Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity
Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity
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DOI：
10.1007/s00222-018-0799-8
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发表时间：
2016-10
影响因子：
3.1
通讯作者：
J. Luk;Jared Speck
中科院分区：
文献类型：
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作者：
J. Luk;Jared Speck
本文研究了二维可压缩Euler方程在除Chaubergin气体外的任何物理正压状态方程下的Cauchy问题。我们证明了著名的现象冲击形成简单的平面波解决方案，从光滑的初始数据，是稳定的扰动下的初始数据，打破了平面对称性。此外，我们提供了一个尖锐的渐近描述的奇异性形成。我们的工作的新特点是，扰动的解决方案被允许有小的，但非零涡，即使在激波的位置。因此，我们的研究结果提供了第一个建设性的描述附近的奇点压缩形成的涡。具体地说，涡度保持一致有界，而涡度除以密度则表现出更有规律的行为：相对于标准笛卡尔坐标，比值保持一致的Lipschitz。为了控制涡度，我们依靠新的几何和分析见解的联盟，补充了Christodoulou在无涡区域激波形成的开创性，尖锐的证明中使用的见解。特别是，我们依赖于一个新的制定的可压缩欧拉方程（在同伴文章中导出）表现出显着的结构。为了得到估计，我们构造了一个适应于声学特性（对应于声波传播）的程函函数以及一组相关的几何坐标和微分算子。由于方程组的显着结构，可以使用相同的坐标和微分算子来分析涡量，其特性与声学特性是横向的。特别是，我们的工作提供了第一个建设性的描述冲击波的形成没有对称性的假设，在一个系统中的多个速度。
We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial data, is stable under perturbations of the initial data that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression. Specifically, the vorticity remains uniformly bounded, while the vorticity divided by the density exhibits even more regular behavior: the ratio remains uniformly Lipschitz relative to the standard Cartesian coordinates. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations (derived in a companion article) exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.