-
Previous Article
L∞ resolvent bounds for steady Boltzmann's Equation
- KRM Home
- This Issue
-
Next Article
Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three
The stability of contact discontinuity for compressible planar magnetohydrodynamics
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China |
This paper is concerned with the planar magnetohydrodynamicswith initial data whose behaviors at far fields $x\rightarrow \pm\infty$ are different. Motivated by the relationship between planar magnetohydrodynamics and Navier-Stokes, we can prove that the solutions to the planar magnetohydrodynamics tend time-asymptotically to a viscous contact wave which is constructed from a contact discontinuity solution of the Riemann problemon Euler system. This result is proved by the method of elementary energy estimates.
References:
| [1] |
F. V. Atkinson and L. A. Peletier,
Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal., 54 (1974), 373-392.
doi: 10.1007/BF00249197. |
| [2] |
G. Q. Chen and D. H. Wang,
Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
| [3] |
G. Q. Chen and D. H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
| [4] |
Q. Chen and Z. Tan,
Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451.
doi: 10.1016/j.na.2010.02.019. |
| [5] |
B. Ducomet and E. Feireisl,
The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
| [6] |
J. S. Fan, S. Jiang and G. Nakamura,
Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708.
doi: 10.1007/s00220-006-0167-1. |
| [7] |
H. Freistühler and P. Szmolyan,
Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128.
doi: 10.1137/S0036141093247366. |
| [8] |
J. Goodman,
Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
| [9] |
X. P. Hu and D. H. Wang,
Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
| [10] |
F. M. Huang, J. Li and A. Matsumura,
Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.
doi: 10.1007/s00205-009-0267-0. |
| [11] |
F. M. Huang, A. Matsumura and X. D. Shi,
On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004), 193-210.
|
| [12] |
F. M. Huang, A. Matsumura and Z. P. Xin,
Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.
doi: 10.1007/s00205-005-0380-7. |
| [13] |
F. M. Huang, Y. Wang and T. Yang,
Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.
doi: 10.3934/krm.2010.3.685. |
| [14] |
F. M. Huang, Z. P. Xin and T. Yang,
Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.
doi: 10.1016/j.aim.2008.06.014. |
| [15] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A, Math. Sci., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
| [16] |
H. L. Li, X. Y. Xu and J. W. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
| [17] |
T. P. Liu and Z. P. Xin,
Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
doi: 10.1007/BF01466726. |
| [18] |
B. Q. Lv and B. Huang,
On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.
doi: 10.1088/0951-7715/28/2/509. |
| [19] |
J. Nash,
Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.
|
| [20] |
X. K. Pu and B. L. Guo,
Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.
doi: 10.1007/s00033-012-0245-5. |
| [21] |
X. H. Qin, T. Wang and Y. Wang,
Global stability of wave patterns for compressible Navier-Stokes system with free boundary, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1192-1214.
doi: 10.1016/S0252-9602(16)30062-5. |
| [22] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
| [23] |
A. Vasseur and Y. Wang,
The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method, SIAM J. Math. Anal., 47 (2015), 4350-4359.
doi: 10.1137/15M1023439. |
| [24] |
D. H. Wang,
Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.
doi: 10.1137/S0036139902409284. |
| [25] |
X. Ye and J. W. Zhang,
On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit, J. Differential Equations, 260 (2016), 3927-3961.
doi: 10.1016/j.jde.2015.10.049. |
show all references
References:
| [1] |
F. V. Atkinson and L. A. Peletier,
Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal., 54 (1974), 373-392.
doi: 10.1007/BF00249197. |
| [2] |
G. Q. Chen and D. H. Wang,
Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
| [3] |
G. Q. Chen and D. H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
| [4] |
Q. Chen and Z. Tan,
Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451.
doi: 10.1016/j.na.2010.02.019. |
| [5] |
B. Ducomet and E. Feireisl,
The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
| [6] |
J. S. Fan, S. Jiang and G. Nakamura,
Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708.
doi: 10.1007/s00220-006-0167-1. |
| [7] |
H. Freistühler and P. Szmolyan,
Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128.
doi: 10.1137/S0036141093247366. |
| [8] |
J. Goodman,
Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
| [9] |
X. P. Hu and D. H. Wang,
Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
| [10] |
F. M. Huang, J. Li and A. Matsumura,
Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.
doi: 10.1007/s00205-009-0267-0. |
| [11] |
F. M. Huang, A. Matsumura and X. D. Shi,
On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004), 193-210.
|
| [12] |
F. M. Huang, A. Matsumura and Z. P. Xin,
Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.
doi: 10.1007/s00205-005-0380-7. |
| [13] |
F. M. Huang, Y. Wang and T. Yang,
Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.
doi: 10.3934/krm.2010.3.685. |
| [14] |
F. M. Huang, Z. P. Xin and T. Yang,
Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.
doi: 10.1016/j.aim.2008.06.014. |
| [15] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A, Math. Sci., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
| [16] |
H. L. Li, X. Y. Xu and J. W. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
| [17] |
T. P. Liu and Z. P. Xin,
Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
doi: 10.1007/BF01466726. |
| [18] |
B. Q. Lv and B. Huang,
On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.
doi: 10.1088/0951-7715/28/2/509. |
| [19] |
J. Nash,
Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.
|
| [20] |
X. K. Pu and B. L. Guo,
Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.
doi: 10.1007/s00033-012-0245-5. |
| [21] |
X. H. Qin, T. Wang and Y. Wang,
Global stability of wave patterns for compressible Navier-Stokes system with free boundary, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1192-1214.
doi: 10.1016/S0252-9602(16)30062-5. |
| [22] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
| [23] |
A. Vasseur and Y. Wang,
The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method, SIAM J. Math. Anal., 47 (2015), 4350-4359.
doi: 10.1137/15M1023439. |
| [24] |
D. H. Wang,
Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.
doi: 10.1137/S0036139902409284. |
| [25] |
X. Ye and J. W. Zhang,
On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit, J. Differential Equations, 260 (2016), 3927-3961.
doi: 10.1016/j.jde.2015.10.049. |
| [1] |
Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic and Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685 |
| [2] |
Toyohiko Aiki, Adrian Muntean. Large time behavior of solutions to a moving-interface problem modeling concrete carbonation. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1117-1129. doi: 10.3934/cpaa.2010.9.1117 |
| [3] |
Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096 |
| [4] |
Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419 |
| [5] |
Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 |
| [6] |
Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure and Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959 |
| [7] |
Feng Li, Erik Lindgren. Large time behavior for a nonlocal nonlinear gradient flow. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022079 |
| [8] |
Wancheng Sheng, Tong Zhang. Structural stability of solutions to the Riemann problem for a scalar nonconvex CJ combustion model. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 651-667. doi: 10.3934/dcds.2009.25.651 |
| [9] |
Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 |
| [10] |
Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure and Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 |
| [11] |
María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201 |
| [12] |
Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 707-727. doi: 10.3934/dcds.2019029 |
| [13] |
Jiamin Zhu, Emmanuel Trélat, Max Cerf. Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1347-1388. doi: 10.3934/dcdsb.2016.21.1347 |
| [14] |
Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks and Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749 |
| [15] |
Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481 |
| [16] |
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 |
| [17] |
Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 |
| [18] |
Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091 |
| [19] |
Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 |
| [20] |
Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 |
2021 Impact Factor: 1.398
Tools
Metrics
Other articles
by authors
[Back to Top]