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Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
4
votes
A function which belongs on a concrete Besov Space
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
$$\exists C,\forall x,h,\quad
\v …
2
votes
Accepted
Exponential stability in nonlinear differential equations
Here is a statement, due to Lagrange. Take a square system $\dot x=f(x)$ with $f$
of class $C^2$
such that $f(0)=0$ and the spectrum of $df(0)$ is contained in
{$z\in \mathbb C , \Re{z}\le -\delta$} …
5
votes
Hörmander’s propagation of singularities in two variables
The Propagation-of-Singularities Theorem is telling you that for a real-principal type operator $P$, and a given equation $Pu=f$,
$$
WF(u)\backslash WF(f)\quad \text{is invariant by the flow of $H_p$} …
2
votes
Replacing large-dimensional ODE systems with one PDE
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 …
1
vote
Estimating the flow when we know the vector field
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 …