• Previous Article
    Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
  • KRM Home
  • This Issue
  • Next Article
    Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum
February  2018, 11(1): 107-118. doi: 10.3934/krm.2018006

Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics

Department of Mathematics, Sookmyung Women's University, Seoul 140-742, Korea

Received  July 2015 Revised  March 2017 Published  August 2017

Fund Project: This work was supported by the Foundation Sciences Mathématiques de Paris as a postdoctoral fellowship, and by an AMS-Simons Travel Grant. The author was also supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2013R1A6A3A03020506). The author thanks Professor Kevin Zumbrun for valuable comments on stability issues for MHD.

We investigate a non-contraction property of large perturbations around intermediate entropic shock waves and contact discontinuities for the three-dimensional planar compressible isentropic magnetohydrodynamics (MHD). To do that, we take advantage of criteria developed by the author and Vasseur in [6], and non-contraction property is measured by pseudo distance based on relative entropy.

Citation: Moon-Jin Kang. Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics. Kinetic & Related Models, 2018, 11 (1) : 107-118. doi: 10.3934/krm.2018006
References:
[1]

B. BarkerJ. Humpherys and K. Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, J. Differential Equations, 249 (2010), 2175-2213.  doi: 10.1016/j.jde.2010.07.019.  Google Scholar

[2]

B. BarkerO. Lafitte and K. Zumbrun, Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 447-498.  doi: 10.1016/S0252-9602(10)60058-6.  Google Scholar

[3]

I.-L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math., 42 (1989), 815-844.  doi: 10.1002/cpa.3160420606.  Google Scholar

[4]

H. Freistühler and Y. Trakhinin, On the viscous and inviscid stability of magnetohydrodynamic shock waves, Phys. D: Nonlinear Phenomena, 237 (2008), 3030-3037.  doi: 10.1016/j.physd.2008.07.003.  Google Scholar

[5]

O. GuésG. MétivierM. Williams and K. Zumbrun, Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations, Arch. Ration. Mech. Anal., 197 (2010), 1-87.  doi: 10.1007/s00205-009-0277-y.  Google Scholar

[6]

M.-J. Kang and A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws, Arch. Ration. Mech. Anal., 222 (2016), 343-391.  doi: 10.1007/s00205-016-1003-1.  Google Scholar

[7]

N. Leger and A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations, Arch. Ration. Mech. Anal., 201 (2011), 271-302.  doi: 10.1007/s00205-011-0431-1.  Google Scholar

[8]

M. Lewicka, $L^1$ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J., 49 (2000), 1515-1537.  doi: 10.1512/iumj.2000.49.1899.  Google Scholar

[9]

M. Lewicka and K. Trivisa, On the $L^1$ well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179 (2002), 133-177.  doi: 10.1006/jdeq.2000.4000.  Google Scholar

[10]

G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations, 211 (2005), 61-134.  doi: 10.1016/j.jde.2004.06.002.  Google Scholar

[11]

A. Vasseur, Relative entropy and contraction for extremal shocks of Conservation Laws up to a shift, Contemporary Mathematics of the AMS, 666 (2016), 385-404.  doi: 10.1090/conm/666/13296.  Google Scholar

[12]

K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48 (1999), 937-992.  doi: 10.1512/iumj.1999.48.1765.  Google Scholar

show all references

References:
[1]

B. BarkerJ. Humpherys and K. Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, J. Differential Equations, 249 (2010), 2175-2213.  doi: 10.1016/j.jde.2010.07.019.  Google Scholar

[2]

B. BarkerO. Lafitte and K. Zumbrun, Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 447-498.  doi: 10.1016/S0252-9602(10)60058-6.  Google Scholar

[3]

I.-L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math., 42 (1989), 815-844.  doi: 10.1002/cpa.3160420606.  Google Scholar

[4]

H. Freistühler and Y. Trakhinin, On the viscous and inviscid stability of magnetohydrodynamic shock waves, Phys. D: Nonlinear Phenomena, 237 (2008), 3030-3037.  doi: 10.1016/j.physd.2008.07.003.  Google Scholar

[5]

O. GuésG. MétivierM. Williams and K. Zumbrun, Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations, Arch. Ration. Mech. Anal., 197 (2010), 1-87.  doi: 10.1007/s00205-009-0277-y.  Google Scholar

[6]

M.-J. Kang and A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws, Arch. Ration. Mech. Anal., 222 (2016), 343-391.  doi: 10.1007/s00205-016-1003-1.  Google Scholar

[7]

N. Leger and A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations, Arch. Ration. Mech. Anal., 201 (2011), 271-302.  doi: 10.1007/s00205-011-0431-1.  Google Scholar

[8]

M. Lewicka, $L^1$ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J., 49 (2000), 1515-1537.  doi: 10.1512/iumj.2000.49.1899.  Google Scholar

[9]

M. Lewicka and K. Trivisa, On the $L^1$ well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179 (2002), 133-177.  doi: 10.1006/jdeq.2000.4000.  Google Scholar

[10]

G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations, 211 (2005), 61-134.  doi: 10.1016/j.jde.2004.06.002.  Google Scholar

[11]

A. Vasseur, Relative entropy and contraction for extremal shocks of Conservation Laws up to a shift, Contemporary Mathematics of the AMS, 666 (2016), 385-404.  doi: 10.1090/conm/666/13296.  Google Scholar

[12]

K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48 (1999), 937-992.  doi: 10.1512/iumj.1999.48.1765.  Google Scholar

[1]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[2]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[3]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[4]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[5]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[6]

Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001

[7]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[8]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020397

[9]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226

[10]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85

[11]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[12]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

[13]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[14]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[15]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[16]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[17]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[18]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[19]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[20]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (67)
  • HTML views (149)
  • Cited by (2)

Other articles
by authors

[Back to Top]