• Previous Article
    Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment
  • DCDS-B Home
  • This Issue
  • Next Article
    Quasi-periodic solutions for a class of beam equation system
January  2020, 25(1): 55-80. doi: 10.3934/dcdsb.2019172

Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

1. 

School of Mathematics and Statistics, Xidian University, Xi'an 710126, China

2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: Bo You

Received  November 2018 Published  July 2019

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459, 11801427, 11871389), the Natural Science Foundation of Shaanxi Province (2018JQ1009, 2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088).

The main objective of this paper is to study the long-time behavior of solutions for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping for $ r>4. $ Inspired by the the methods of $ \ell $-trajectories in [27], we will prove the existence of a finite dimensional pullback attractor and a pullback exponential attractor, which gives another way of considering the long-time behavior of the non-autonomous evolutionary equations.

Citation: Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
References:
[1]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.  Google Scholar

[3]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.  Google Scholar

[4]

X. J. Cai and Q. S. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[7]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.  Google Scholar

[8]

A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.  Google Scholar

[9] J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar
[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar

[11]

B. Q. Dong and Y. Jia, Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations, Nonlinear Anal. Real World Appl., 30 (2016), 41-58.  doi: 10.1016/j.nonrwa.2015.10.011.  Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994.  Google Scholar

[13]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[14]

M. Efendiev 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.  Google Scholar

[15]

F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398.  doi: 10.1023/A:1021937715194.  Google Scholar

[16]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997.  Google Scholar

[17]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376.  doi: 10.1007/s00205-002-0234-5.  Google Scholar

[18]

Y. JiaX. W. Zhang and B. Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747.  doi: 10.1016/j.nonrwa.2010.11.006.  Google Scholar

[19]

Z. H. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.  doi: 10.1016/j.na.2012.04.014.  Google Scholar

[20]

Z. H. Jiang and M. X. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102.  doi: 10.1002/mma.1540.  Google Scholar

[21]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar

[22]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.  Google Scholar

[23]

F. LiB. You and Y. Xu, Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4267-4284.   Google Scholar

[24]

Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[25]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[26]

J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518.  doi: 10.1006/jdeq.1996.0080.  Google Scholar

[27]

J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, J. Differential Equations, 181 (2002), 243-279.  doi: 10.1006/jdeq.2001.4087.  Google Scholar

[28]

C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Lett., 57 (2016), 126-131.  doi: 10.1016/j.aml.2016.01.016.  Google Scholar

[29] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[30]

R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269.  doi: 10.1016/j.jde.2006.03.004.  Google Scholar

[31]

G. R. Sell, Global attractor for the three dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[32]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[33]

X. L. Song and Y. R. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.  doi: 10.3934/dcds.2011.31.239.  Google Scholar

[34]

X. L. Song and Y. R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351.  doi: 10.1016/j.jmaa.2014.08.044.  Google Scholar

[35]

X. L. SongF. Liang and J. Su, Exponential attractor for the three dimensional Navier-Stokes equation with nonlinear damping, Journal of Pure and Applied Mathematics: Advances and Applications, 14 (2015), 27-39.   Google Scholar

[36]

X. L. Song, F. Liang and J. H. Wu, Pullback $\mathcal{D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), Paper No. 145, 15 pp. doi: 10.1186/s13661-016-0654-z.  Google Scholar

[37]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[38]

L. YangM. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar

[39]

B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410.  doi: 10.1515/ans-2013-0208.  Google Scholar

[40]

Z. J. ZhangX. L. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.  Google Scholar

[41]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

show all references

References:
[1]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.  Google Scholar

[3]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.  Google Scholar

[4]

X. J. Cai and Q. S. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[7]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.  Google Scholar

[8]

A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.  Google Scholar

[9] J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar
[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar

[11]

B. Q. Dong and Y. Jia, Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations, Nonlinear Anal. Real World Appl., 30 (2016), 41-58.  doi: 10.1016/j.nonrwa.2015.10.011.  Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994.  Google Scholar

[13]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[14]

M. Efendiev 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.  Google Scholar

[15]

F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398.  doi: 10.1023/A:1021937715194.  Google Scholar

[16]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997.  Google Scholar

[17]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376.  doi: 10.1007/s00205-002-0234-5.  Google Scholar

[18]

Y. JiaX. W. Zhang and B. Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747.  doi: 10.1016/j.nonrwa.2010.11.006.  Google Scholar

[19]

Z. H. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.  doi: 10.1016/j.na.2012.04.014.  Google Scholar

[20]

Z. H. Jiang and M. X. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102.  doi: 10.1002/mma.1540.  Google Scholar

[21]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar

[22]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.  Google Scholar

[23]

F. LiB. You and Y. Xu, Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4267-4284.   Google Scholar

[24]

Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[25]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[26]

J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518.  doi: 10.1006/jdeq.1996.0080.  Google Scholar

[27]

J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, J. Differential Equations, 181 (2002), 243-279.  doi: 10.1006/jdeq.2001.4087.  Google Scholar

[28]

C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Lett., 57 (2016), 126-131.  doi: 10.1016/j.aml.2016.01.016.  Google Scholar

[29] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[30]

R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269.  doi: 10.1016/j.jde.2006.03.004.  Google Scholar

[31]

G. R. Sell, Global attractor for the three dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[32]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[33]

X. L. Song and Y. R. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.  doi: 10.3934/dcds.2011.31.239.  Google Scholar

[34]

X. L. Song and Y. R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351.  doi: 10.1016/j.jmaa.2014.08.044.  Google Scholar

[35]

X. L. SongF. Liang and J. Su, Exponential attractor for the three dimensional Navier-Stokes equation with nonlinear damping, Journal of Pure and Applied Mathematics: Advances and Applications, 14 (2015), 27-39.   Google Scholar

[36]

X. L. Song, F. Liang and J. H. Wu, Pullback $\mathcal{D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), Paper No. 145, 15 pp. doi: 10.1186/s13661-016-0654-z.  Google Scholar

[37]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[38]

L. YangM. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar

[39]

B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410.  doi: 10.1515/ans-2013-0208.  Google Scholar

[40]

Z. J. ZhangX. L. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.  Google Scholar

[41]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

[1]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[2]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[3]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[4]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[5]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[6]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[8]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[9]

Nikolai Botkin, Varvara Turova, Barzin Hosseini, Johannes Diepolder, Florian Holzapfel. Tracking aircraft trajectories in the presence of wind disturbances. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021010

[10]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[11]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[12]

Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021018

[13]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[14]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[15]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[16]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[17]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[18]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[19]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[20]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (313)
  • HTML views (370)
  • Cited by (0)

Other articles
by authors

[Back to Top]