论文标题
截短的快速慢性PDE的截短的Galerkin离散化的几何分析
Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities
论文作者
论文摘要
我们考虑一个快速偏差的部分微分方程（PDE），在快速变量中具有反应 - 扩散动力学，以及由有界域上的差分运算符驱动的慢变量。假设反应项的跨临界正常形式并将慢变量视为动态分叉参数，我们分析了PDE的光谱Galerkin近似的快速子系统分叉点。我们通过爆炸分析使用几何底线化来表征有限维圆锥形odes的不变歧管。除了至关重要的近似程序外，我们还在爆破分析过程中使域动态动态。最后，我们详细说明了我们的结果近似于无限维度问题。在我们的分析中，我们发现出现在入口和退出爆炸图中的PDE是准线性的自由边界值问题，而在中央/缩放图中，我们获得了PDE，PDE通常在经典的反应 - 扩散问题中遇到了具有有限时间奇异性的解决方案。
We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the reaction term and viewing the slow variable as a dynamic bifurcation parameter, we analyze the passage through the fast subsystem bifurcation point for the spectral Galerkin approximation of the PDE. We characterize the invariant manifolds for the finite-dimensional Galerkin ODEs using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, we also make the domain dynamic during the blow-up analysis. Finally, we elaborate in which sense our results approximate the infinite-dimensional problem. Within our analysis, we find that the PDEs appearing in entry and exit blow-up charts are quasi-linear free boundary value problems, while in the central/scaling chart we obtain a PDE, which is often encountered in classical reaction-diffusion problems exhibiting solutions with finite-time singularities.
