# Commuting flows problem for non-Lipschitz vector fields

Let $$X$$ be a continuous vector field on a (say compact) manifold $$M$$, if $$X$$ has ODE uniqueness then we can define its associated flow $$\mathcal F_X:\mathbb R\times M\to M$$ uniquely given by $$\mathcal F_X(0;x)=x$$ and $$\frac d{dt}\mathcal F_X(t,x)=X(F_X(t,x))$$.

When $$X,Y$$ are two Lipschitz vector fields on (say compact) manifold $$M$$, we know if $$[X,Y]=0$$ then $$\mathcal F_X^t\circ \mathcal F_Y^s=\mathcal F_Y^s\circ\mathcal F_X^t$$ for all $$s,t\in\mathbb R$$.

A proof, for example, can be done by using approximation of smooth manifold, and apply formula $$\mathcal F_X^{-t}\mathcal F_Y^{-s}\mathcal F_X^t\mathcal F_Y^sx-x=\int_0^t\int_0^s([X,Y](f_{t,\tau,\sigma}))(\mathcal F_X^{t-\sigma}\mathcal F_Y^\tau x)d\tau d\sigma$$ where $$f_{t,\tau,\sigma}=\mathcal F_X^{-t}\mathcal F_Y^{-\tau}\mathcal F_X^\sigma:\mathbb R^n\to\mathbb R^n$$ (Lemma A.8 in this paper). The approximation works because when $$X_k\to X,Y_k\to Y$$ in Lipschitz space, then $$[X_k,Y_k]\xrightarrow{L^\infty}0$$ and $$\mathcal F_{X_k}\xrightarrow{Lips}\mathcal F_{X}$$, $$\mathcal F_{Y_k}\xrightarrow{Lips}\mathcal F_Y$$, two $$L^\infty$$-functions are multiplicable, and composition of two Lipschitz function are still Lipschitz.

When vector fields are below Lipschitz, for example, if $$X,Y$$ are log-Lipschitz, the Lie bracket is still well-defined as distribution. Let $$\alpha>\frac12$$, if $$X,Y\in C^\alpha$$, then $$[X,Y]$$ is a $$C^{\alpha-1}$$-distribution based on the fact that a $$C^\alpha$$ function can multiply with a $$C^{\alpha-1}$$-distribution.

My question is, if $$X,Y$$ are non-Lipschitz and both has ODE uniqueness, do we still have $$\mathcal F_X^t\circ \mathcal F_Y^s=\mathcal F_Y^s\circ\mathcal F_X^t$$?

When the thing goes below Lipschitz, the approximation breaks down because we cannot multiply two $$C^{-\epsilon}$$-functions whenever $$\epsilon>0$$: When $$f\in C^{1-\epsilon}$$ and $$X\in C^{-\epsilon}$$ then $$Xf$$ is undefined. And from here we know a flow log-Lipschitz vector field is merely $$C^{1-\epsilon}$$ when the time is small.

I try a lot to use approximation and apply the vanishing Lie bracket condition, but most of the time the bad regularity of composition make things undefined in a function space.

One attempt to make the vector field'' defined is to consider the generator of $$C_0$$-semigroup. Consider $$\{\mathcal F_X^t\}_{t\in\mathbb R}$$ as a $$C_0$$-group operator $$S_X(t):C^0(M)\to C^0(M)$$, given by $$S_X(t)f=f\circ\mathcal F_X^t$$. So we know $$\{S_X(t)\}$$ is one parameter subgroup of bounded linear operators on $$C^0(M)$$ which is strongly continuous. By functional analysis, we know $$\frac d{dt}|_{t=0}S(t)$$ gives a unbounded operator which is densely defined closed.

Then fix $$s\in\mathbb R$$, the family $$\{S_Y(-s)S_X(t)S_Y(s)\}_{t\in\mathbb R}$$ is a $$C_0$$-group as well. Say its infinitesimal generator be $$T_s:D(T_s)\subset C^0(M)\to C^0(M)$$. Can we define $$\frac d{ds}T_s$$ and show that it equals'' to $$[X,Y]$$?