doi: 10.3934/dcds.2021132
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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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021132
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[22]

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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