arXiv 0812.11842008-12-05math.AP

一类混合双曲-抛物方程组粘性剖面的不变流形

Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems

Stefano Bianchini, Laura V. Spinolo

备注：: 10 pages

摘要：

本文研究了一类混合双曲抛物方程组在一个空间变量中的粘性分布（行波和定常解）。对于包含可压缩Navier-Stokes方程的一类系统，这些剖面满足形式为\label{e:ab} dU / dt = F（U）/ z（U）的奇异常微分方程。这里U取$R^d$中的值，而$F: R^d \to R^d$是一个常规函数。实值函数$z (U)$也是正则的，但是方程是奇异的，因为$z (U)$可以达到值0。我们专注于一个足够小的邻域点$\bar U$满足$F(\bar U) = \vec 0$，$z (\bar U) =0$。从双曲抛物系统的研究应用的角度来看，这意味着限制到系统的小的总变化。讨论了如何推广中心流形和一致稳定流形的概念。此外，我们给出的条件，确保如果$z (U) \neq 0$在$t=0$，然后$z (U) \neq 0$在每个$t$。我们提供了一个例子表明，如果$z(U)$在有限时间内变为零，那么在一般情况下，方程\eqref{e:ab}的解$U$不是连续可微的。

英文摘要：

We are concerned with viscous profiles (travelling waves and steady solutions) for mixed hyperbolic-parabolic systems in one space variable. For a class of systems including the compressible Navier Stokes equation, these profiles satisfy a singular ordinary differential equation in the form \label{e:ab} dU / dt = F(U)/ z (U) . Here U takes values in $R^d$ and $F: R^d \to R^d$ is a regular function. The real valued function $z (U)$ is as well regular, but the equation is singular because $z (U)$ can attain the value 0. We focus on a small enough neighbourhood of a point $\bar U$ satisfying $F(\bar U) = \vec 0$, $z (\bar U) =0$. From the point of view of the applications to the study of hyperbolic-parabolic systems this means restricting to systems with small total variation. We discuss how to extend the notions of center manifold and of uniformly stable manifold. Also, we give conditions ensuring that if $z (U) \neq 0$ at $t=0$ then $z (U) \neq 0$ at every $t$. We provide an example showing that if $z(U)$ becomes zero in finite time then in general the solution $U$ of equation \eqref{e:ab} is not continuously differentiable.

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。

摘要：

英文摘要：

相关论文