发表机构
Georgia Institute of Technology; Vanderbilt University（佐治亚理工学院; 范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明两个互补的小数据稳定性定理，分别给出球对称3D可压缩欧拉方程激波形成解的MGHD唯一存在性及带$1/r$密度尾部数据的全时空光滑全局存在性。
AI 中文摘要
我们证明了一对互补的小数据稳定性定理，描述了等熵、球对称$3D$可压缩欧拉方程经典解在全局未来和过去的结构。我们允许任何具有正声速的状态方程，但我们的激波形成结果不适用于Chaplygin气体。我们的第一个定理涉及光滑柯西数据的开集，这些数据是速度为零且密度为正常数的平凡数据的扰动。扰动数据具有有限动能，扰动密度等于正常数加上一个带符号的“渐近平坦”尾部。我们的主定理完整描述了数据的极大全局双曲发展（MGHD）的存在性和唯一性。MGHD的边界包含延伸到空间无穷远的超曲面，在这些超曲面上我们的解发展出梯度奇点。这是任何多维拟线性波动系统中激波形成解的第一个MGHD存在性-唯一性-稳定性结果。虽然先前的工作已经产生了“候选”MGHD的局部部分，但MGHD的存在性和唯一性都不能从局部考虑中推断出来。我们的第二个定理考虑具有相反符号的$1/r$密度尾部的类似数据。我们证明了在整个时空中的全局存在性，包括未来和过去。解在零无穷远处的渐近行为不是线性的，而是被声锥相对于平坦闵可夫斯基声锥的对数弯曲所扭曲，导致对数增强的色散。这是任何不满足零条件和弱零条件的$3D$拟线性波动系统在全时空中的第一个小数据全局存在性结果。稳定的主要机制是全局时空稀疏效应，与$1/r$尾部相关。
英文摘要
We prove a pair of complementary small-data stability theorems describing the global future and past structure of classical solutions to the isentropic, spherically symmetric $3D$ compressible Euler equations. We allow any equation of state with positive sound speed, except that our shock-formation results do not apply to the Chaplygin gas. Our first theorem concerns open sets of smooth Cauchy data that are perturbations of trivial data with vanishing velocity and constant positive density. The perturbed data have finite kinetic energy, and the perturbed density is equal to a positive constant plus a signed ``asymptotically flat'' tail. Our main theorem yields a complete description of the existence and uniqueness of the maximal globally hyperbolic development (MGHD) of the data. The boundary of the MGHD contains hypersurfaces that extend to spatial infinity on which our solutions develop gradient-singularities. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-$D$ quasilinear wave system. While prior works have yielded local portions of ``candidate'' MGHDs, neither the existence nor the uniqueness of an MGHD can be inferred from local considerations. Our second theorem considers analogous data with the opposite sign of the $1/r$ density tail. We prove global existence throughout spacetime, to both the future and past. The solution's asymptotic behavior towards null infinity is not linear, but rather is distorted by a logarithmic bending of the sound cones away from the flat Minkowski ones, leading to logarithmically enhanced dispersion. This is the first small-data global existence result in the entire spacetime for any $3D$ quasilinear wave system that fails to satisfy the null condition and the weak null condition. The main mechanism of stabilization is a global-in-spacetime rarefaction effect, tied to the $1/r$ tail.
