A map $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.

Motivated by conversations on this questions we ask:

Is the flow of a Harmonic vector field on $\mathbb{R}^n$, a harmonic function?

More precisely, assume that every component of the vector field $x'=f(x)$ is a harmonic map. Is it true to say that for every $t$, $\phi_t$ is a Harmonic map? Note that the converse is true. That is the vector field is Harmonic if its flow is harmonic.

A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.

Motivated by conversations on this questions we ask:

Is the flow of a Harmonic vector field on $\mathbb{R}^n$, a harmonic function?

More precisely, assume that every component of the vector field $x'=f(x)$ is a harmonic map. Is it true to say that for every $t$, $\phi_t$ is a Harmonic map? Note that the converse is true. Namely, a vector field is Harmonic if its flow is harmonic.