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arXiv · 1104.2571

Ill-posedness of degenerate dispersive equations

David M. Ambrose·Gideon Simpson·J. Douglas Wright·Dennis G. Yang
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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 $H^2$).

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David M. Ambrose, Gideon Simpson, J. Douglas Wright, Dennis G. Yang. 2011-04-13. Ill-posedness of degenerate dispersive equations. https://doi.org/10.1088/0951-7715%2F25%2F9%2F2655

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