February  2022, 42(2): 643-677. doi: 10.3934/dcds.2021132

Pullback attractors for 2D MHD equations on time-varying domains

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

2. 

Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

*Corresponding author: Xiaoya Song (songxy29math@163.com)

Received  May 2020 Revised  April 2021 Published  February 2022 Early access  September 2021

Fund Project: D. Cao was supported by NSFC (grant No. 11831009) and Chinese Academy of Sciences by grant QYZDJ-SSW-SYS021; X. Song was supported by China Postdoctoral Science Foundation (grant No. 2020M672566); C. Sun was supported by NSFC (grant No. 11522109, 11871169)

In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any $ n $-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure $ p $ and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.

Citation: Daomin Cao, Xiaoya Song, Chunyou Sun. Pullback attractors for 2D MHD equations on time-varying domains. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 643-677. doi: 10.3934/dcds.2021132
References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. 

[2]

C. T. Anh and D. T. Son, Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains, Ann. Polon. Math., 113 (2015), 129-154.  doi: 10.4064/ap113-2-2.

[3]

D. N. Bock, On the Navier-Stokes equations in noncylindrical domains, J. Differential Equations, 25 (1977), 151-162.  doi: 10.1016/0022-0396(77)90197-8.

[4]

M. L. BernardiG. Bonfanti and F. Luterotti, Abstract Schrödinger-type differential equations with variable domain, J. Math. Anal. Appl., 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.

[5]

S. Bonaccorsi and G. Guatteri, A variational approach to evolution problems with variable domains, J. Differential Equations, 175 (2001), 51-70.  doi: 10.1006/jdeq.2000.3959.

[6]

L. BoudinC. Grandmont and A. Moussa, Global existence of solutions to the incompressible Navier-Stokes-Vlasov equations in a time-dependent domain, J. Differential Equations, 262 (2017), 1317-1340.  doi: 10.1016/j.jde.2016.10.012.

[7]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[8]

P. CannarsaG. Da Prato and J. P. Zolésto, The damped wave equation in a moving domain, J. Differential Equations, 85 (1990), 1-16.  doi: 10.1016/0022-0396(90)90086-5.

[9]

D. N. ChebanP. E. Kloeden and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144. 

[10]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-514. 

[11]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[12]

T. CaraballoG. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[15]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[16]

T. CaraballoJ. Real and P. E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.  doi: 10.1515/ans-2006-0304.

[17]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

[18]

D. R. da CostaC. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211.  doi: 10.1103/PhysRevE.83.066211.

[19]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticit et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[20]

H. Flanders, Differentiation under the integral sign, Amer. Math. Monthly, 80 (1973), 615-627.  doi: 10.1080/00029890.1973.11993339.

[21]

J. Ferreira and M. L. Santos, Asymptotic behaviour for wave equations with memory in a noncylindrical domains, Commun. Pure Appl. Anal., 2 (2003), 511-520.  doi: 10.3934/cpaa.2003.2.511.

[22]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[23]

C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.

[24]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319. 

[25]

N. JamesA. Ilyasse and D. Stevan, Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process, Internat. J. Control, 86 (2013), 1467-1478.  doi: 10.1080/00207179.2013.786187.

[26]

P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Differential Equations Appl., 6 (2000), 33-52.  doi: 10.1080/10236190008808212.

[27]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[28]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[29]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[30]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.

[31]

P. E. KloedenJ. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.

[32]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limits Non Linéaires, Dunod-Gauthier Villars, Paris, 1969.

[33]

J. LímacoL. A. Medeiros and E. Zuazua, Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains, Seventh Workshop on Partial Differential Equations, Part II, Rio de Janeiro, 2001, Mat. Contemp., 23 (2002), 49-70. 

[34]

T. F. MaP. Marín-Rubio and C. M. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.

[35]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528. 

[36] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. 
[37]

G. R. Sell, Non-autonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[38]

H. M. Soner and S. E. Shreve, A free boundary problem related to singular stochastic control: The parabolic case, Comm. Partial Differential Equations, 16 (1991), 373-424.  doi: 10.1080/03605309108820763.

[39]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.

[40]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.

