关于三维Euler方程自相似解的Luo-Hou方法的注记
Remark on Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations
摘要：
本文证明了在适当的爆破剖面衰减条件下，Luo-Hou对三维Euler方程轴对称解的自相似解的证明导致了解的平凡性。爆破剖面的方程简化为一个超定的偏微分方程组偏微分方程组，其衰减的唯一解是平凡解。我们还提出了一个推广罗厚的答案。使用消失的正常速度在边界处，我们表明，这种广义自相似anomaly也导致一个平凡的解决方案。这些结果表明，自相似轮廓可能只在以自相似速率收缩到边界圆的时间相关区域内有效，也可能在自相似轮廓空间无穷远处的不同边界条件下有效。
英文摘要：
In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
