全球动力学、爆炸和比安奇宇宙学
Global Dynamics, Blow-Up, and Bianchi Cosmology
摘要：
几何学、拓扑学和数学物理学中的许多中心问题都引出了关于常微分方程和偏微分方程解的长时间动态的问题。例子从广义相对论的爱因斯坦场方程到抛物型的准线性反应-平流-扩散方程。具体问题涉及到均衡的收敛，周期，同宿，异宿解的存在，以及全局吸引子的存在和几何结构。另一方面，许多解在有限时间内产生奇点。在试图将解扩展到奇点之外之前，必须对奇点进行详细分析，或者理解它们的几何形状与全局有界解。在这方面，我们还着眼于全面定性描述的爆破和成长的现象。
英文摘要：
Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
