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Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves
Approximation of the trajectory attractor of the 3D MHD System
1. | Department of Mathematics and Computer Science, University of Dschang, Cameroon |
References:
[1] |
J. P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044. |
[2] |
J. M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[3] |
T. Caraballo, J. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.
doi: 10.1016/S0362-546X(00)00216-9. |
[4] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[5] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in pipes and channels, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[6] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa- Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[7] |
S. Chen, D. D. Holm, L. G. Margolin, and R. Zhang, Direct numerical simulations of the Navier-Stokes-alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C.R. Acad. Sci. Paris Series I, 10 (1995), 1309-1314 .
doi: 10.1016/S0021-7824(97)89978-3. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 10 (1997), 913-964 .
doi: 10.1016/S0021-7824(97)89978-3. |
[10] |
V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[11] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002. |
[12] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of trajectory attractors of 3D Navier-Stokes-$\alpha$ model as alpha approaches 0, Mat. Sb., 198 (2007), 3-36.
doi: 10.1070/SM2007v198n12ABEH003902. |
[13] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52.
doi: 10.3934/dcds.2007.17.481. |
[14] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, Trajectory attractor approximation of the 3D Navier-Stokes by a Leray-$\alpha$ model, Doklady Mathematics, 71 (2005), 92-95. |
[15] |
G. Deugoue, P. A. Razafimandimby and M. Sango, On the 3D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Processes and their Applications, 122 (2012), 2211-2248.
doi: 10.1016/j.spa.2012.03.002. |
[16] |
Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642. |
[17] |
G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[18] |
C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 153 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
[19] |
C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[20] |
A. 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. |
[21] |
J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 28pp.
doi: 10.1063/1.2360145. |
[22] |
J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968. |
[23] |
J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969. |
[24] |
M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923.
doi: 10.1016/j.physd.2010.01.009. |
[25] |
V. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 8 (2000), 375-403.
doi: 10.1023/A:1008608431399. |
[26] |
P. D. Mininni, D. C. Montgomery and A. G. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model, Phys. Rev. E, 71 (2005), 046304.
doi: 10.1103/PhysRevE.71.046304. |
[27] |
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. |
[28] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001. |
[29] |
R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997. |
[30] |
M. I. Vishik and V. V. Chepyzhov, Trajectory attractor and global attractors of three-dimensional Navier-Stokes systems, Mathematical Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[31] |
Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued processes, J. Differential Equations, 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
show all references
References:
[1] |
J. P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044. |
[2] |
J. M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[3] |
T. Caraballo, J. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.
doi: 10.1016/S0362-546X(00)00216-9. |
[4] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[5] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in pipes and channels, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[6] |
S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa- Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[7] |
S. Chen, D. D. Holm, L. G. Margolin, and R. Zhang, Direct numerical simulations of the Navier-Stokes-alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C.R. Acad. Sci. Paris Series I, 10 (1995), 1309-1314 .
doi: 10.1016/S0021-7824(97)89978-3. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 10 (1997), 913-964 .
doi: 10.1016/S0021-7824(97)89978-3. |
[10] |
V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[11] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002. |
[12] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of trajectory attractors of 3D Navier-Stokes-$\alpha$ model as alpha approaches 0, Mat. Sb., 198 (2007), 3-36.
doi: 10.1070/SM2007v198n12ABEH003902. |
[13] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52.
doi: 10.3934/dcds.2007.17.481. |
[14] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, Trajectory attractor approximation of the 3D Navier-Stokes by a Leray-$\alpha$ model, Doklady Mathematics, 71 (2005), 92-95. |
[15] |
G. Deugoue, P. A. Razafimandimby and M. Sango, On the 3D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Processes and their Applications, 122 (2012), 2211-2248.
doi: 10.1016/j.spa.2012.03.002. |
[16] |
Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642. |
[17] |
G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[18] |
C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 153 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
[19] |
C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[20] |
A. 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. |
[21] |
J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 28pp.
doi: 10.1063/1.2360145. |
[22] |
J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968. |
[23] |
J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969. |
[24] |
M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923.
doi: 10.1016/j.physd.2010.01.009. |
[25] |
V. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 8 (2000), 375-403.
doi: 10.1023/A:1008608431399. |
[26] |
P. D. Mininni, D. C. Montgomery and A. G. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model, Phys. Rev. E, 71 (2005), 046304.
doi: 10.1103/PhysRevE.71.046304. |
[27] |
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. |
[28] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001. |
[29] |
R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997. |
[30] |
M. I. Vishik and V. V. Chepyzhov, Trajectory attractor and global attractors of three-dimensional Navier-Stokes systems, Mathematical Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[31] |
Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued processes, J. Differential Equations, 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
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