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

Abstract

We discuss the problem of computing the distributional divergence of a vector field of the form h(w)B, where h is a smooth scalar function and B is a BV vector field, knowing the distributional divergence of all vector fields w_iB, where w_i are the components of w. We present partial results on this problem, conjectures, and links with other problems related to the SBV regularity of solutions of Hamilton-Jacobi equations and systems of conservation laws, and a conjecture recently made by Bressan.

Keywords

• Chain
• Divergence
• $BV$-like
• Vector
• Fields
• Applications
• Partial
• Results
• Problems