Middle frequency band and remark on Koch–Tataru’s iteration space

Abstract

When we wanted to further investigate the results of Bourgain–Pavlović, Koch–Tataru and Li–Xiao–Yang, we encountered the problem of stability of the parameter flow. For initial data in some space [Formula: see text], norm inflation implies that the solution does not belong to [Formula: see text]. For [Formula: see text], Auscher–Dubois–Tchamitchian proved that Koch–Tataru’s solution is stable in their defined space [Formula: see text]. In this paper, we consider Koch–Tataru’s iteration space and construct a non-Gauss flow function to show that [Formula: see text]-stability may have meaning different to [Formula: see text], which supports Chemin and Gallagher’s point: well-posedness and norm inflation may have no conflict.