利用整体不变流形理解小粘性Burgers方程的亚稳性
Using global invariant manifolds to understand metastability in Burgers equation with small viscosity
摘要：
利用不变流形描述了具有小粘性的Burgers方程解的长时间行为。特别是，几何解释被称为亚稳态的现象，这在本上下文中意味着，解决方案花了很长的时间附近的家庭的解决方案称为扩散N波最终收敛到一个稳定的自相似扩散波。更确切地说，它示出，在相似性，或缩放，在代数加权$L^2$空间中的变量，自相似扩散波对应于一个一维的全球中心流形的固定解决方案。通过这些不动点中的每一个，存在一个一维的，全局的，吸引的，不变流形对应于扩散N-波。因此，亚稳态对应于一个快速的瞬态，其中解决方案接近这个“亚稳态”的扩散N波流形，然后沿着这个流形缓慢衰减，最后，收敛到自相似扩散波。
英文摘要：
The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this "metastable" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.
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