Title: Nonlinear stability of the Larson-Penston collapse Show Chinese title

Title: 非线性稳定性 of the Larson-Penston collapse

Abstract: We prove nonlinear stability of the Larson-Penston family of self-similarly collapsing solutions to the isothermal Euler-Poisson system. Our result applies to radially symmetric perturbations and it is the first full nonlinear stability result for radially imploding compressible flows. At the heart of the proof is the ground state character of the Larson-Penston solution, which exhibits important global monotonicity properties used throughout the proof. One of the key challenges is the proof of mode-stability for the non self-adjoint spectral problem which arises when linearising the dynamics around the Larson-Penston collapsing solution. To exclude the presence of complex growing modes other than the trivial one associated with time translation symmetry, we use a high-order energy method in low and high frequency regimes, for which the monotonicity properties are crucially exploited, and use rigorous computer-assisted techniques in the intermediate regime. In addition, the maximal dissipativity of the linearised operator is proven on arbitrary large backward light cones emanating from the singular point using the global monotonicity of the Larson-Penston solutions. Such a flexibility in linear analysis also facilitates nonlinear analysis and allows us to identify the exact number of derivatives necessary for the nonlinear stability statement. The proof is based on a two-tier high-order weighted energy method which ties bounds derived from the Duhamel formula to quasilinear top order estimates. To prove global existence we further use the Brouwer fixed point theorem to identify the final collapse time, which suppresses the trivial instability caused by the time-translation symmetry of the system. Show Chinese abstract

Abstract: 我们证明了等温Euler-Poisson系统自相似坍缩解的Larson-Penston族的非线性稳定性。 我们的结果适用于径向对称扰动，并且是对于径向内爆可压缩流的第一个完整的非线性稳定性结果。 证明的核心是Larson-Penston解的基态特性，它表现出在整个证明过程中使用的重要全局单调性性质。 其中一个关键挑战是当在Larson-Penston坍缩解周围线性化动力学时出现的非自伴谱问题的模态稳定性证明。 为了排除除了与时间平移对称性相关的平凡模态以外的复数增长模态，我们在低频和高频区域使用高阶能量方法，其中单调性性质被关键地利用，并在中间区域使用严格计算机辅助技术。 此外，利用Larson-Penston解的全局单调性，在任意大的从奇点发出的后向光锥上证明了线性化算子的最大耗散性。 这种在线性分析中的灵活性也促进了非线性分析，并使我们能够确定非线性稳定性陈述所需的导数的确切数量。 证明基于一种双层高阶加权能量方法，该方法将从Duhamel公式得出的边界与准线性最高阶估计联系起来。 为了证明全局存在性，我们进一步使用Brouwer不动点定理来确定最终坍缩时间，这抑制了由系统的时间平移对称性引起的平凡不稳定性。