A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below H1/2H1/2

We propose a priori estimates for a weak solution to the derivative nonlinear Schrödinger equation (DNLS) on torus with small L2L2-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions in Hs(T)Hs(T) with some s<1/2s<1/2 in a relatively weak sense. Furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of DNLS using the Fourier–Lesbesgue type as auxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bellet and Staffilani [12] have already seen them.