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We study the global Cauchy problem of the quasilinear Schr\"odingerequations, for which KENIG et al. (Invent Math, 2004; Adv Math, 2006) obtainedshort time local wellposedness with large data by pseudo-differentialtechniques and viscosity methods, while MARZUOLA et al. (Adv Math, 2012; KyotoJ Math, 2014; Arch Ration Mech Anal, 2021) improved the results by dispersivearguments. In this paper, we introduce the generalized energy method that canclose the bounds by combining momentum and energy estimates and derive theresults by viscosity methods. The whole arguments basically only involveintegration by parts and Sobolev embedding inequalities, just like theclassical local existence theorem for semilinear Schr\"odinger equations. Forquadratic interaction problem with small data, we derive low regularity localwellposedness in the same function spaces as in the works of Kenig et al. Forcubic interaction problem, we obtain the same low regularity results as inMarzuola et al. (Kyoto J Math, 2014).

Local wellposedness for the quasilinear Schr\"odinger equations via the generalized energy method