# American Institute of Mathematical Sciences

November  2014, 13(6): 2407-2443. doi: 10.3934/cpaa.2014.13.2407

## 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

Received  November 2013 Revised  April 2014 Published  July 2014

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary.

Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method.

An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.
Citation: Davide Catania, Marcello D'Abbicco, Paolo Secchi. Stability of the linearized MHD-Maxwell free interface problem. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2407-2443. doi: 10.3934/cpaa.2014.13.2407
##### 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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##### 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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