Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

1 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).

share|improve this answer
    
Do you have any references for this? It is quite hard to follow your comment. For example, what is $X_A(t)$? –  Matthias Ludewig May 25 '12 at 15:14
    
@Kofi. $X_A$ was the same thing as $W_A$, I just edited my answer fixing this (sorry for the confusion). Unfortunately I do not know a reference where your statement is explicitly proved. What I wrote should be enough for the case of first order derivatives; for higher derivatives you need also the formula of variation of arbitrary constants (higher order derivatives solve a inhomogeneous linear equation). If you find difficulties in proving it I can try to write more details. –  Alberto Abbondandolo May 25 '12 at 17:15
1  
Alberto, maybe Kofi just asks for a reference for the lemma, in which case we do have it (as Lemma 1.1 springerlink.com/content/kek5k4da1h33a444/fulltext.pdf ) –  Pietro Majer May 25 '12 at 18:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.