We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the jump of the tangential component of the velocity satisfies $|[v\cdot\tau]| 2\sqrt{2}\,c$, then the problem is weakly stable, and we are able to derive a wave-type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well-posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces.

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关于可压缩涡旋片的演化方程

我们关注的是二维空间维度中可压缩无粘性流体的欧拉方程的超音速涡旋片。对于常系数问题，我们推导出涡旋片不连续前沿的演化方程。这是一个二阶伪微分方程。与经典稳定性分析一致，如果速度的切向分量的跳跃满足 $|[v\cdot\tau]| 2\sqrt{2}\,c$，那么问题是弱稳定的，我们能够推导出解的波型先验能量估计，而不会丢失数据的规律性。然后我们通过证明在加权 Sobolev 空间中解的存在性来证明问题的适定性。