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A remark on the transport equation with b ∈ BV and d i v x b ∈ B M O

Paweł Subko — 2014

Colloquium Mathematicae

We investigate the transport equation ∂ t u ( t , x ) + b ( t , x ) · D x u ( t , x ) = 0 . Our result improves the classical criteria of uniqueness of weak solutions in the case of irregular coefficients: b ∈ BV, d i v x b ∈ B M O . 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.

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