A remark on the transport equation with $b\in {\rm BV}$ and ${\rm div}_{x}\, b\in {\rm BMO}$
Tom 135 / 2014
Colloquium Mathematicum 135 (2014), 113-125
MSC: Primary 35F25; Secondary 35A02.
DOI: 10.4064/cm135-1-9
Streszczenie
We investigate the transport equation $\partial _t u(t,x) + b(t,x)\cdot D_x u(t,x) = 0$. Our result improves the classical criteria of uniqueness of weak solutions in the case of irregular coefficients: $b\in {\rm BV}$, ${\rm div}_x\, b\in {\rm BMO}$. To obtain our result we use a procedure similar to DiPerna and Lions's one developed for Sobolev vector fields. We apply renormalization theory for BV vector fields and logarithmic type inequalities to obtain energy estimates.
