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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 & 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],, \emph{Comm. Partial Differential Equations}, 14 (1989), 173.  doi: 10.1080/03605308908820595.  Google Scholar

[2]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications,, Oxford University Press, (2007).  doi: 10.1093/acprof:oso/9780199211234.001.0001.  Google Scholar

[3]

A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD,, in \emph{Handbook of Mathematical Fluid Dynamics} (eds. S. Friedlander and D. Serre), (2002), 545.  doi: 10.1016/S1874-5792(02)80013-1.  Google Scholar

[4]

I.B. Bernstein, E.A. Frieman, M.D. Kruskal and R.M. Kulsrud, An energy principle for hydromagnetic stability problems,, \emph{Proc. Roy. Soc. A}, 244 (1958), 17.  doi: 10.1098/rspa.1958.0023.  Google Scholar

[5]

J.-F. Coulombel, Weakly stable multidimensional shocks,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 21 (2004), 401.  doi: 10.1016/j.anihpc.2003.04.001.  Google Scholar

[6]

J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 41 (2008), 85.   Google Scholar

[7]

D. Ebin, The equations of motion of a perfect fluid with free boundary are not well-posed,, \emph{Comm. Partial Differential Equations}, 12 (1987), 1175.  doi: 10.1080/03605308708820523.  Google Scholar

[8]

J.P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas,, Cambridge University Press, (2010).   Google Scholar

[9]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, \emph{Commun. Pure Appl. Math.}, 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar

[10]

D. Lannes, Well-posedness of the water-waves equations,, \emph{J. Amer. Math. Soc.}, 18 (2005), 605.  doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[11]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary,, \emph{Ann. of Math. (2)}, 162 (2005), 109.  doi: 10.4007/annals.2005.162.109.  Google Scholar

[12]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary,, \emph{Comm. Math. Phys.}, 260 (2005), 319.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[13]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[14]

G. Métivier, Stability of multidimensional shocks,, in \emph{Advances in the theory of shock waves} (eds. H. Freist\, 47 (2001), 25.  doi: 10.1007/978-1-4612-0193-9_2.  Google Scholar

[15]

A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD,, \emph{Quart. Appl. Math.}, 72 (2014), 549.  doi: 10.1090/S0033-569X-2014-01346-7.  Google Scholar

[16]

A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized problem for MHD contact discontinuities,, \emph{J. Differential Equations}, 258 (2015), 2531.  doi: 10.1016/j.jde.2014.12.018.  Google Scholar

[17]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, \emph{Trans. Amer. Math. Soc.}, 291 (1985), 167.  doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar

[18]

P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem,, \emph{Interface Free Bound.}, 15 (2013), 323.  doi: 10.4171/IFB/305.  Google Scholar

[19]

P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem,, \emph{Non\-linearity}, 27 (2014), 105.  doi: 10.1088/0951-7715/27/1/105.  Google Scholar

[20]

Y. Trakhinin, On existence of compressible current-vortex sheets: variable coefficients linear analysis,, \emph{Arch. Ration. Mech. Anal.}, 177 (2005), 331.  doi: 10.1007/s00205-005-0364-7.  Google Scholar

[21]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 245.  doi: 10.1007/s00205-008-0124-6.  Google Scholar

[22]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition,, \emph{Comm. Pure Appl. Math.}, 62 (2009), 1551.  doi: 10.1002/cpa.20282.  Google Scholar

[23]

Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD,, \emph{J. Differential Equations}, 249 (2010), 2577.  doi: 10.1016/j.jde.2010.06.007.  Google Scholar

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],, \emph{Comm. Partial Differential Equations}, 14 (1989), 173.  doi: 10.1080/03605308908820595.  Google Scholar

[2]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications,, Oxford University Press, (2007).  doi: 10.1093/acprof:oso/9780199211234.001.0001.  Google Scholar

[3]

A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD,, in \emph{Handbook of Mathematical Fluid Dynamics} (eds. S. Friedlander and D. Serre), (2002), 545.  doi: 10.1016/S1874-5792(02)80013-1.  Google Scholar

[4]

I.B. Bernstein, E.A. Frieman, M.D. Kruskal and R.M. Kulsrud, An energy principle for hydromagnetic stability problems,, \emph{Proc. Roy. Soc. A}, 244 (1958), 17.  doi: 10.1098/rspa.1958.0023.  Google Scholar

[5]

J.-F. Coulombel, Weakly stable multidimensional shocks,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 21 (2004), 401.  doi: 10.1016/j.anihpc.2003.04.001.  Google Scholar

[6]

J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions,, \emph{Ann. Sci. \'Ecole Norm. Sup. (4)}, 41 (2008), 85.   Google Scholar

[7]

D. Ebin, The equations of motion of a perfect fluid with free boundary are not well-posed,, \emph{Comm. Partial Differential Equations}, 12 (1987), 1175.  doi: 10.1080/03605308708820523.  Google Scholar

[8]

J.P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas,, Cambridge University Press, (2010).   Google Scholar

[9]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, \emph{Commun. Pure Appl. Math.}, 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar

[10]

D. Lannes, Well-posedness of the water-waves equations,, \emph{J. Amer. Math. Soc.}, 18 (2005), 605.  doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[11]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary,, \emph{Ann. of Math. (2)}, 162 (2005), 109.  doi: 10.4007/annals.2005.162.109.  Google Scholar

[12]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary,, \emph{Comm. Math. Phys.}, 260 (2005), 319.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[13]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[14]

G. Métivier, Stability of multidimensional shocks,, in \emph{Advances in the theory of shock waves} (eds. H. Freist\, 47 (2001), 25.  doi: 10.1007/978-1-4612-0193-9_2.  Google Scholar

[15]

A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD,, \emph{Quart. Appl. Math.}, 72 (2014), 549.  doi: 10.1090/S0033-569X-2014-01346-7.  Google Scholar

[16]

A. Morando, Y. Trakhinin and P. Trebeschi, Well-posedness of the linearized problem for MHD contact discontinuities,, \emph{J. Differential Equations}, 258 (2015), 2531.  doi: 10.1016/j.jde.2014.12.018.  Google Scholar

[17]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, \emph{Trans. Amer. Math. Soc.}, 291 (1985), 167.  doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar

[18]

P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem,, \emph{Interface Free Bound.}, 15 (2013), 323.  doi: 10.4171/IFB/305.  Google Scholar

[19]

P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem,, \emph{Non\-linearity}, 27 (2014), 105.  doi: 10.1088/0951-7715/27/1/105.  Google Scholar

[20]

Y. Trakhinin, On existence of compressible current-vortex sheets: variable coefficients linear analysis,, \emph{Arch. Ration. Mech. Anal.}, 177 (2005), 331.  doi: 10.1007/s00205-005-0364-7.  Google Scholar

[21]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 245.  doi: 10.1007/s00205-008-0124-6.  Google Scholar

[22]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition,, \emph{Comm. Pure Appl. Math.}, 62 (2009), 1551.  doi: 10.1002/cpa.20282.  Google Scholar

[23]

Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD,, \emph{J. Differential Equations}, 249 (2010), 2577.  doi: 10.1016/j.jde.2010.06.007.  Google Scholar

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