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I asked this question on stackexchange: Let us consider the Cauchy problem for the transport equation $$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,\cdot)=\varphi_0, $$

where $\text{div}(b)=0$. It is well known that for the homogeneous transport equation

$$ \partial_t \varphi + b\cdot \nabla \varphi= 0 \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,\cdot)=\varphi_0, $$

there exists a map $X\in C([0,T]$,{$\phi:\mathbb{R}^3\to\mathbb{R} \text{ measurable } \})$ which is measure preserving, i.e. $\lambda\circ X(t)=\lambda$ for all $t\in \mathbb{R}$ and if $\varphi_0$ is measurable then $\varphi(t,x)=\varphi_0(X(t,x))$ is the unique solution of the homogeneous transport equation. This is due to DiPerna-Lions.
Now I'm interested in the inhomogeneous case. Are there any sufficient conditions on $f$ to assure the same for the inhomogeneous transport equation at least with $\lambda\circ X\leq \lambda$?

https://math.stackexchange.com/questions/4418757/existence-of-measure-preserving-lagrange-flow-for-inhomogeneous-transport-equati

Since I got no answer, I'm asking it here. I would be grateful for any comment!

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    $\begingroup$ What do you expect as an answer? Certainly you cannot have $\varphi(t,x) = \varphi_0(X(t,x))$: even in the classical case you need to account for the integral $\int_0^T f(X(s,x)) ~ds$ to in the solution. But the method of characteristics will definitely get you a measure preserving flow when $\mathrm{div}b = 0$ (if $b$ and $f$ are sufficiently smooth this can be done by calculus), and the addition of an inhomogeneity has no impact on the flow. $\endgroup$
    – Willie Wong
    Commented Apr 4, 2022 at 15:19
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    $\begingroup$ DiPerna-Lions is mainly important in that they extended the existence of this mapping $X$ from the classical setting where the method of characteristics have sure footing to the lower regularity setting. $\endgroup$
    – Willie Wong
    Commented Apr 4, 2022 at 15:21
  • $\begingroup$ I'm interested in the lower regularity setting. Is the formula by the method of characteristics still valid in this case, so do we have $\varphi(t,x)=\varphi_0(X(t,x))+\int_0^T f(X(s,x))ds$ ? $\endgroup$
    – user99432
    Commented Apr 4, 2022 at 16:15
  • $\begingroup$ Is there any literature where I can find the inhomogeneous case for $\varphi_0\in L^p$ and $f\in L^1L^p$? $\endgroup$
    – user99432
    Commented Apr 4, 2022 at 19:15
  • $\begingroup$ unfortunately: I don't know. It seems plausible. $\endgroup$
    – Willie Wong
    Commented Apr 4, 2022 at 19:29

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