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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. 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.  Google Scholar [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.  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. 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.  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

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##### 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.  Google Scholar [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.  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. 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.  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
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