Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ where we have assumed $f \in L^1_tW^{1,p}_x \cap L^\infty$.

Does the following hold in a weak sense?

$$\begin{cases} \frac{d}{dt} \nabla \Phi(x,t) = \nabla_1 f(\Phi(x,t),t)\nabla \Phi(x,t) \quad t>0 \\ \nabla\Phi(x,0) = 1 \quad x \in \mathbb{R}^N \end{cases}$$ and $$\begin{cases} \frac{d}{dt} J \Phi(x,t) = \mathrm{div} f(\Phi(x,t),t)J \Phi(x,t) \quad t>0 \\ \nabla\Phi(x,0) = 1 \quad x \in \mathbb{R}^N \end{cases}$$ where $Jf = \det \nabla f$.

In Derivative and Jacobian determinant of solution of ODE, the same question was asked assuming $f$ smooth.


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