具有正宇宙学常数的爱因斯坦 - 欧拉系统的湍流、范数膨胀和伦德尔不稳定性
Ultraviolet Cascades, Norm Inflation, and the Rendall Instability for the Einstein-Euler System with Positive Cosmological Constant
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中文总结 AI 辅助
研究具有正宇宙学常数的爱因斯坦 - 欧拉方程微扰，首次探讨一维外伦德尔不稳定性，发现其导致流体能量密度的正向湍流级联和范数膨胀，是小数据湍流首例，还能驱动晚期宇宙结构形成。
中文摘要 AI 辅助
我们对具有正宇宙学常数且状态方程为\(p = K\rho\)（\(K\in(\frac{1}{3},1)\)）的爱因斯坦 - 欧拉方程的空间均匀且正交解的微扰进行了数值研究。近期数值和分析工作表明，在这些微扰时空中，流体密度梯度会形成陡峭梯度并在接近未来类时无穷时爆炸，即‘伦德尔不稳定性’。本文首次研究了一维空间外这种不稳定性的起始。结果表明，伦德尔不稳定性导致流体能量密度出现正向湍流级联和\(H^{s}\)范数膨胀，这是具有正宇宙学常数的场方程解的小数据湍流的首个例子，还发现该不稳定性可驱动晚期宇宙中的结构形成。
英文摘要
We numerically study perturbations of spatially homogeneous and orthogonal solutions to the Einstein-Euler equations with positive cosmological constant and linear equation of state $p=Kρ$ for $K \in (\frac{1}{3},1)$. Recent numerical and analytic work has demonstrated that the fluid density gradient in these perturbed spacetimes can form steep gradients and blow up as future timelike infinity is approached, a phenomenon now known as the `Rendall instability'. In this article, we study the onset of this instability outside of one spatial dimension for the first time. Our results provide compelling evidence that the Rendall instability causes a forward ultraviolet cascade and $H^{s}$ norm inflation in the energy density of the fluid from small $U(1)$-symmetric perturbations. This is, to the best of our knowledge, the first example of small data norm inflation for solutions of the field equations with a positive cosmological constant.
发表机构
- Institute of Applied and Computational Mathematics FORTH（FORTH应用与计算数学研究所）
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