Dispersion vs. anti-diffusion: well-posedness in variable coefficient and quasilinear equations of KdV-type

TL;DR

This paper investigates the conditions under which dispersive effects can ensure well-posedness in variable coefficient and quasilinear KdV-type equations, despite the presence of anti-diffusion that can cause ill-posedness.

Contribution

It establishes a specific condition guaranteeing well-posedness for linear equations with variable coefficients and anti-diffusion, and extends these results to certain quasilinear equations.

Findings

Dispersive effects can dominate backwards diffusion, ensuring well-posedness.

A condition is identified that guarantees well-posedness despite anti-diffusion.

Violating this condition leads to ill-posedness due to dominant anti-diffusion.

Abstract

We study the well-posedness of the initial value problem on periodic intervals for linear and quasilinear evolution equations for which the leading-order terms have three spatial derivatives. In such equations, there is a competition between the dispersive effects which stem from the leading-order term, and anti-diffusion which stems from the lower-order terms with two spatial derivatives. We show that the dispersive effects can dominate the backwards diffusion: we find a condition which guarantees well-posedness of the initial value problem for linear, variable coefficient equations of this kind, even when such anti-diffusion is present. In fact, we show that even in the presence of localized backwards diffusion, the dispersion will in some cases lead to an overall effect of parabolic smoothing. By contrast, we also show that when our condition is violated, the backwards diffusion can dominate the dispersive effects, leading to an ill-posed initial value problem. We use these results on linear evolution equations as a guide when proving well-posedness of the initial value problem for some quasilinear equations which also exhibit this competition between dispersion and anti-diffusion: a Rosenau-Hyman compacton equation, the Harry Dym equation, and equations which arise in the numerical analysis of finite difference schemes for dispersive equations. For these quasilinear equations, the well-posedness theorem requires that the initial data be uniformly bounded away from zero.