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February  2016, 9(1): 289-313. doi: 10.3934/dcdss.2016.9.289

## On local existence of MHD contact discontinuities

 1 DICATAM, Sezione di Matematica, Università di Brescia, Via Valotti, 9, 25133 Brescia 2 Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk, Russian Federation 3 Dipartimento di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia

Received  September 2014 Revised  February 2015 Published  December 2015

We present a recent result [23] for the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows. This is a necessary step to prove a local-in-time existence theorem [24] for the original nonlinear free boundary problem provided that the Rayleigh-Taylor sign condition is satisfied at each point of the initial discontinuity. The uniqueness of a solution to this problem follows already from the basic a priori estimate deduced for the linearized problem.
Citation: 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
##### References:
 [1] S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations, 14 (1989), 173-230. doi: 10.1080/03605308908820595. [2] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-order Systems and Applications, Oxford University Press, Oxford, 2007. [3] A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, in Handbook of Mathematical Fluid Dynamics, vol. 1 (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, 2002, 545-652. doi: 10.1016/S1874-5792(02)80013-1. [4] J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup. (4), 41 (2008), 85-139. [5] J.-F. Coulombel, A. Morando, P. Secchi and P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275. doi: 10.1007/s00220-011-1340-8. [6] D. Ebin, The equations of motion of a perfect fluid with free boundary are not well-posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201. doi: 10.1080/03605308708820523. [7] J. Fang and L. Zhang, Two-dimensional magnetohydrodynamics simulations of young type Ia supernova remnants, Mon. Not. R. Astron. Soc., 424 (2012), 2811-2820. doi: 10.1111/j.1365-2966.2012.21405.x. [8] O. L. Filippova, Stability of plane MHD shock waves in an ideal gas, Fluid Dyn., 26 (1991), 897-904. doi: 10.1007/BF01056793. [9] J. P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139195560. [10] K. Ilin and Y. Trakhinin, On stability of Alfvén discontinuities, Math. Methods Appl. Sci., 32 (2009), 307-329. doi: 10.1002/mma.1039. [11] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [12] B. Kwon, Structural conditions for full MHD equations, Quart. Appl. Math., 67 (2009), 593-600. doi: 10.1090/S0033-569X-09-01139-6. [13] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4. [14] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [15] L. D. Landau, E. M. Lifshiz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. [16] H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194. doi: 10.4007/annals.2005.162.109. [17] H. Lindblad, Well-posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392. doi: 10.1007/s00220-005-1406-6. [18] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [19] G. Métivier, Stability of multidimensional shocks, in Advances in the Theory of Shock Waves (eds. H. Freistühler and A. Szepessy), Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser, Boston, 2001, 25-103. [20] 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. [21] A. Morando, Y. Trakhinin and P. Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl., 347 (2008), 502-520. doi: 10.1016/j.jmaa.2008.06.002. [22] A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math., 72 (2014), 549-587. doi: 10.1090/S0033-569X-2014-01346-7. [23] A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571. doi: 10.1016/j.jde.2014.12.018. [24] A. Morando, Y. Trakhinin and P. Trebeschi, Local existence of MHD contact discontinuities, work in progress. [25] P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interface Free Bound., 15 (2013), 323-357. doi: 10.4171/IFB/305. [26] P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), 105-169. doi: 10.1088/0951-7715/27/1/105. [27] Y. Trakhinin, A complete 2D stability analysis of fast MHD shocks in an ideal gas, Comm. Math. Phys., 236 (2003), 65-92. doi: 10.1007/s00220-002-0791-3. [28] Y. Trakhinin, On existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331-366. doi: 10.1007/s00205-005-0364-7. [29] Y. Trakhinin, On the existence of incompressible current-vortex sheets: Study of a linearized free boundary value problem, Math. Methods Appl. Sci., 28 (2005), 917-945. doi: 10.1002/mma.600. [30] Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. doi: 10.1007/s00205-008-0124-6. [31] Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math., 62 (2009), 1551-1594. doi: 10.1002/cpa.20282. [32] Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, J. Differential Equations, 249 (2010), 2577-2599. doi: 10.1016/j.jde.2010.06.007. [33] T. Yanagisawa and A. Matsumura, The fixed boundary value problems for the equations of ideal magnetohydrodynamics with a perfectly conducting wall condition, Comm. Math. Phys., 136 (1991), 119-140. doi: 10.1007/BF02096793.

