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A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $
On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol
1. | Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk, Russian Federation |
References:
[1] |
S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, (French) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems], 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.
doi: 10.1093/acprof:oso/9780199211234.001.0001. |
[3] |
A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, in Handbook of Mathematical Fluid Dynamics (eds. S. Friedlander and D. Serre), vol. 1, North-Holland, Amsterdam, (2002), 545-652.
doi: 10.1016/S1874-5792(02)80013-1. |
[4] |
I.B. Bernstein, E.A. Frieman, M.D. Kruskal and R.M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. A, 244 (1958), 17-40.
doi: 10.1098/rspa.1958.0023. |
[5] |
J.-F. Coulombel, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 401-443.
doi: 10.1016/j.anihpc.2003.04.001. |
[6] |
J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup. (4), 41 (2008), 85-139. |
[7] |
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. |
[8] |
J.P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2010. Information on this title at: http://www.cambridge.org/9780521879576. |
[9] |
H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-296.
doi: 10.1002/cpa.3160230304. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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.
doi: 10.1007/978-1-4612-0193-9_2. |
[15] |
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. |
[16] |
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. |
[17] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.1090/S0002-9947-1985-0797053-4. |
[18] |
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. |
[19] |
P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Non-linearity, 27 (2014), 105-169.
doi: 10.1088/0951-7715/27/1/105. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
show all references
References:
[1] |
S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, (French) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems], 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.
doi: 10.1093/acprof:oso/9780199211234.001.0001. |
[3] |
A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, in Handbook of Mathematical Fluid Dynamics (eds. S. Friedlander and D. Serre), vol. 1, North-Holland, Amsterdam, (2002), 545-652.
doi: 10.1016/S1874-5792(02)80013-1. |
[4] |
I.B. Bernstein, E.A. Frieman, M.D. Kruskal and R.M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. A, 244 (1958), 17-40.
doi: 10.1098/rspa.1958.0023. |
[5] |
J.-F. Coulombel, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 401-443.
doi: 10.1016/j.anihpc.2003.04.001. |
[6] |
J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup. (4), 41 (2008), 85-139. |
[7] |
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. |
[8] |
J.P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2010. Information on this title at: http://www.cambridge.org/9780521879576. |
[9] |
H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-296.
doi: 10.1002/cpa.3160230304. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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.
doi: 10.1007/978-1-4612-0193-9_2. |
[15] |
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. |
[16] |
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. |
[17] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.1090/S0002-9947-1985-0797053-4. |
[18] |
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. |
[19] |
P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Non-linearity, 27 (2014), 105-169.
doi: 10.1088/0951-7715/27/1/105. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
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