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

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.

Keywords: Non-contraction, intermediate entropic discontinuity, shock, contact discontinuity, planar isentropic MHD.

Mathematics Subject Classification: 35L65, 35L67, 35B35, 35B40.

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

References:

[1]	B. Barker, J. 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.
[2]	B. Barker, O. 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.
[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.
[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.
[5]	O. Gués, G. Métivier, M. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	B. Barker, J. 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.
[2]	B. Barker, O. 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.
[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.
[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.
[5]	O. Gués, G. Métivier, M. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic and Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047
[2]	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
[3]	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
[4]	Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[5]	Luca Dieci, Cinzia Elia, Dingheng Pi. Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3091-3112. doi: 10.3934/dcdsb.2017165
[6]	Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[7]	Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
[8]	John Bissell, Brian Straughan. Discontinuity waves as tipping points: Applications to biological & sociological systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1911-1934. doi: 10.3934/dcdsb.2014.19.1911
[9]	Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3887-3909. doi: 10.3934/cpaa.2021136
[10]	Alessandro Morando, Yuri Trakhinin, Paola Trebeschi. On local existence of MHD contact discontinuities. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 289-313. doi: 10.3934/dcdss.2016.9.289
[11]	Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569
[12]	Luca Dieci, Cinzia Elia. Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1039-1068. doi: 10.3934/dcdss.2016041
[13]	I-Kun Chen, Daisuke Kawagoe. Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography. Inverse Problems and Imaging, 2019, 13 (2) : 337-351. doi: 10.3934/ipi.2019017
[14]	Sebastian Reich, Paul J. Rozdeba. Posterior contraction rates for non-parametric state and drift estimation. Foundations of Data Science, 2020, 2 (3) : 333-349. doi: 10.3934/fods.2020016
[15]	Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic and Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559
[16]	Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005
[17]	Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4575-4598. doi: 10.3934/cpaa.2020207
[18]	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
[19]	Volker W. Elling. Shock polars for non-polytropic compressible potential flow. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1581-1594. doi: 10.3934/cpaa.2022032
[20]	Junxiong Jia, Jigen Peng, Jinghuai Gao. Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption. Inverse Problems and Imaging, 2021, 15 (2) : 201-228. doi: 10.3934/ipi.2020061

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Moon-Jin Kang
on Google Scholar
Moon-Jin Kang

2020 Impact Factor: 1.432

Tools

Article outline

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Moon-Jin Kang
on Google Scholar
Moon-Jin Kang

[Back to Top]