# Renormalization for Transport Equations with SBD velocity field

In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in SBD(R^d)$ with $div(B)\ll \mathcal{L}^d$ and locally bounded functions $w$

$$B \cdot \nabla w = c \mathcal{L}^d \implies B \cdot \nabla (h(w)) = ch'(w)\mathcal{L}^d \text{ for any } h\in C^1(R). \qquad (1)$$

This follows from their Theorem 5.1 since $B \cdot \nabla w = c \mathcal{L}^d$ together with the assumption on the divergence of $B$ implies $div(wB)\ll\mathcal{L}^d$.

To prove renormalization for transport equations, they then consider vector fields $b(t,x):(0,T)\times R^d \to R^d$ with $b_t(\cdot)=b(t,\cdot) \in SBD(R^d)$ and $div(b_t)\ll \mathcal{L}^{d}$ for a.e. $t$ and set $B(t,x)=(1,b_t(x))$. Then they claim that $$B \cdot \nabla_{t,x} w = c \mathcal{L}^{d+1} \implies B \cdot \nabla_{t,x}(h(w)) = c h'(w) \mathcal{L}^{d+1}. \qquad (2)$$ Since by the definition of $B$ the left-handside reduces to $\partial_t w + b_t \cdot \nabla_x w$, this gives renormalization for transport equations.

However, I do not see why $(2)$ holds, since we can not deduce $div(wb_t)\ll \mathcal{L}^d$ as we could in the stationary case in $(1)$, since we have an extra derivative in time which we know nothing about: $$c\mathcal{L}^{d+1}=B\cdot \nabla_{t,x} w = \partial_t w + div_x(wb_t) - w div(b_t)$$ Why can we apply Theorem 5.1 anyway?