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Stability of the linearized MHD-Maxwell free interface problem
1. | DICATAM, Mathematical Division, University of Brescia, Via Valotti, 9, 25133 Brescia, Italy, Italy |
2. | Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, 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, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007. |
[3] |
I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. London. Ser. A., 244 (1958), 17-40. |
[4] |
D. Catania, Existence and stability for the 3d linearized constant-coefficient incompressible current-vortex sheets, Int. J. Differ. Equ., (2013), 1-13. |
[5] |
G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369-408.
doi: 10.1007/s00205-007-0070-8. |
[6] |
J. F. Coulombel, A. Morando, P. Secchi and P. Trebeschi, A priori estimate for 3-D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275.
doi: 10.1007/s00220-011-1340-8. |
[7] |
J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.
doi: 10.1512/iumj.2004.53.2526. |
[8] |
J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup., 41 (2008), 85-139. |
[9] |
D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587.
doi: 10.1007/s00220-010-1028-5. |
[10] |
D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.
doi: 10.1090/S0894-0347-07-00556-5. |
[11] |
D. Coutand and S. Shkoller, A simple proof of well-posedness of the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449.
doi: 10.3934/dcdss.2010.3.429. |
[12] |
D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., to appear.
doi: 10.1002/cpa.20344. |
[13] |
J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004. |
[14] |
H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. |
[15] |
D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic).
doi: 10.1090/S0894-0347-05-00484-4. |
[16] |
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. |
[17] |
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968. |
[18] |
N. Mandrik and Y. Trakhinin, Influence of vacuum electric field on the stability of a plasma-vacuum interface, Commun. Math. Sci., 12 (2014), 1065-1100. |
[19] |
A. Morando, P. Secchi and P. Trebeschi, Regularity of solutions to characteristic initial-boundary value problems for symmetrizable systems, J. Hyperbolic Differ. Equ., 6 (2009), 753-808.
doi: 10.1142/S021989160900199X. |
[20] |
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. |
[21] |
A. Morando, Y. Trakhinin and P. Trebeschi, The linearized plasma-vacuum interface problem in ideal incompressible mhd, Proc. of Hyp2012, to appear, 2013. |
[22] |
A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible mhd, Quart. Appl. Math., to appear, 2013.
doi: 10.4171/IFB/305. |
[23] |
A. Morando and P. Secchi, Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary, J. Hyperbolic Differ. Equ., 8 (2011), 37-99.
doi: 10.1142/S021989161100238X. |
[24] |
A. Morando and P. Secchi, Regularity of weakly well-posed characteristic boundary value problems, Int. J. Differ. Equ. (2010), 39 pages.
doi: 10.1155/2010/524736. |
[25] |
T. Nishitani and M. Takayama, Regularity of solutions to non-uniformly characteristic boundary value problems for symmetric systems, Comm. Partial Differential Equations, 25 (2000), 987-1018.
doi: 10.1080/03605300008821539. |
[26] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.2307/1999902. |
[27] |
P. Secchi, Linear symmetric hyperbolic systems with characteristic boundary, Math. Methods Appl. Sci., 18 (1995), 855-870.
doi: 10.1002/mma.1670181103. |
[28] |
P. Secchi, The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity, Differential Integral Equations, 9 (1996), 671-700. |
[29] |
P. Secchi, A symmetric positive system with nonuniformly characteristic boundary, Differ. Integral Equ., 11 (1998), 605-621. |
[30] |
P. Secchi, Full regularity of solutions to a nonuniformly characteristic boundary value problem for symmetric positive systems, Adv. Math. Sci. Appl., 10 (2000), 39-55. |
[31] |
P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound., 15 (2013), 323-357.
doi: 10.4171/IFB/305. |
[32] |
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. |
[33] |
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. |
[34] |
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. |
[35] |
Y. Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differential Equations, 9 (2012), 469-509.
doi: 10.1142/S0219891612500154. |
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, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007. |
[3] |
I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. London. Ser. A., 244 (1958), 17-40. |
[4] |
D. Catania, Existence and stability for the 3d linearized constant-coefficient incompressible current-vortex sheets, Int. J. Differ. Equ., (2013), 1-13. |
[5] |
G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369-408.
doi: 10.1007/s00205-007-0070-8. |
[6] |
J. F. Coulombel, A. Morando, P. Secchi and P. Trebeschi, A priori estimate for 3-D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275.
doi: 10.1007/s00220-011-1340-8. |
[7] |
J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.
doi: 10.1512/iumj.2004.53.2526. |
[8] |
J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup., 41 (2008), 85-139. |
[9] |
D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587.
doi: 10.1007/s00220-010-1028-5. |
[10] |
D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.
doi: 10.1090/S0894-0347-07-00556-5. |
[11] |
D. Coutand and S. Shkoller, A simple proof of well-posedness of the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449.
doi: 10.3934/dcdss.2010.3.429. |
[12] |
D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., to appear.
doi: 10.1002/cpa.20344. |
[13] |
J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004. |
[14] |
H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. |
[15] |
D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic).
doi: 10.1090/S0894-0347-05-00484-4. |
[16] |
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. |
[17] |
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968. |
[18] |
N. Mandrik and Y. Trakhinin, Influence of vacuum electric field on the stability of a plasma-vacuum interface, Commun. Math. Sci., 12 (2014), 1065-1100. |
[19] |
A. Morando, P. Secchi and P. Trebeschi, Regularity of solutions to characteristic initial-boundary value problems for symmetrizable systems, J. Hyperbolic Differ. Equ., 6 (2009), 753-808.
doi: 10.1142/S021989160900199X. |
[20] |
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. |
[21] |
A. Morando, Y. Trakhinin and P. Trebeschi, The linearized plasma-vacuum interface problem in ideal incompressible mhd, Proc. of Hyp2012, to appear, 2013. |
[22] |
A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible mhd, Quart. Appl. Math., to appear, 2013.
doi: 10.4171/IFB/305. |
[23] |
A. Morando and P. Secchi, Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary, J. Hyperbolic Differ. Equ., 8 (2011), 37-99.
doi: 10.1142/S021989161100238X. |
[24] |
A. Morando and P. Secchi, Regularity of weakly well-posed characteristic boundary value problems, Int. J. Differ. Equ. (2010), 39 pages.
doi: 10.1155/2010/524736. |
[25] |
T. Nishitani and M. Takayama, Regularity of solutions to non-uniformly characteristic boundary value problems for symmetric systems, Comm. Partial Differential Equations, 25 (2000), 987-1018.
doi: 10.1080/03605300008821539. |
[26] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.2307/1999902. |
[27] |
P. Secchi, Linear symmetric hyperbolic systems with characteristic boundary, Math. Methods Appl. Sci., 18 (1995), 855-870.
doi: 10.1002/mma.1670181103. |
[28] |
P. Secchi, The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity, Differential Integral Equations, 9 (1996), 671-700. |
[29] |
P. Secchi, A symmetric positive system with nonuniformly characteristic boundary, Differ. Integral Equ., 11 (1998), 605-621. |
[30] |
P. Secchi, Full regularity of solutions to a nonuniformly characteristic boundary value problem for symmetric positive systems, Adv. Math. Sci. Appl., 10 (2000), 39-55. |
[31] |
P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound., 15 (2013), 323-357.
doi: 10.4171/IFB/305. |
[32] |
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. |
[33] |
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. |
[34] |
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. |
[35] |
Y. Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differential Equations, 9 (2012), 469-509.
doi: 10.1142/S0219891612500154. |
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