[42]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[43]

Y. Yong and Q. Jiu, Energy equality and uniqueness of weak solutions to MHD equations in $L^{\infty}(0, T;L^{n}(\Omega))$, Acta Math. Sin., Engl. Ser., 25 (2009), 803-814.  doi: 10.1007/s10114-009-7214-8.

[44]

F. Zhou and C. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3767-3792.  doi: 10.3934/dcdsb.2016120.

[45]

F. ZhouC. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.

show all references

References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. 

[2]

C. T. Anh and D. T. Son, Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains, Ann. Polon. Math., 113 (2015), 129-154.  doi: 10.4064/ap113-2-2.

[3]

D. N. Bock, On the Navier-Stokes equations in noncylindrical domains, J. Differential Equations, 25 (1977), 151-162.  doi: 10.1016/0022-0396(77)90197-8.

[4]

M. L. BernardiG. Bonfanti and F. Luterotti, Abstract Schrödinger-type differential equations with variable domain, J. Math. Anal. Appl., 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.

[5]

S. Bonaccorsi and G. Guatteri, A variational approach to evolution problems with variable domains, J. Differential Equations, 175 (2001), 51-70.  doi: 10.1006/jdeq.2000.3959.

[6]

L. BoudinC. Grandmont and A. Moussa, Global existence of solutions to the incompressible Navier-Stokes-Vlasov equations in a time-dependent domain, J. Differential Equations, 262 (2017), 1317-1340.  doi: 10.1016/j.jde.2016.10.012.

[7]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[8]

P. CannarsaG. Da Prato and J. P. Zolésto, The damped wave equation in a moving domain, J. Differential Equations, 85 (1990), 1-16.  doi: 10.1016/0022-0396(90)90086-5.

[9]

D. N. ChebanP. E. Kloeden and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144. 

[10]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-514. 

[11]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[12]

T. CaraballoG. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[15]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[16]

T. CaraballoJ. Real and P. E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.  doi: 10.1515/ans-2006-0304.

[17]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

[18]

D. R. da CostaC. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211.  doi: 10.1103/PhysRevE.83.066211.

[19]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticit et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[20]

H. Flanders, Differentiation under the integral sign, Amer. Math. Monthly, 80 (1973), 615-627.  doi: 10.1080/00029890.1973.11993339.

[21]

J. Ferreira and M. L. Santos, Asymptotic behaviour for wave equations with memory in a noncylindrical domains, Commun. Pure Appl. Anal., 2 (2003), 511-520.  doi: 10.3934/cpaa.2003.2.511.

[22]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[23]

C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.

[24]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319. 

[25]

N. JamesA. Ilyasse and D. Stevan, Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process, Internat. J. Control, 86 (2013), 1467-1478.  doi: 10.1080/00207179.2013.786187.

[26]

P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Differential Equations Appl., 6 (2000), 33-52.  doi: 10.1080/10236190008808212.

[27]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[28]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[29]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[30]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.

[31]

P. E. KloedenJ. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.

[32]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limits Non Linéaires, Dunod-Gauthier Villars, Paris, 1969.

[33]

J. LímacoL. A. Medeiros and E. Zuazua, Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains, Seventh Workshop on Partial Differential Equations, Part II, Rio de Janeiro, 2001, Mat. Contemp., 23 (2002), 49-70. 

[34]

T. F. MaP. Marín-Rubio and C. M. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.

[35]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528. 

[36] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. 
[37]

G. R. Sell, Non-autonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[38]

H. M. Soner and S. E. Shreve, A free boundary problem related to singular stochastic control: The parabolic case, Comm. Partial Differential Equations, 16 (1991), 373-424.  doi: 10.1080/03605309108820763.

[39]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.

[40]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.

[42]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[43]

Y. Yong and Q. Jiu, Energy equality and uniqueness of weak solutions to MHD equations in $L^{\infty}(0, T;L^{n}(\Omega))$, Acta Math. Sin., Engl. Ser., 25 (2009), 803-814.  doi: 10.1007/s10114-009-7214-8.

[44]

F. Zhou and C. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3767-3792.  doi: 10.3934/dcdsb.2016120.

[45]

F. ZhouC. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.

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