Kavli Affiliate: Yi Zhou
| First 5 Authors: Jie Shao, Yi Zhou, , ,
| Summary:
We study the global Cauchy problem of the quasilinear Schr"odinger
equations, for which KENIG et al. (Invent Math, 2004; Adv Math, 2006) obtained
short time local wellposedness with large data by pseudo-differential
techniques and viscosity methods, while MARZUOLA et al. (Adv Math, 2012; Kyoto
J Math, 2014; Arch Ration Mech Anal, 2021) improved the results by dispersive
arguments. In this paper, we introduce the generalized energy method that can
close the bounds by combining momentum and energy estimates and derive the
results by viscosity methods. The whole arguments basically only involve
integration by parts and Sobolev embedding inequalities, just like the
classical local existence theorem for semilinear Schr"odinger equations. For
quadratic interaction problem with small data, we derive low regularity local
wellposedness in the same function spaces as in the works of Kenig et al. For
cubic interaction problem, we obtain the same low regularity results as in
Marzuola et al. (Kyoto J Math, 2014).
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