• Previous Article
    Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data
  • CPAA Home
  • This Issue
  • Next Article
    Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities
November  2016, 15(6): 2007-2021. doi: 10.3934/cpaa.2016025

Uniform global existence and convergence of Euler-Maxwell systems with small parameters

1. 

20 Rue de Vialle, Lamothe, 43100, France

Received  April 2015 Revised  April 2016 Published  September 2016

The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.
Citation: Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model, SIAM J. Appl. Math., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Alì, L. Chen, A. Jungel and Y.-J. Peng, The zero-electron-mass limit in the hydrodynamic models for plasmas, Nonlinear Analysis TMA, 72 (2010), 4410-4427. doi: 10.1016/j.na.2010.02.016.

[3]

C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci., 14 (2004), 393-415. doi: 10.1142/S0218202504003283.

[4]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195.

[5]

Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Comm. Math. Sci., 1 (2003), 437-447.

[6]

G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319. doi: 10.1016/j.jde.2008.05.015.

[7]

J. Y. Chemin, Fluides Parfaits Incompressibles, Astérisque No. 230, 1995.

[8]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984.

[9]

G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport theory and statistical physics, 29 (2000), 311-331. doi: 10.1080/00411450008205877.

[10]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Transactions Amer. Math. Soc., 359 (2007), 637-648. doi: 10.1090/S0002-9947-06-04028-1.

[11]

P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of computational physics, 231 (2012), 1917-1946. doi: 10.1016/j.jcp.2011.11.011.

[12]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case, J. Hyper. Diff. Equations, 8 (2011), 375-413. doi: 10.1142/S0219891611002421.

[13]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566. doi: 10.1006/jdeq.1995.1172.

[14]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Annales Scientifiques de l'ENS, 47 (2014), 469-503.

[15]

Y. Guo, A. D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D,, preprint, ().  doi: 10.4007/annals.2016.183.2.1.

[16]

B. Hanouzet and R. Natalini, Global existence of smooth solutions for partial dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 169 (2003), 89-117. doi: 10.1007/s00205-003-0257-6.

[17]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[18]

A. D. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein-Gordon equations in 3D, J. Eur. Math. Soc., 16 (2014), 2355-2431. doi: 10.4171/JEMS/489.

[19]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703. doi: 10.1142/S0218202594000388.

[20]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033. doi: 10.1080/03605309908821456.

[21]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Math. Appl., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[23]

C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semi-conductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360. doi: 10.1142/S0218202500000215.

[24]

C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dyn. Syst., 5 (1999), 449-455. doi: 10.3934/dcds.1999.5.449.

[25]

C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 447-461. doi: 10.1007/s00030-012-0159-0.

[26]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New-York, 1984. doi: 10.1007/978-1-4612-1116-7.

[27]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equations, Arch. Ration. Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.

[28]

P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[29]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Annals of Mathematics, 28-B (2007), 583-602. doi: 10.1007/s11401-005-0556-3.

[30]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Communications in Partial Differential Equations, 33 (2008), 349-376. doi: 10.1080/03605300701318989.

[31]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 349-376. doi: 10.1137/070686056.

[32]

Y. J. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970. doi: 10.1137/100786927.

[33]

Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pure Appl., 103 (2015), 39-67. doi: 10.1016/j.matpur.2014.03.007.

[34]

Y. J. Peng, Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters, SIAM J. Math. Anal., 47 (2015), 1355-1376. doi: 10.1137/140983276.

[35]

Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, AN (2015), http://dx.doi.org/10.2016/j.anihpc.2015.03.006

[36]

Y. J. Peng and V. Wasiolek, Uniform global existence and parabolic limit for partially dissipative hyperbolic Systems,, preprint., ().  doi: 10.1016/j.jde.2016.01.019.

[37]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.

[38]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[39]

B. Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal., 42 (2005), 211-250.

[40]

B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal., 184 (2007), 121-183. doi: 10.1007/s00205-006-0034-4.

[41]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-267. doi: 10.4310/MAA.2011.v18.n3.a1.

[42]

W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors, SIAM J. Appl. Math., 64 (2004), 1737-1748. doi: 10.1137/S0036139903427404.

[43]

W. A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188.

show all references

References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-Dimensional Euler-Poisson model, SIAM J. Appl. Math., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Alì, L. Chen, A. Jungel and Y.-J. Peng, The zero-electron-mass limit in the hydrodynamic models for plasmas, Nonlinear Analysis TMA, 72 (2010), 4410-4427. doi: 10.1016/j.na.2010.02.016.

[3]

C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci., 14 (2004), 393-415. doi: 10.1142/S0218202504003283.

[4]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195.

[5]

Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Comm. Math. Sci., 1 (2003), 437-447.

