@Jochen Glueck Yes, this is correct. However, I have strong doubts on the existence of a non-trivial $X$, so I mention what I qualified as a crude answer. Note that you can be logarithmically close to $L^2$, playing a bit with the second integral in the series of inequalities. Also you may bluntly assume some condition of support for the Fourier transform of $u$, providing a closed space of very regular functions (in that case, you are done with the first integral, with a proper choice of $\lambda$).

@user111164 Well, I think that this requires some work: taking a close look at Depauw's example, you will see that his construction of a vector field $V$ so that $curl V$ does not have uniqueness is rather explicit (as well as Aizenman's). Then you have to check $curl(\phi V)$ where $\phi$ is a cutoff function and carefully follow what happens in the transition region where $0<\phi <1$. Some "good" choices of $\phi$ may simplify matters.

It seems that Aizenman's article deals with a non-uniqueness result and not with the existence of two different flows. The Depauw paper provides an Eulerian perspective, i.e. replaces the ODE by a first-order PDE. On the other hand, to get 3 in your question, it is quite likely that Aizenman's vector field $X$ is three-dimensional and thus $X=\text{curl} V$. To get 3, it should be enough to take $\tilde X=\text{curl}(\phi V)$ where $\phi$ is a cutoff function.

I don't think so. Maybe checking the Donoghue book on monotone matrix functions could be useful since convex functions are functions whose derivative is increasing.

@AsyaRorschach I choose $t$ small enough to stay in the coordinate chart. On the other hand, although I did not use a Lipschitz constant for $X_1$, I was glad to have the Lipschitz assumption on $X_1$ to ensure the existence of a flow. You can make various variations on the above inequalities and control $\psi_2(t,m_2)-\psi_1(t,m_1)$, which would make the role of $X_1, X_2$ more symmetrical. Also, you can get flows with an Osgood-type hypothesis "logarithmically close" to Lipschitz, e.g. for log-Lipschitz vector fields and I believe that the reasoning should be similar to the above arguments.

A nice reference for monotone-matrix functions is the book MR0486556 by Donoghue, William F., Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207. Springer-Verlag,

$[X,\Delta]=0.$

King of strange a question: given a complex number $z$, you can always find a polynomial $P_z$ such that $P(z)=0$, taking e.g. $P_z(X)=X-z$. If you want to impose a degree $n$, take $P_z(X)=(X-z) Q(X)$, where $Q$ is any polynomial with degree $n-1$.