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The Propagation-of-Singularities Theorem is telling you that for a real-principal type operator $P$, and a given equation $Pu=f$, $$ WF(u)\backslash WF(f)\quad \text{is invariant by the flow of $H_p$} …

You are talking about $B^{0,\infty}_\infty$. Take a function $u$ in the Zygmund class $B^{1,\infty}_\infty$, which the vector space of $L^\infty$ functions such that $$\exists C,\forall x,h,\quad \v …

Here is a statement, due to Lagrange. Take a square system $\dot x=f(x)$ with $f$ of class $C^2$ such that $f(0)=0$ and the spectrum of $df(0)$ is contained in {$z\in \mathbb C , \Re{z}\le -\delta$} …

Let me single out a situation which goes the other way around: how a system of ODE is describing the propagation of singularities for a principal type PDE. Take a linear (pseudo)differential operato …

Let $X=\sum_{1\le j\le n}a_j(x)\partial_{x_j}$ be a Lipschitz-continuous vector field on some open subset of $\mathbb R^n$. The flow is then Lipschitz-continuous: it is a consequence of Gronwall's in …