Ill-posedness of degenerate dispersive equations

TL;DR

This paper demonstrates that certain degenerate dispersive PDEs, including the K(2,2) and degenerate Airy equations, are ill-posed, with numerical and analytical evidence showing solutions can behave unpredictably even with small initial data.

Contribution

It provides the first rigorous proof of ill-posedness for the degenerate Airy equation and numerical evidence for the ill-posedness of the K(2,2) equation.

Findings

Numerical simulations show small initial data can lead to large solutions.

Existence of self-similar solutions implies ill-posedness.

The results are rigorously proven for the degenerate Airy equation.

Abstract

In this article we provide numerical and analytical evidence that some degenerate dispersive partial differential equations are ill-posed. Specifically we study the K(2,2) equation $u_t = (u^2)_{xxx} + (u^2)_{x}$ and the "degenerate Airy" equation $u_t = 2 u u_{xxx}$. For K(2,2) our results are computational in nature: we conduct a series of numerical simulations which demonstrate that data which is very small in $H^2$ can be of unit size at a fixed time which is independent of the data's size. For the degenerate Airy equation, our results are fully rigorous: we prove the existence of a compactly supported self-similar solution which, when combined with certain scaling invariances, implies ill-posedness (also in $H^2$).