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| Divergence and curl operators are basic operators in vector analysis. It is often necessary to control certain norms of gradient of a vector field and itself by certain norms of its divergence, curl, boundary term and quantities related to topology of the domain. In this paper, we review $L^p(p>1)$ theory and $C^lpha$ theory of the vector field. For a vector field with compact support, $L^1$-norm of gradient of the vector field can't be controlled by $L^1$-norm of its divergence and curl. But surprisingly, for the divergence-free vector field, Bourgain and Brezis have established some estimates similar to Gagliardo-Nirenberg inequality. Consequently, it stimulates many subsequent works. This paper introduces various results in this field, including some results of the author and his co-author, and gives some interesting problems. |
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| Keywords:Applied mathematics, div-curl systems, differential forms, $L^1$-data, best constant |
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