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We have the continuity equation

$$ \partial_t \, P(\mathbf{x},t) = -\nabla \cdot[ \, P(\mathbf{x},t) \, \mathbf{v}(\mathbf{x},t) \, ] $$

where the positive scalar function $P$ is advected by the smooth field $\mathbf{v}$ (everything can be considered continuous and with continuous derivatives).

Imagine that the field $\mathbf{v}(\mathbf{x})$ is prescribed and constant in time. How do we get the steady (i.e. time independent) solution $P(\mathbf{x})$? Namely, we want to solve

$$ \nabla \cdot[ \, P(\mathbf{x}) \, \mathbf{v}(\mathbf{x}) \, ] = 0 \, . $$

Note: this means that the field $P \mathbf{v}$ is solenoidal and there should be a vector potential $P \mathbf{v} = \nabla \times \mathbf{A}$. Therefore, the function $P$ is a sort of "integrating factor" but for the existence of the vector potential (instead of the more usual notion of integrating factor for non-exact 1-forms that involves a scalar potential). Maybe the language of differential forms can help understanding the problem (i.e. looking at $q$ as a 0-form and $\mathbf{w}$ as a 1-form.. or a 2-form, since the divergence should be the external derivative for 2-forms in three-dimensional space). Partial answers and insights of any kind are well accepted.

Quillo
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    Is $\mathbf w$ nowhere-vanishing? If $\mathbf w$ vanishes — say when $z=0$ — but $\nabla\cdot\mathbf w$ is nonzero, then you are in trouble. Yes, it's natural to frame this in terms of $2$-forms. – Ted Shifrin Jun 18 '20 at 04:48
  • Thank you! let's say $\mathbf{w}$ is non vanishing. If $\mathbf{w}=0$ only in some isolated points, which is the source of trouble? – Quillo Jun 18 '20 at 08:20
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    Try $\mathbf w = z \mathbf k$, which vanishes on the $xy$-plane. The vector field $\mathbf w = x\mathbf i + y\mathbf j + z\mathbf k$ vanishes only at the origin. – Ted Shifrin Jun 18 '20 at 16:55
  • For nowhere-vanishing vector fields, try $\mathbf w = (e^{f(x,y)},e^{g(x,y)},0)$ with $f_{xy}\ne g_{xy}$. I don't believe you can find $q$ in this case, but I haven't worked out all the details. – Ted Shifrin Jun 19 '20 at 21:11
  • Related question about the steady state of the diffusion-advection in periodic boundary conditions: https://math.stackexchange.com/q/3759470/532409 – Quillo Jul 16 '20 at 21:46

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