论文标题
关于Sobolev Vector Fields发散链规则的失败
On the failure of the chain rule for the divergence of Sobolev vector fields
论文作者
论文摘要
我们构建了一大批不可压缩的向量字段，具有sobolev的规律性，在dimension $ d \ geq 3 $中，链条规则问题具有负面答案。 In particular, for any renormalization map $β$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = {\rm div}\, h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $ {\ rm div} \，（ρu）= 0 $ ar $ {$ {$ {\ rm div} \，（β（p）u）= t $，提供$ 1/p + 1/\ tilde p \ geq 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1/（d-1 + 1 + 1/（d-d-1）$，$ {\ rm div} \，（ρu）
We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $β$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = {\rm div}\, h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $ρ\in L^p$ for which ${\rm div}\, (ρu) =0$ but ${\rm div}\, (β(ρ) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$
