Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data
Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data
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DOI：
10.1007/s40818-022-00141-6
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发表时间：
2021-06
期刊：
影响因子：
2.8
通讯作者：
T. Buckmaster;Theodore D. Drivas;S. Shkoller;V. Vicol
中科院分区：
文献类型：
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作者：
T. Buckmaster;Theodore D. Drivas;S. Shkoller;V. Vicol
流体动力学中的一个基本问题涉及到从光滑的初始数据形成不连续的冲击波。我们证明了从光滑的初值出发，具有方位对称性的二维欧拉方程的光滑解形成了第一奇性，即所谓的预激波。在这个预冲击附近的解决方案示出有一个分数级数展开系数计算的数据。使用这种精确的描述的预冲击，我们证明了一个不连续的shockkinestandering发展后的预冲击。在具有方位对称性和由激波前膨胀决定的正则性的一类熵解中，这种正则激波解是唯一的。在激波阵面发展的同时，另两个尖点型奇点的特征面也从预激波中出现。朗道和利夫希茨[，第九章，§96]称这些曲面为弱不连续，他们证明了沿着这些曲面的导数的某种奇异行为。我们证明，沿着最慢的表面，所有的流体变量，除了熵有单侧尖点从激波一侧，和正常的速度是减少其运动的方向，因此，我们称这个表面弱稀疏波。沿着表面随着流体速度运动，密度和熵形成单侧尖点，而压力和法向速度保持不变，因此，我们称这种表面为弱接触间断。
A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-calledpre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that adiscontinuous shockinstantaneously develops after the pre-shock. Thisregular shock solutionis shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termedweak discontinuitiesby Landau & Lifschitz [, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy haveone-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface aweak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy formone-sided cusps while the pressure and normal velocity remain; as such, we term this surface aweak contact discontinuity.