21世紀COEプログラム : 特異性から見た非線構造の数学

加藤 淳:COE シンポジウム「非線型分散方程式」一般講演

Existence and uniqueness of the solution to the modified Schrödinger maps

Abstract

We consider the local well-posedness of the Cauchy problem for the modified Schrödinger map (MSM) under the low regularity assumption. The modified Schrödinger map, which is a system of derivative nonlinear Schrödinger equations, is derived from the Schrödinger map from R×R2 to the unit sphere S2 choosing an appropriate gauge change. In two space dimensions, the scaling suggests that the critical space for the local well-posedness of the Cauchy problem of MSM is L2 (R2), which corresponds to the energy class for the original Schrödinger map. MSM is derived by Nahmod-Stefanov-Uhlenbeck and they proved the existence of unique solution to MSM for the data in the class Hs (R2) with s > 1 by using the energy method. Using the Strichartz type estimates, which were introduced by Koch-Tzvetkov in the context of the Benjamin-Ono equation, we prove the existence of the solution to MSM for the data in Hs (R2) with s > 1/2. We also prove the uniqueness of the solution in the class H1(R2).

21世紀COEプログラム : 特異性から見た非線構造の数学