Seminar Analysis: Nikolay Gusev (SISSA Trieste)

On the chain rule for the divergence operator in $\mathbb{R}^2$

Suppose that b:R^d→R^d is a vector field, β:R→R  is a smoth function and u:R²→R is a scalar field. If both u and b are smooth then the following formula holds: div(β(u))b = β(u) - (β'(u) - uβ'(u)) div (b) + β'(u) div(ub). Generalizations of this formula when u∈L^∞ and b belongs to Sobolev space or has bounded variation were studied by R. Di Perna, P.-L. Lions, L. Ambrosio, C. De Lellis, J. Maly and other authors. I will present a new result in this direction for d=2, which was obtained recently in collaboration with S. Bianchini. In particular our result holds when b is steady nearly incompressible BV vector field.

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