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##### References:
 [1] S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations, 14 (1989), 173-230. doi: 10.1080/03605308908820595. [2] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-order Systems and Applications, Oxford University Press, Oxford, 2007. [3] A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, in Handbook of Mathematical Fluid Dynamics, vol. 1 (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, 2002, 545-652. doi: 10.1016/S1874-5792(02)80013-1. [4] J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup. (4), 41 (2008), 85-139. [5] J.-F. Coulombel, A. Morando, P. Secchi and P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275. doi: 10.1007/s00220-011-1340-8. [6] D. Ebin, The equations of motion of a perfect fluid with free boundary are not well-posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201. doi: 10.1080/03605308708820523. [7] J. Fang and L. Zhang, Two-dimensional magnetohydrodynamics simulations of young type Ia supernova remnants, Mon. Not. R. Astron. Soc., 424 (2012), 2811-2820. doi: 10.1111/j.1365-2966.2012.21405.x. [8] O. L. Filippova, Stability of plane MHD shock waves in an ideal gas, Fluid Dyn., 26 (1991), 897-904. doi: 10.1007/BF01056793. [9] J. P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139195560. [10] K. Ilin and Y. Trakhinin, On stability of Alfvén discontinuities, Math. Methods Appl. Sci., 32 (2009), 307-329. doi: 10.1002/mma.1039. [11] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [12] B. Kwon, Structural conditions for full MHD equations, Quart. Appl. Math., 67 (2009), 593-600. doi: 10.1090/S0033-569X-09-01139-6. [13] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4. [14] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [15] L. D. Landau, E. M. Lifshiz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. [16] H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194. doi: 10.4007/annals.2005.162.109. [17] H. Lindblad, Well-posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392. doi: 10.1007/s00220-005-1406-6. [18] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [19] G. Métivier, Stability of multidimensional shocks, in Advances in the Theory of Shock Waves (eds. H. Freistühler and A. Szepessy), Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser, Boston, 2001, 25-103. [20] 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. [21] A. Morando, Y. Trakhinin and P. Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl., 347 (2008), 502-520. doi: 10.1016/j.jmaa.2008.06.002. [22] A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math., 72 (2014), 549-587. doi: 10.1090/S0033-569X-2014-01346-7. [23] A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571. doi: 10.1016/j.jde.2014.12.018. [24] A. Morando, Y. Trakhinin and P. Trebeschi, Local existence of MHD contact discontinuities, work in progress. [25] P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interface Free Bound., 15 (2013), 323-357. doi: 10.4171/IFB/305. [26] P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), 105-169. doi: 10.1088/0951-7715/27/1/105. [27] Y. Trakhinin, A complete 2D stability analysis of fast MHD shocks in an ideal gas, Comm. Math. Phys., 236 (2003), 65-92. doi: 10.1007/s00220-002-0791-3. [28] Y. Trakhinin, On existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331-366. doi: 10.1007/s00205-005-0364-7. [29] Y. Trakhinin, On the existence of incompressible current-vortex sheets: Study of a linearized free boundary value problem, Math. Methods Appl. Sci., 28 (2005), 917-945. doi: 10.1002/mma.600. [30] Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. doi: 10.1007/s00205-008-0124-6. [31] Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math., 62 (2009), 1551-1594. doi: 10.1002/cpa.20282. [32] Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, J. Differential Equations, 249 (2010), 2577-2599. doi: 10.1016/j.jde.2010.06.007. [33] T. Yanagisawa and A. Matsumura, The fixed boundary value problems for the equations of ideal magnetohydrodynamics with a perfectly conducting wall condition, Comm. Math. Phys., 136 (1991), 119-140. doi: 10.1007/BF02096793.
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