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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 |
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.
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. Bernardi, G. 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. Boudin, C. 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. Cannarsa, G. 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. Cheban, P. 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. Cui, X. 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. Caraballo, G. 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. Carvalho, J. A. Langa, J. 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. Caraballo, J. 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. Caraballo, J. 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 Costa, C. 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. James, A. 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. Kloeden, P. 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. Kloeden, J. 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. Límaco, L. 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. Ma, P. 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. Song, C. 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. Zhou, C. 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. Bernardi, G. 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. Boudin, C. 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. Cannarsa, G. 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. Cheban, P. 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. Cui, X. 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. Caraballo, G. 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. Carvalho, J. A. Langa, J. 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. Caraballo, J. 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. Caraballo, J. 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 Costa, C. 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. James, A. 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. Kloeden, P. 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. Kloeden, J. 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. Límaco, L. 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. Ma, P. 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. Song, C. 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. Zhou, C. 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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