Abstract
We present a recent result [23] for the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows. This is a necessary step to prove a local-in-time existence theorem [24] for the original nonlinear free boundary problem provided that the Rayleigh-Taylor sign condition is satisfied at each point of the initial discontinuity. The uniqueness of a solution to this problem follows already from the basic a priori estimate deduced for the linearized problem.
Highlights
1.1 Free boundary problem for MHD contact discontinuitiesWe consider the equations of ideal compressible MHD:∂tρ + div = 0, ∂t(ρv) + div + ∇q =∂tH − ∇ × (v×H) = ∂t ρe + |H
We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for 2D planar flows provided that the Rayleigh-Taylor sign condition [∂p/∂N ] < 0 on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity
This paper is a natural completion of our previous analysis (Morando, Trakhinin, Trebeschi in J Differential Equations 258:2531–2571, 2015) where the wellposedness in Sobolev spaces of the linearized problem was proved under the Rayleigh-Taylor sign condition satisfied at each point of the unperturbed discontinuity
Summary
We consider the equations of ideal compressible MHD:. |v|2 total energy, and E = E(ρ, S) internal energy. According to the classification of strong discontinuities in MHD [7], for contact discontinuities there is no plasma flow across the discontinuity and the magnetic field on both its sides is nowhere tangent to the discontinuity In view of these requirements, the Rankine-Hugoniot conditions imply the boundary conditions [7]. It is not evident that the formally introduced piecewise smooth solutions to the MHD equations satisfying the boundary conditions (8) must necessarily exist, at least locally in time, for any initial data. Providing the existence and uniqueness of a smooth solution (U +, U −, φ) on some time interval [0, T ] to the free boundary problem (5), (8), (10) These conditions will be additional ones to (6) and (9) satisfied at t = 0. Provided that the initial data (10) satisfy the Rayleigh-Taylor sign condition (11) together with (6) and (9) (as well as appropriate compatibility conditions)
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