On the Chain Rule for the Divergence of Bv like Vector Fields: Applications, Partial Results, Open Problems

In this paper we study the distributional divergence of vector fields U in R of the form U = wB, where w is scalar function and B is a weakly differentiable vector field (or more in general the divergence of tensor fields of the form w⊗B). In particular we are interested in a kind of chain rule property, relating the divergence of h(w)B to the divergence of wB. In some sense, if one replaces “divergence” by “derivative” this problem is reminiscent to the problem of writing a chain rule for weakly differentiable functions, a theme that has been investigated in several papers (we mention for instance Vol’pert’s paper [36] and [3] in the BV setting). However, the “divergence” problem seems to be much harder than the “derivative” problem, due to much stronger cancellation effects. For instance it may happen that U ∈ Lloc has distributional divergence f ∈ Lloc, but f 6= 0 L -a.e. on {U = 0}. This cannot happen for distributional derivatives, see (16). The problem of writing a chain rule for vector fields U = wB arises in a natural way when one studies the well-posedness of the PDE D · (wB) = c, for instance when B has a space-time structure. Indeed one can use h(s) = s± to establish uniqueness and comparison principles, very much like in Kruzhkov’s theory of scalar conservation laws (see [29]). When B belongs to a Sobolev spaceW 1,p loc and w ∈ L q loc, with p, q dual exponents, the chain rule has been established in [25], obtaining