Question

Suppose I have a smooth vector field that has the form $$ X(y) = \sum_j \lambda_j y^j \partial_j + \text{higher order terms}$$ for $\lambda_j>0$. Let $\Phi_t$ be the flow of $X$. Then it follows that $\Phi_t(y) \longrightarrow 0$ for $y$ near $0$ as $t \longrightarrow - \infty$.

I am now looking estimates on the $y$-derivatives. Precisely, suppose that $K$ is a compact neighborhood of $0$ that lies in the unstable manifold near the point $0$. I would like to have a statement like "For every multiindex $\alpha$, there exists a constant $C>0$ such that $$ \sup_{y \in K} |D^\alpha_y \Phi_t(y)| \leq C e^{t\lambda}$$ for all $t<0$ and $y \in K$, where $\lambda$ is the smallest eigenvalue of the linearization of $X$ at $0$"

Is some statement like this true? Where to find it or how do I prove it?

Answer 1

This is certainly true if you choose $\lambda$ to be strictly smaller than the smaller eigenvalue of $DX(0)$. You may prove it inductively, by noticing that for a given $y$ the function $t\mapsto D^{\alpha}_y \Phi_t(y)$ solves a linear equation.

For instance, the first step goes as follows: the path of matrices $W(t):= D_y^{\alpha} \Phi_t(y)$ solves the ODE $$ W'(t) = DX(\Phi_t(y)) W(t), \quad W(0)=I, $$ where $\|DX(\Phi_t(y)) - DX(0)\| \leq C_0 e^{\lambda_0 t}$ for all $t\leq 0$. Then for every $\lambda_1<\lambda_0$ you can find $C_1$ such that $\|DX(\Phi_t(y))\| \leq C_1 e^{\lambda_1 t}$ for all $t\leq 0$.

A useful lemma for proving this and getting the uniformity you need is the following: given a continuous bounded path of matrices $t\mapsto A(t)$, $t\geq 0$, denote by $W_A(t)$ the solution of the linear Cauchy problem $$ W_A'(t) = A(t) W_A(t), \quad W_A(0) = I. $$ Assume that $\|W_A(t)W_A(s)^{-1}\|\leq c e^{\lambda (t-s)}$ for every $t\geq s\geq 0$. Then for every continuous bounded path of matrices $t\mapsto H(t)$, $t\geq 0$, there holds $$ \| W_{A+H}(t)W_{A+H}(s)^{-1}\|\leq c e^{\mu (t-s)}, \quad \forall t\geq s\geq 0, $$ with $\mu := \lambda + c \|H\|_{\infty}$.

(Sorry if here I switched to positive time, that's just because I am more used to work with stable manifolds).