Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions
This paper is devoted to the study of the Cauchy problem for the nonlinear Schrödinger equation \(i \partial_ tu = - (1/2) \Delta u + F(u, \overline u, \nabla u, \nabla \overline u)\) in the borderline case of space dimension 3, with \(F\) a quadratic function of its arguments and \(\partial F/ \partial (\nabla u)\) pure imaginary. The authors prove the existence of small almost global solutions, namely of solutions with a lifespan \(T = \exp [C (\varphi)/ \varepsilon]\) for initial data \(u(0) = \varepsilon \varphi\) small of order \(\varepsilon\) in a suitable norm. This paper continues previous work by the second author proving the existence of small global solutions of the same problem, either in dimension 4, or in dimension 3 with \(F\) at least linear in \(\nabla u\), \(\nabla \overline u\). The method relies on energy estimates and Sobolev inequalities associated with a Lie algebra of vector fields which commute with the linear part of the equation, in the same spirit as in Klainerman's work on the nonlinear wave equation. In contrast with the previous works, however, it requires the use of half integer derivatives.
quadratic nonlinear Schrödinger equations
scientific article; zbMATH DE number 784104
scientific article