We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma–vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh–Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash–Moser iteration scheme. Our result is uniform in the speed of light and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.

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非相对论和相对论理想可压缩磁流体动力学中自由边界问题的适定性

我们在两个和三个空间维度上考虑非相对论和相对论理想可压缩磁流体动力学的自由边界问题，总压力在等离子体 - 真空界面上消失。我们在总压力的瑞利-泰勒符号条件下建立了这个非线性特征双曲问题解的局部时间存在性和唯一性。该证明基于线性化问题的各向异性 Sobolev 空间中的某些温和估计以及 Nash-Moser 迭代方案的修改。我们的结果在光速方面是均匀的，并且似乎是移动边界上总压力为零的理想可压缩磁流体动力学中自由边界问题的第一个适定结果。