On the ill-posedness of the 5th-order Gardner equation

2018/10/24 by Alejo, Miguel A., Cardoso, Eleomar
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.10434

Abstract

We present ill-posedness results for the initial value problem (IVP) of the 5th-order Gardner equation. We use new breather solutions discovered for this higher order Gardner equation to measure the regularity of the Cauchy problem in Sobolev spaces H^s. We find the sharp Sobolev index under which the local well-posedness of the problem is lost, meaning that the dependence of 5th order Gardner solutions upon initial data fails to be continuous.

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