Existence locale et effet régularisant précisés pour des équations non linéaires de type Schrödinger

Abstract

In this paper, we consider the Cauchy problem in the usual Sobolev spaces for some nonlinear equations of the form

that is, equations which are of Schrödinger type. We study the local existence and the smoothing effect of the solutions, following C. E. Kenig, G. Ponce and L. Vega, and extend some of their results. The nonlinearity $F$ is a smooth function which vanishes to the 3rd order at 0 and the operator $\mathcal{L}$ has the form $\mathcal{L}={\sum }_{j\le k}{\partial }_{x_j}^2-{\sum }_{j>k}{\partial }_{x_j}^2$. It extends the Laplace operator but is not elliptic in general. We prove the local existence, the uniqueness and the smoothing effect given any $u_0\in H^s({\mathbb{R}}^n)$ with $s>n/2+3$. The proof follows the same plan as that of Kenig, Ponce and Vega, [5]. We improve the estimates by using the paradifferential calculus of J.-M. Bony.