[6]

G. Carbou, B. Hanouzet and R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319. doi: 10.1016/j.jde.2008.05.015.

[7]

J. Y. Chemin, Fluides Parfaits Incompressibles, Astérisque No. 230, 1995.

[8]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984.

[9]

G. Q. Chen, J. W. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport theory and statistical physics, 29 (2000), 311-331. doi: 10.1080/00411450008205877.

[10]

J. F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Transactions Amer. Math. Soc., 359 (2007), 637-648. doi: 10.1090/S0002-9947-06-04028-1.

[11]

P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of computational physics, 231 (2012), 1917-1946. doi: 10.1016/j.jcp.2011.11.011.

[12]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system : the relaxation case, J. Hyper. Diff. Equations, 8 (2011), 375-413. doi: 10.1142/S0219891611002421.

[13]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566. doi: 10.1006/jdeq.1995.1172.

[14]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Annales Scientifiques de l'ENS, 47 (2014), 469-503.

[15]

Y. Guo, A. D. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D,, preprint, ().  doi: 10.4007/annals.2016.183.2.1.

[16]

B. Hanouzet and R. Natalini, Global existence of smooth solutions for partial dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 169 (2003), 89-117. doi: 10.1007/s00205-003-0257-6.

[17]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[18]

A. D. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein-Gordon equations in 3D, J. Eur. Math. Soc., 16 (2014), 2355-2431. doi: 10.4171/JEMS/489.

[19]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703. doi: 10.1142/S0218202594000388.

[20]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033. doi: 10.1080/03605309908821456.

[21]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Math. Appl., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[23]

C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semi-conductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360. doi: 10.1142/S0218202500000215.

[24]

C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dyn. Syst., 5 (1999), 449-455. doi: 10.3934/dcds.1999.5.449.

[25]

C. Lin and J. F. Coulombel, The strong relaxation limit of the multidimensional Euler equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 447-461. doi: 10.1007/s00030-012-0159-0.

[26]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New-York, 1984. doi: 10.1007/978-1-4612-1116-7.

[27]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equations, Arch. Ration. Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.

[28]

P. A. Markowich, C. A. Ringhofer and C. Shmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[29]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Annals of Mathematics, 28-B (2007), 583-602. doi: 10.1007/s11401-005-0556-3.

[30]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Communications in Partial Differential Equations, 33 (2008), 349-376. doi: 10.1080/03605300701318989.

[31]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 349-376. doi: 10.1137/070686056.

[32]

Y. J. Peng, S. Wang and Q. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970. doi: 10.1137/100786927.

[33]

Y. J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pure Appl., 103 (2015), 39-67. doi: 10.1016/j.matpur.2014.03.007.

[34]

Y. J. Peng, Uniformly global smooth solutions and convergence of Euler-Poisson systems with small parameters, SIAM J. Math. Anal., 47 (2015), 1355-1376. doi: 10.1137/140983276.

[35]

Y. J. Peng and V. Wasiolek, Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, AN (2015), http://dx.doi.org/10.2016/j.anihpc.2015.03.006

[36]

Y. J. Peng and V. Wasiolek, Uniform global existence and parabolic limit for partially dissipative hyperbolic Systems,, preprint., ().  doi: 10.1016/j.jde.2016.01.019.

[37]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.

[38]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[39]

B. Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal., 42 (2005), 211-250.

[40]

B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal., 184 (2007), 121-183. doi: 10.1007/s00205-006-0034-4.

[41]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-267. doi: 10.4310/MAA.2011.v18.n3.a1.

[42]

W. A. Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors, SIAM J. Appl. Math., 64 (2004), 1737-1748. doi: 10.1137/S0036139903427404.

[43]

W. A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188.

[1]

Jiang Xu, Wen-An Yong. Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1319-1332. doi: 10.3934/dcds.2009.25.1319

[2]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[3]

Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure and Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963

[4]

Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure and Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365

[5]

Dayton Preissl, Christophe Cheverry, Slim Ibrahim. Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system. Kinetic and Related Models, 2021, 14 (6) : 1035-1079. doi: 10.3934/krm.2021042

[6]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[7]

Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic and Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002

[8]

Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313

[9]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156

[10]

Yachun Li, Qiufang Shi. Global existence of the entropy solutions to the isentropic relativistic Euler equations. Communications on Pure and Applied Analysis, 2005, 4 (4) : 763-778. doi: 10.3934/cpaa.2005.4.763

[11]

Seung-Yeal Ha, Myeongju Kang, Bora Moon. Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics. Kinetic and Related Models, 2021, 14 (6) : 1003-1033. doi: 10.3934/krm.2021036

[12]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002

[13]

Nuno J. Alves, Athanasios E. Tzavaras. The relaxation limit of bipolar fluid models. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 211-237. doi: 10.3934/dcds.2021113

[14]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001

[15]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046

[16]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136

[17]

Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211

[18]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[19]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133

[20]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (176)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]