Microscopic conservation laws for the derivative nonlinear Schrodinger equation

Using the perturbation determinant introduced by B. Simon, we show the microscopic conservation laws for the Schwarz solutions of the derivative nonlinear Schrodinger equation (DNLS) with small mass, which can be used to show uniformly boundedness and local smoothing estimate of DNLS in the lower regularity space. This is joint work with Xingdong Tang. Lastly, we will show some progresses about DNLS.