Construction of 2-peakon Solutions and Ill-Posedness for the Novikov equation

2017/08/18 by Himonas, A. Alexandrou, Holliman, Curtis, Kenig, Carlos
#37K10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35Q53

paper · doi:10.48550/arxiv.1708.05759

Abstract

For the Novikov equation, on both the line and the circle, we construct a 2-peakon solution with an asymmetric antipeakon-peakon initial profile whose H^s-norm for s<3/2 is arbitrarily small. Immediately after the initial time, both the antipeakon and peakon move in the positive direction, and a collision occurs in arbitrarily small time. Moreover, at the collision time the H^s-norm of the solution becomes arbitrarily large when 5/43/2, these results put together establish 3/2 as the critical index of well-posedness for this equation. The case s=3/2 remains an open question.

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