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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.