Quadratic derivative nonlinear Schrödinger equations in two space dimensions
In this paper, the authors study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions \(i \partial_t u+\frac{1}{2}\Delta u=N(\nabla u,\nabla u)\) with initial data \(u(0,x) = u_0 (x)\), \(x \in\mathbb R^2\), where the quadratic nonlinearity has the form \(N (\nabla u, \nabla v) =\sum_{ k,l=1,2}\lambda_{kl}\partial_k u\partial_l v\) with \(\lambda \in\mathbb C\). The authors prove that, if the initial data \(u_0 \in H^6 \cap H^{3,3}\) is small enough in the norm, then the solution of the Cauchy problem exists globally in time. Furthermore, the existence of the usual scattering states is proved. The proof depends on the energy type estimates, smoothing property by Doi, and method of normal forms by Shatah. The problem of this type has been studied by many authors and has long history since the early 1980's. The difficulty of the problem originates in the fact that the nonlinear perturbation is a long range one: by this, we mean that it can be written as the product of a derivative of u and of a potential whose \(L^\infty\) space-norm is not time integrable at infinity. We refer the reader to the introduction of this paper for a detailed discussion of the related results. Here we just mention an earlier work of \textit{J.-M. Delort} [Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations. Mém. Soc. Math. Fr., Nouv. Sér. 91 (2002; Zbl 1008.35072)], where the global result was obtained for general quadratic nonlinearities, but with more restricted assumption on the data (i.e., \(u_0 \in H^{10} \cap H^{0,10}\)).
scientific article; zbMATH DE number 5909284
scientific article