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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 |
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. |
show all references
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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