Biot-Savart Law In Cylinder Domains

It is well-known that the mathematical theory on fluid dynamics is the focus of nonlinear partial differential equations.Under many circumstantces,it is necessary to establish the relationship among the velocity field and its divergence and curl fields of fluids,this is the well known div-curl lemma.When it comes to incompressible fluids,i.e.,the divergence field vanishes,the div-curl lemma is ususally called as Biot-Savart law.This paper is devoted to considering the Biot-Savart law in two types of cylinder domains and establishing the L^p(1<p<+?)estimates between the curl field and the velocity field when the divergence field vanishes.Two cylinder domains considered in this paper are O_?={R²\B₁(0)}×T and O₂={B₂(0)\B₁(0)}×T.Our main problems can be classified as the L^ptheory of elliptic equation of second order in unbounded domain O_?and bounded domain O₂respectively.Based on the method of variable separation,the theories of Fourier series and modified Bessel functions as well as elliptic estimates,we finish the proof of main problems.