发表机构
School of Mathematics and Statistics, Wuhan University; School of Mathematical Sciences, Fudan University（武汉大学数学与统计学院; 复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半球壳中定常相对论Euler方程的跨音速激波问题，通过打靶法和单调性证明球对称解的存在唯一性，并在无背景限制下证明三维扰动的存在稳定性，采用变形张量、涡量解耦及球面投影坐标处理奇异性。
AI 中文摘要
我们研究了半球壳中定常相对论Euler方程的跨音速激波问题。通过打靶法以及出口压力与激波位置之间的单调性，我们证明了球对称跨音速激波解的存在性和唯一性。此外，在不对背景跨音速激波施加任何限制的情况下，我们证明了在出口压力的三维扰动下跨音速激波的存在性和稳定性。关键技术涉及利用变形张量和涡量将定常相对论Euler方程中的双曲型和椭圆型分量解耦。为了处理坐标奇异性，采用了结合球坐标和球极投影的“球面投影坐标”。随后，适当重新表述Rankine-Hugoniot条件以确定激波前沿，并为一阶非局部变形-涡量系统导出激波前沿上的边界条件。
英文摘要
We investigates the transonic shock problem for steady relativistic Euler equations in hemispherical shells. We show the existence and uniqueness of spherically symmetric transonic shock solutions via the shooting method and the monotonicity between the exit pressure and shock position. Furthermore, without any restrictions on background transonic shocks, we prove the existence and stability of transonic shocks under three dimensional perturbations of the exit pressure. The key techniques involve the decoupling the hyperbolic and elliptic components in steady relativistic Euler equations using the deformation tensor and vorticity. To address the coordinate singularities, the ``spherical projection coordinates" combining the spherical coordinates and the stereographic projection is employed. Subsequently, the Rankine-Hugoniot conditions are appropriately reformulated to determine the shock front and derive the boundary conditions on the shock front for a first-order nonlocal deformation-curl system.
Comments37 pages, 2 figures
