# American Institute of Mathematical Sciences

• Previous Article
Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary
• CPAA Home
• This Issue
• Next Article
A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2$
July  2016, 15(4): 1371-1399. doi: 10.3934/cpaa.2016.15.1371

## 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

Received  December 2015 Revised  January 2015 Published  April 2015

We consider the plasma-vacuum interface problem in a classical statement when in the plasma region the flow is governed by the equations of ideal compressible magnetohydrodynamics, while in the vacuum region the magnetic field obeys the div-curl system of pre-Maxwell dynamics. The local-in-time existence and uniqueness of the solution to this problem in suitable anisotropic Sobolev spaces was proved in [17], provided that at each point of the initial interface the plasma density is strictly positive and the magnetic fields on either side of the interface are not collinear. The non-collinearity condition appears as the requirement that the symbol associated to the interface is elliptic. We now consider the case when this symbol is not elliptic and study the linearized problem, provided that the unperturbed plasma and vacuum non-zero magnetic fields are collinear on the interface. We prove a basic a priori $L^2$ estimate for this problem under the (generalized) Rayleigh-Taylor sign condition $[\partial q/\partial N]<0$ on the jump of the normal derivative of the unperturbed total pressure satisfied at each point of the interface. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh-Taylor sign condition leads to Rayleigh-Taylor instability.
Citation: Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371
##### 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.
 [1] N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079 [2] Steinar Evje, Kenneth H. Karlsen. Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1 (4) : 639-673. doi: 10.3934/nhm.2006.1.639 [3] Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885 [4] Kunquan Li, Yaobin Ou. Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 487-522. doi: 10.3934/dcdsb.2021052 [5] Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763 [6] J. I. Díaz, J. F. Padial. On a free-boundary problem modeling the action of a limiter on a plasma. Conference Publications, 2007, 2007 (Special) : 313-322. doi: 10.3934/proc.2007.2007.313 [7] King-Yeung Lam, Daniel Munther. Invading the ideal free distribution. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3219-3244. doi: 10.3934/dcdsb.2014.19.3219 [8] Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893 [9] Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1 [10] Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072 [11] Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133 [12] Robert Stephen Cantrell, Chris Cosner, Yuan Lou. Evolution of dispersal and the ideal free distribution. Mathematical Biosciences & Engineering, 2010, 7 (1) : 17-36. doi: 10.3934/mbe.2010.7.17 [13] Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086 [14] Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459 [15] Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure and Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161 [16] Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic and Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005 [17] Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic and Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047 [18] Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2661-2681. doi: 10.3934/dcdsb.2021153 [19] Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111 [20] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

2020 Impact Factor: 1.916