Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid

Previous Article
On the mild Itô formula in Banach spaces
DCDS-B Home
This Issue
Next Article
Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes

August 2018, 23(6): 2193-2216. doi: 10.3934/dcdsb.2018231

Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid

Guowei Liu ^, and Rui Xue

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Guowei Liu

Received November 2016 Revised April 2018 Published August 2018 Early access July 2018

Fund Project: The first author is partially supported by the National NSFC grant No.11771284 and Weng Hongwu Academic Innovation Research Fund of Peking University and Original Research Fund of Peking University.

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid on 2D bounded domains. We show existence of the pullback exponential attractor introduced by Langa, Miranville and Real [27], moreover, give existence of the global pullback attractor with finite fractal dimension and reveal the relationship between the global pullback attractor and the pullback exponential attractor. These results improve our previous associated results in papers [29,40] for the non-Newtonian fluid.

Keywords: Non-Newtonian fluid, pullback exponential attractor, global pullback attractor.

Mathematics Subject Classification: Primary: 35B41, 35Q35; Secondary: 76A05, 76D03.

Citation: Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
[3]	H. Bellout and F. Bloom, J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464.
[4]	F. Bloom and W. Hao, Regularization of a non-Newtonian system in unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3.
[5]	F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence of a maximal compact attractor, Nonlinear Anal., 43 (2001), 743-766. doi: 10.1016/S0362-546X(99)00232-1.
[6]	F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.
[7]	H. Bellout and F. Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, Springer, Cham, 2014. doi: 10.1007/978-3-319-00891-2.
[8]	S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hillard-Navier-Stokes system in 2D, Dyn. Partial Differ Equ., 11 (2014), 1-38. doi: 10.4310/DPDE.2014.v11.n1.a1.
[9]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002.
[10]	R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053.
[11]	A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.
[12]	A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 12 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141.
[13]	B. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluds in $\mathbb{R}^2$, J. Math. Anal. Appl., 298 (2004), 667-676. doi: 10.1016/j.jmaa.2004.05.032.
[14]	B. Dong and Z. Chen, Time decay rates of non-Newtonian flows in $\mathbb{R}^n_+$, J. Math. Anal. Appl., 324 (2006), 820-833. doi: 10.1016/j.jmaa.2005.12.070.
[15]	A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994.
[16]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy Soc. Edinburgh Sect. A., 135 (2005), 703-730. doi: 10.1017/S030821050000408X.
[17]	M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013. doi: 10.1090/surv/192.
[18]	P. Fabrie and A. Miranville, Exponential attractors for nonautonomous first-order evolution equation, Discrete Contin. Dyn. Syst., 4 (1998), 225-240. doi: 10.3934/dcds.1998.4.225.
[19]	Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.
[20]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in Ⅴ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010.
[21]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst., 34 (2014), 203-227. doi: 10.3934/dcds.2014.34.203.
[22]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.
[23]	P. E. Kloden, J. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.
[24]	J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[25]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969.
[26]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[27]	J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.
[28]	G. Liu, C. Zhao and J. Cao, $H^4$-boundedness of pullback attractor for a 2D non-Newtonian fluid flow, Front. Math. China, 8 (2013), 1377-1390. doi: 10.1007/s11464-013-0250-9.
[29]	G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech., 19 (2017), 623-643. doi: 10.1007/s00021-016-0299-9.
[30]	J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.
[31]	A. Miranville, Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22. doi: 10.1016/S0893-9659(98)00004-4.
[32]	M. Pokorný, Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201.
[33]	J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[34]	G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.
[35]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[36]	C. Zhao and Y. Li, $H^2$-compact attractor for a non-Newtonian system in two-dimensional unbound domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006.
[37]	C. Zhao and S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.
[38]	C. Zhao, Y. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031.
[39]	C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302.
[40]	C. Zhao, G. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2.
[41]	C. Zhao, G. Liu and R. An, Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays, Differ. Equ. Dyn. Syst., 25 (2017), 39-64. doi: 10.1007/s12591-014-0231-9.

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
[3]	H. Bellout and F. Bloom, J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464.
[4]	F. Bloom and W. Hao, Regularization of a non-Newtonian system in unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3.
[5]	F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence of a maximal compact attractor, Nonlinear Anal., 43 (2001), 743-766. doi: 10.1016/S0362-546X(99)00232-1.
[6]	F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.
[7]	H. Bellout and F. Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, Springer, Cham, 2014. doi: 10.1007/978-3-319-00891-2.
[8]	S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hillard-Navier-Stokes system in 2D, Dyn. Partial Differ Equ., 11 (2014), 1-38. doi: 10.4310/DPDE.2014.v11.n1.a1.
[9]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002.
[10]	R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053.
[11]	A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.
[12]	A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 12 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141.
[13]	B. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluds in $\mathbb{R}^2$, J. Math. Anal. Appl., 298 (2004), 667-676. doi: 10.1016/j.jmaa.2004.05.032.
[14]	B. Dong and Z. Chen, Time decay rates of non-Newtonian flows in $\mathbb{R}^n_+$, J. Math. Anal. Appl., 324 (2006), 820-833. doi: 10.1016/j.jmaa.2005.12.070.
[15]	A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994.
[16]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy Soc. Edinburgh Sect. A., 135 (2005), 703-730. doi: 10.1017/S030821050000408X.
[17]	M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013. doi: 10.1090/surv/192.
[18]	P. Fabrie and A. Miranville, Exponential attractors for nonautonomous first-order evolution equation, Discrete Contin. Dyn. Syst., 4 (1998), 225-240. doi: 10.3934/dcds.1998.4.225.
[19]	Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.
[20]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in Ⅴ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010.
[21]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst., 34 (2014), 203-227. doi: 10.3934/dcds.2014.34.203.
[22]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.
[23]	P. E. Kloden, J. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.
[24]	J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[25]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969.
[26]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[27]	J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.
[28]	G. Liu, C. Zhao and J. Cao, $H^4$-boundedness of pullback attractor for a 2D non-Newtonian fluid flow, Front. Math. China, 8 (2013), 1377-1390. doi: 10.1007/s11464-013-0250-9.
[29]	G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech., 19 (2017), 623-643. doi: 10.1007/s00021-016-0299-9.
[30]	J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.
[31]	A. Miranville, Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22. doi: 10.1016/S0893-9659(98)00004-4.
[32]	M. Pokorný, Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201.
[33]	J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[34]	G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.
[35]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[36]	C. Zhao and Y. Li, $H^2$-compact attractor for a non-Newtonian system in two-dimensional unbound domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006.
[37]	C. Zhao and S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.
[38]	C. Zhao, Y. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031.
[39]	C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302.
[40]	C. Zhao, G. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2.
[41]	C. Zhao, G. Liu and R. An, Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays, Differ. Equ. Dyn. Syst., 25 (2017), 39-64. doi: 10.1007/s12591-014-0231-9.

[1]	M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[2]	Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
[3]	Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[4]	Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[5]	Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[6]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[7]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[8]	Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[9]	Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[10]	Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036
[11]	Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2699-2722. doi: 10.3934/dcdsb.2020028
[12]	I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
[13]	José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329
[14]	Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[15]	Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[16]	Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure and Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
[17]	Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143
[18]	Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[19]	Theodore Tachim Medjo. Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088
[20]	Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Guowei Liu Rui Xue
on Google Scholar
Guowei Liu Rui Xue

2021 Impact Factor: 1.497

Tools

Article outline

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Guowei Liu Rui Xue
on Google Scholar
Guowei Liu Rui Xue

[Back to Top]