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

Stability of the linearized MHD-Maxwell free interface problem

Davide Catania 1, , Marcello D'Abbicco 1, and  Paolo Secchi 2,

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.
Keywords: Maxwell equations, characteristic free boundary, nonuniformly characteristic boundary., plasma-vacuum interface, Ideal compressible magneto-hydrodynamics.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35Q35, 35L50, 76E17, 76E25, 35R35, 76B0.
Citation: Davide Catania, Marcello D'Abbicco, Paolo Secchi. Stability of the linearized MHD-Maxwell free interface problem. Communications on Pure & 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.  Google Scholar

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S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007.  Google Scholar

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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.  Google Scholar

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D. Catania, Existence and stability for the 3d linearized constant-coefficient incompressible current-vortex sheets, Int. J. Differ. Equ., (2013), 1-13.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup., 41 (2008), 85-139.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum,, \emph{Arch. Ration. Mech. Anal.}, ().  doi: 10.1002/cpa.20344.  Google Scholar

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J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004. Google Scholar

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H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[21]

A. Morando, Y. Trakhinin and P. Trebeschi, The linearized plasma-vacuum interface problem in ideal incompressible mhd, Proc. of Hyp2012, to appear, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.2307/1999902.  Google Scholar

[27]

P. Secchi, Linear symmetric hyperbolic systems with characteristic boundary, Math. Methods Appl. Sci., 18 (1995), 855-870. doi: 10.1002/mma.1670181103.  Google Scholar

[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.  Google Scholar

[29]

P. Secchi, A symmetric positive system with nonuniformly characteristic boundary, Differ. Integral Equ., 11 (1998), 605-621.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differential Equations, 9 (2012), 469-509. doi: 10.1142/S0219891612500154.  Google Scholar

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.  Google Scholar

[2]

S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007.  Google Scholar

[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.  Google Scholar

[4]

D. Catania, Existence and stability for the 3d linearized constant-coefficient incompressible current-vortex sheets, Int. J. Differ. Equ., (2013), 1-13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. École Norm. Sup., 41 (2008), 85-139.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum,, \emph{Arch. Ration. Mech. Anal.}, ().  doi: 10.1002/cpa.20344.  Google Scholar

[13]

J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004. Google Scholar

[14]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Morando, Y. Trakhinin and P. Trebeschi, The linearized plasma-vacuum interface problem in ideal incompressible mhd, Proc. of Hyp2012, to appear, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.2307/1999902.  Google Scholar

[27]

P. Secchi, Linear symmetric hyperbolic systems with characteristic boundary, Math. Methods Appl. Sci., 18 (1995), 855-870. doi: 10.1002/mma.1670181103.  Google Scholar

[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.  Google Scholar

[29]

P. Secchi, A symmetric positive system with nonuniformly characteristic boundary, Differ. Integral Equ., 11 (1998), 605-621.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differential Equations, 9 (2012), 469-509. doi: 10.1142/S0219891612500154.  Google Scholar

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