三维可压缩欧拉流中从折痕产生的柯西视界

中文摘要 AI 辅助

我们研究带有动态熵和非平凡涡度的三维可压缩欧拉方程的近平面对称数据的开集。在我们之前的工作中，我们证明了解会沿奇异边界产生梯度奇异性，该奇异边界是一个从称为折痕的余维2类空子流形发出的超曲面。在本文中，我们证明了一个称为柯西视界的零超曲面从折痕发出，并沿与奇异边界横向的方向传播。解在柯西视界处保持光滑，即使它“感受到”折痕处梯度奇异性的影响。柯西视界、折痕和奇异边界的并集构成了数据的最大全局双曲发展（MGHD）边界的连通部分。大致而言，MGHD是光滑数据演化为经典解的“最大”方式。已知例子表明，保证MGHD唯一性的唯一方法是证明其整个边界上成立某些结构性质。我们证明了所需性质的局部版本在柯西视界和奇异边界处成立。我们的工作首次描述了三维中无对称性假设的O(1)大小的柯西视界部分的形成、结构和稳定性。动态熵和涡度拉伸的存在（二者在二维等熵欧拉等更简单情形中不存在）带来了重大困难，我们用新技术解决了这些困难。我们的方法依赖于动态适配折痕形状的新时空叶状结构，以及基于双零叶状结构和我们前期工作中推导的能量恒等式的偏微分方程框架。总体而言，我们的技术使我们能够处理折痕缺乏严格凸性的具有挑战性的新解 regime。

英文摘要

We study open sets of nearly plane symmetric data for the $3D$ compressible Euler equations with dynamic entropy and non-trivial vorticity. In our prior work, we proved that the solutions develop a gradient singularity along a singular boundary, which is a hypersurface that emanates from a co-dimension 2, spacelike submanifold called the crease. In the present paper, we prove that a null hypersurface, called a Cauchy horizon, emanates from the crease, and propagates in a direction transverse to the singular boundary. The solution remains smooth up to the Cauchy horizon, even though it ``feels'' the influence of the gradient singularity at the crease. The union of the Cauchy horizon, the crease, and the singular boundary, make up a connected portion of the boundary of a maximal globally hyperbolic development (MGHD) of the data. Roughly, an MGHD is the ``largest'' way that smooth data can evolve into a classical solution. Known examples show that the only way one can guarantee uniqueness of an MGHD is by proving that certain structural properties hold along its entire boundary. We prove that a local version of the needed properties are satisfied along the Cauchy horizon and singular boundary. Our work provides the first description of the formation, structure, and stability of an $O(1)$-size portion of the Cauchy horizon in $3D$ without symmetry assumptions. The presence of dynamic entropy and vorticity stretching, both of which are absent in simpler settings such as $2D$ isentropic Euler, introduces significant difficulties that we resolve with novel techniques. Our approach relies on new foliations of spacetime dynamically adapted to the shape of the crease, and a PDE framework based on double-null foliations and the energy identities we derived in earlier work. Collectively, our techniques allow us to handle a challenging new solution regime where the crease lacks strict convexity.

发表机构

• Georgia Institute of Technology（佐治亚理工学院）
• Vanderbilt University（范德堡大学）

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