December  2016, 21(10): 3767-3792. doi: 10.3934/dcdsb.2016120

Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case

1. 

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

Received  January 2016 Revised  March 2016 Published  November 2016

The purpose of this article is to analyze the dynamics of the following complex Ginzburg-Landau equation \begin{align*} \partial_{t}u-(\lambda+i\alpha)\Delta u+(\kappa+i\beta)|u|^{p-2}u-\gamma u=f(t) \end{align*} on non-cylindrical domains, which are obtained by diffeomorphic transformation of a bounded base domain, without any upper restriction on $p>2$, only with some restriction on $\beta/\kappa$. We establish the existence and uniqueness of strong and weak solutions as well as some energy inequalities for this equation on variable domains. Moreover the existence of a $\mathscr{D}$-pullback attractor is established for the process generated by the weak solutions under a slightly weaker condition that the measure of the spatial domains in the past is uniformly bounded above.
Citation: Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
References:
[1]

G. P. Agrawal, Optical pulse propagation in doped fiber amplifiers, Phys. Rev. A, 44 (1991), 7493-7501. doi: 10.1103/PhysRevA.44.7493.

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[3]

M. Bartuccelli, E. Constantin, C. R. Doering, J. D. Gibbon, M. Gisselfalt and G. P. Agrawal, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J.

[4]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

[5]

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

[6]

D. M. Cao, C. Y. Sun and M. H. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[7]

T. Caraballo, G. Łukaszewicz 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.

[8]

A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.

[9]

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

[10]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

[11]

P. Clément, N. Okazawa, M. Sobajima and T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, J. Differential Equations, 253 (2012), 1250-1263. doi: 10.1016/j.jde.2012.05.002.

[12]

H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292.

[13]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1089. doi: 10.1103/RevModPhys.65.851.

[14]

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.

[15]

J. Q. Duan and V. J. Ervin, On nonlinear amplitude evolution under stochastic forcing, Appl. Math. Comput., 109 (2000), 59-65. doi: 10.1016/S0096-3003(99)00016-8.

[16]

J. Q. Duan, H. V. Ly and E. S. Titi, The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation, Z. Ang. Math. Phys., 47 (1996), 432-455. doi: 10.1007/BF00916648.

[17]

M. Efendiev, A. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X.

[18]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.

[19]

B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial Differential Equations and Their Numberical Solution, Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9543.

[20]

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.

[21]

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.

[22]

S. Jimbo and Y. Morita, Ginzburg-Landau equation with magnetic effect in a thin domain, Calc. Var. Partial Differential Equations, 15 (2002), 325-352. doi: 10.1007/s005260100130.

[23]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.

[24]

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.

[25]

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.

[26]

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.

[27]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985. doi: 10.1007/978-1-4757-4317-3.

[28]

F. Li and B. You, Global attractors for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 415 (2014), 14-24. doi: 10.1016/j.jmaa.2014.01.059.

[29]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.

[30]

N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-Laplacian, J. Differential Equations, 182 (2002), 541-576. doi: 10.1006/jdeq.2001.4097.

[31]

N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl., 267 (2002), 247-263. doi: 10.1006/jmaa.2001.7770.

[32]

X. K. Pu and B. L. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92 (2011), 318-334. doi: 10.1080/00036811.2011.614601.

[33]

G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems (Montecatini terme, 1994), 208-315, Lecture Notes in Math. 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241.

[34]

C. Y. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.

[35]

C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous wave dynamics with memory-Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761. doi: 10.3934/dcdsb.2008.9.743.

[36]

C. Y. Sun, Y. P. Xiao, Z. T. Tang and Y. Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domains,, submitted., (). 

[37]

C. Y. Sun and Y. B. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177.

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[39]

B. You, Y. R. Hou, F. Li and J. P. Jiang, Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1801-1814. doi: 10.3934/dcdsb.2014.19.1801.

[40]

F. Zhou and C. Y. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: the monotonicity case,, work in progress., (). 

show all references

References:
[1]

G. P. Agrawal, Optical pulse propagation in doped fiber amplifiers, Phys. Rev. A, 44 (1991), 7493-7501. doi: 10.1103/PhysRevA.44.7493.

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[3]

M. Bartuccelli, E. Constantin, C. R. Doering, J. D. Gibbon, M. Gisselfalt and G. P. Agrawal, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J.

[4]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

[5]

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

[6]

D. M. Cao, C. Y. Sun and M. H. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[7]

T. Caraballo, G. Łukaszewicz 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.

[8]

A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.

[9]

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

[10]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

[11]

P. Clément, N. Okazawa, M. Sobajima and T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, J. Differential Equations, 253 (2012), 1250-1263. doi: 10.1016/j.jde.2012.05.002.

[12]

H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292.

[13]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1089. doi: 10.1103/RevModPhys.65.851.

[14]

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.

[15]

J. Q. Duan and V. J. Ervin, On nonlinear amplitude evolution under stochastic forcing, Appl. Math. Comput., 109 (2000), 59-65. doi: 10.1016/S0096-3003(99)00016-8.

[16]

J. Q. Duan, H. V. Ly and E. S. Titi, The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation, Z. Ang. Math. Phys., 47 (1996), 432-455. doi: 10.1007/BF00916648.

[17]

M. Efendiev, A. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X.

[18]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.

[19]

B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial Differential Equations and Their Numberical Solution, Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9543.

[20]

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.

[21]

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.

[22]

S. Jimbo and Y. Morita, Ginzburg-Landau equation with magnetic effect in a thin domain, Calc. Var. Partial Differential Equations, 15 (2002), 325-352. doi: 10.1007/s005260100130.

[23]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.

[24]

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.

[25]

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.

[26]

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.

[27]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985. doi: 10.1007/978-1-4757-4317-3.

[28]

F. Li and B. You, Global attractors for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 415 (2014), 14-24. doi: 10.1016/j.jmaa.2014.01.059.

[29]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.

[30]

N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-Laplacian, J. Differential Equations, 182 (2002), 541-576. doi: 10.1006/jdeq.2001.4097.

[31]

N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl., 267 (2002), 247-263. doi: 10.1006/jmaa.2001.7770.

[32]

X. K. Pu and B. L. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92 (2011), 318-334. doi: 10.1080/00036811.2011.614601.

[33]

G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems (Montecatini terme, 1994), 208-315, Lecture Notes in Math. 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241.

[34]

C. Y. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.

[35]

C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous wave dynamics with memory-Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761. doi: 10.3934/dcdsb.2008.9.743.

[36]

C. Y. Sun, Y. P. Xiao, Z. T. Tang and Y. Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domains,, submitted., (). 

[37]

C. Y. Sun and Y. B. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177.

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[39]

B. You, Y. R. Hou, F. Li and J. P. Jiang, Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1801-1814. doi: 10.3934/dcdsb.2014.19.1801.

[40]

F. Zhou and C. Y. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: the monotonicity case,, work in progress., (). 

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