s>1. $ We prove the existence of $ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $-global attractor $ \mathcal{A}, $ that is, $ \mathcal{A} $ is compact in $ H^{2\alpha+s}(\mathbb{T}^2) $ and attracts all bounded subsets of $ H^{2\alpha^-+s}(\mathbb{T}^2) $ with respect to the norm of $ H^{2\alpha+s}(\mathbb{T}^2). $ The asymptotic compactness of solutions in $ H^{2\alpha+s}(\mathbb{T}^2) $ is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in $ H^{2\alpha+s}(\mathbb{T}^2), $ whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of $ H^{2\alpha^-+s}(\mathbb{T}^2) $ are all in the topology of $ H^{2\alpha+s}(\mathbb{T}^2). $

Citation: Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021093
References:
[1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975.   Google Scholar
[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[3]

L. C. Berselli, Vanishing viscosity limit and long-time behavior for $2$D quasi-geostrophic equations, Indiana Univ. Math. J., 51 (2002), 905–930. doi: 10.1512/iumj.2002.51.2075.  Google Scholar

[4]

V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617–639. doi: 10.1090/S0033-569X-06-01044-9.  Google Scholar

[5]

J. W. Cholewa and T. Dlotko, Bi-spaces global attractors in abstract parabolic equations, Evolution equations, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 60 (2003), 13–26.  Google Scholar

[6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar
[7]

P. Constantin, M. Coti Zelati and V. Vicol, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298–318. doi: 10.1088/0951-7715/29/2/298.  Google Scholar

[8] P. Constantin and C. Foiaş, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[9]

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

[10]

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

[11]

M. Coti Zelati and P. Kalita, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1857–1873. doi: 10.3934/dcdsb.2017110.  Google Scholar

[12]

T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 531–561. doi: 10.1016/j.jde.2015.02.022.  Google Scholar

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713–718. doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[15]

R. Farwig and C. Qian, Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in $\mathbb{R}^2$, J. Differential Equations, 266 (2019), 6525–6579. doi: 10.1016/j.jde.2018.11.009.  Google Scholar

[16]

C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1–34.  Google Scholar

[17]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117–127. doi: 10.1090/S0002-9939-05-08340-1.  Google Scholar

[18]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267–281. doi: 10.1007/BF00276875.  Google Scholar

[19]

A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094–7137. doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[20]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, , Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[22]

N. Ju, The maximum principle and the global attractor for the dissipative $2$D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161–181. doi: 10.1007/s00220-004-1256-7.  Google Scholar

[23]

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

[24]

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., 4 (1991), 323–347. doi: 10.2307/2939277.  Google Scholar

[25]

C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001.  Google Scholar

[26]

C. J. Niche and M. E. Schonbek, Decay of weak solutions to the $2$D dissipative quasi-geostrophic equation, Comm. Math. Phys., 276 (2007), 93–115. doi: 10.1007/s00220-007-0327-y.  Google Scholar

[27]

C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. Lond. Math. Soc., 91 (2015), 573–595. doi: 10.1112/jlms/jdu085.  Google Scholar

[28]

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

[29]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[30] J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.   Google Scholar
[31]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1–33. Google Scholar

[32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[33]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49–65. doi: 10.1016/j.na.2005.04.034.  Google Scholar

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[35]

S. Wang, Attractors for the $3$D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266–283. doi: 10.1016/0022-247X(92)90078-R.  Google Scholar

[36]

M. Wang and Y. Tang, Long time dynamics of $2$D quasi-geostrophic equations with damping in $L^p$, J. Math. Anal. Appl., 412 (2014), 866–877. doi: 10.1016/j.jmaa.2013.11.019.  Google Scholar

[37]

M. Wang and Y. Tang, On dimension of the global attractor for 2D quasi-geostrophic equations, Nonlinear Anal. Real World Appl., 14 (2013), 1887–1895. doi: 10.1016/j.nonrwa.2012.12.005.  Google Scholar

