A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.

According to Markus' paper A uniqueness theorem for ordinary differential equations involving smooth functions, a $\mathscr C^1$-vector field $X$ can define a flow $\mathscr F_X(t,p):\mathbb R\times M\to M$, that is the unique map satisfies $\frac\partial{\partial t}\mathscr F_X(t,p)=X\circ\mathscr F_X(t,p)$, $\mathscr F_X(0,p)=p$.

But unlike Lipschitz or Holder-$\mathscr C^\gamma$ for $\gamma>1$, where those function is stable under composition, the composition of Zygmund function may not be Zygmund.

For example, $x\log x\circ x\log x=x\log^2x+x\log x\log\log x$, and $x\log ^2x$ is not Zygmund anymore.

I believe the following is true, and my question is, is there any elementary example to:

Show that there is a $\mathscr C^1$-function $f(u,s)$ in $\mathbb R^2$ such that for the uniqueness solution $\frac\partial{\partial t}\phi(t,s)=f(\phi(t,s),s)$, $\phi(0,s)=s$ is not $\mathscr C^1$ in any neighborhood of $0$.

The original question I come up with is, if possible, to find a example for:

Show that there is a $\mathscr C^1$-vector field $X$ such that if $\Phi(t,s)$ is the parameterization (a continuous map homeomorphic to its image) near 0 such that $\partial_t\Phi\in C^0$ and $\partial_t\Phi(t,s)=X(\Phi(t,s))$, then $\Phi\notin\mathscr C^1$ and $\forall s,\ \Phi(\cdot,s)\notin\mathscr C^2_t$ in any neighborhood of $0$.

The idea I have is to make $u$-variable of $f$ behave oscillate as $u\log u$. I try $f(u,s)=u\log u$ and $\phi(1,s)=-s\log s$. I get $\partial_t\phi(1,0)\notin\mathscr C^1$, but this case $\phi$ seems still $\mathscr C^1$ near $(1,0)$, and of course it not restricted locally near 0.

Obstruction also comes from verify how the second differentiation explode: $\frac{\phi(t,s)+\phi(t,s')}2-\phi(t,\frac{s+s'}2)=\int_0^t\frac{f(\phi(u,s),s)+f(\phi(u,s'),s')}2-f(\phi(t,\frac{s+s'}2),\frac{s+s'}2)du$. Suffice to consider $\int_0^tf(\frac{\phi(u,s)+\phi(u,s')}2,\frac{s+s'}2)-f(\phi(u,\frac{s+s'}2),\frac{s+s'}2)du>>|s-s'|$.

But the fact $\phi(0,s)=s$ and $\phi$ is $C^1$ shows that the integral is bounded by $|\int_0^tf(\frac{\phi(u,s)+\phi(u,s')}2,\frac{s+s'}2)-f(\phi(u,\frac{s+s'}2),\frac{s+s'}2)du|\lesssim_\epsilon|s-s'|^{1-\epsilon}t^2$ (since $f\in\mathscr C^{1-\epsilon}$). The computation shows that we are unlike to get the idea if we fix $s$ or $s'$ to be $0$.


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