Weak solutions for the derivative nonlinear Schrödinger equation (Q1347437)

The author studies the initial value problem for the derivative nonlinear Schrödinger equation: \[ \begin{aligned} \partial_tu & =i \partial^2_x u+\partial_x\bigl(T_\lambda(u) u\bigr)x\in\mathbb{R},\;t>0,\\ u(0) & = u_0,\end{aligned} \] where \(T_\lambda(u)=|u|^2-\lambda H(|u|^2)\), \(\lambda>0\) and \(H\) is the Hilbert transform. He shows the existence of global in time \(L^2\)-weak solutions as well as smoothing effect for the equation. To prove the theorem the author uses similar arguments as [\textit{T. Kato}, Adv. Math., Suppl. Stud., 8, 93-128 (1983; Zbl 0549.34001)] for the Korteweg-de Vries equation. He introduces an artificial viscosity term to obtain the following problem: \[ \begin{aligned} \partial_tu & =i\partial^2_xu-\mu \partial^{1 2}_x u+\partial_x^{12} u+\partial_x \bigl(T_\lambda (u)u\bigr)x\in \mathbb{R},\;t>0, \\ u(0) &= u_0. \end{aligned} \] After passing to the limit with \(\mu\searrow 0\), a weak solution of the previous problem is obtained.