Local Well-posedness and Smoothing Effects of Strong Solutions for Nonlinear Schrödinger Equations with Potentials and Magnetic Fields

In this paper, we study the existence and the regularity of local strong solutions for the Cauchy problem of nonlinear Schrödinger equations with time-dependent potentials and magnetic fields. We consider these equations when the nonlinear term is the power type which is, for example, equal to λ|u|p−1u with some 1 ≤ p < ∞, λ ∈ R. We prove local well-posedness of strong solutions under the additional assumption 1 ≤ p < 1 + 4/(n − 4) for space dimension n ≥ 5, and local smoothing effects of it under the additional assumption 1 ≤ p ≤ 1 + 2/(n − 4) for n ≥ 5 without any restrictions on n. ∗Supported by JSPS Research Fellowships for Young Scientists.