[38]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1–13.  Google Scholar

[39]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[40]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367–399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975.   Google Scholar
[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[3]

L. C. Berselli, Vanishing viscosity limit and long-time behavior for $2$D quasi-geostrophic equations, Indiana Univ. Math. J., 51 (2002), 905–930. doi: 10.1512/iumj.2002.51.2075.  Google Scholar

[4]

V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617–639. doi: 10.1090/S0033-569X-06-01044-9.  Google Scholar

[5]

J. W. Cholewa and T. Dlotko, Bi-spaces global attractors in abstract parabolic equations, Evolution equations, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 60 (2003), 13–26.  Google Scholar

[6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar
[7]

P. Constantin, M. Coti Zelati and V. Vicol, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298–318. doi: 10.1088/0951-7715/29/2/298.  Google Scholar

[8] P. Constantin and C. Foiaş, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[9]

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

[10]

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

[11]

M. Coti Zelati and P. Kalita, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1857–1873. doi: 10.3934/dcdsb.2017110.  Google Scholar

[12]

T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 531–561. doi: 10.1016/j.jde.2015.02.022.  Google Scholar

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713–718. doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[15]

R. Farwig and C. Qian, Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in $\mathbb{R}^2$, J. Differential Equations, 266 (2019), 6525–6579. doi: 10.1016/j.jde.2018.11.009.  Google Scholar

[16]

C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1–34.  Google Scholar

[17]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117–127. doi: 10.1090/S0002-9939-05-08340-1.  Google Scholar

[18]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267–281. doi: 10.1007/BF00276875.  Google Scholar

[19]

A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094–7137. doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[20]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, , Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[22]

N. Ju, The maximum principle and the global attractor for the dissipative $2$D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161–181. doi: 10.1007/s00220-004-1256-7.  Google Scholar

[23]

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

[24]

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., 4 (1991), 323–347. doi: 10.2307/2939277.  Google Scholar

[25]

C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001.  Google Scholar

[26]

C. J. Niche and M. E. Schonbek, Decay of weak solutions to the $2$D dissipative quasi-geostrophic equation, Comm. Math. Phys., 276 (2007), 93–115. doi: 10.1007/s00220-007-0327-y.  Google Scholar

[27]

C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. Lond. Math. Soc., 91 (2015), 573–595. doi: 10.1112/jlms/jdu085.  Google Scholar

[28]

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

[29]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[30] J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.   Google Scholar
[31]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1–33. Google Scholar

[32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[33]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49–65. doi: 10.1016/j.na.2005.04.034.  Google Scholar

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[35]

S. Wang, Attractors for the $3$D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266–283. doi: 10.1016/0022-247X(92)90078-R.  Google Scholar

[36]

M. Wang and Y. Tang, Long time dynamics of $2$D quasi-geostrophic equations with damping in $L^p$, J. Math. Anal. Appl., 412 (2014), 866–877. doi: 10.1016/j.jmaa.2013.11.019.  Google Scholar

[37]

M. Wang and Y. Tang, On dimension of the global attractor for 2D quasi-geostrophic equations, Nonlinear Anal. Real World Appl., 14 (2013), 1887–1895. doi: 10.1016/j.nonrwa.2012.12.005.  Google Scholar

[38]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1–13.  Google Scholar

[39]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[40]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367–399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[1]

May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179

[2]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[3]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021023

[4]

T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171

[5]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336

[6]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[7]

Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216

[8]

T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119

[9]

Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133

[10]

Tongtong Liang, Yejuan Wang. Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4697-4726. doi: 10.3934/dcdsb.2020309

[11]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241

[12]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013

[13]

Wenru Huo, Aimin Huang. The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2531-2550. doi: 10.3934/dcdsb.2016059

[14]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[15]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1

[16]

Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure & Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947

[17]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57

[18]

Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023

[19]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025

[20]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (66)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]