A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below $H^{1/2}$

Abstract

We propose a priori estimates for a weak solution to the derivative nonlinear Schrödinger equation (DNLS) on torus with small $L^2$-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions in $H^s(\mathbb{T})$ with some $s<1/2$ 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-Bullet and Staffilani \cite{nors} have already seen them.

41 pages

References in corpus (2)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• On the Cauchy problem for the derivative nonlinear Schroedinger equation with periodic boundary condition

Cited by in corpus (1)

• Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in $H^{\frac{1}{2}}$