## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 34, Number 1 (2005), 37-63.

### Local well-posedness and smoothing effects of strongsolutions for nonlinear Schr\"odinger equations with potentials and magnetic fields

Yoshihisa NAKAMURA and Akihiro SHIMOMURA

#### Abstract

In this paper, we study the existence and the regularity of local strong solutions for the Cauchy problem of nonlinear Schr\"odinger equations with time-dependent potentials and magnetic fields. We consider these equations when the nonlinear term is the critical and/or power type which is, for example, equal to $\lambda |u|^{p-1} u$ with some $1 \le p < \infty$, $\lambda \in {\bf C}$. We prove local well-posedness of strong solutions under the additional assumption $1 \le p < \infty$ for space dimension $n = 4$, $1 \le p \le 1+4/(n-4)$ for $n \ge 5$, and local smoothing effects of it under the additional assumption $1 \le p \le 1+2/(n-4)$ when $n \ge 5$ without any restrictions on $n$.

#### Article information

**Source**

Hokkaido Math. J. Volume 34, Number 1 (2005), 37-63.

**Dates**

First available in Project Euclid: 29 September 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1285766208

**Digital Object Identifier**

doi:10.14492/hokmj/1285766208

**Mathematical Reviews number (MathSciNet)**

MR2130771

**Subjects**

Primary: 35B65: Smoothness and regularity of solutions

Secondary: 35A07 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics

**Keywords**

nonlinear Schr\"odinger equations with time-dependent potentials and magnetic fields
well-posedeness of the Cauchy problem with the subcritical/critical power
local smoothing effects

#### Citation

NAKAMURA, Yoshihisa; SHIMOMURA, Akihiro. Local well-posedness and smoothing effects of strongsolutions for nonlinear Schr\"odinger equations with potentials and magnetic fields. Hokkaido Math. J. 34 (2005), no. 1, 37--63. doi:10.14492/hokmj/1285766208. https://projecteuclid.org/euclid.hokmj/1285766208