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

Abstract

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 leadingorder 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.

On ill-posedness of truncated series models for water waves

“…A study of the competition between dispersion and antidiffusion that the first author and Wright have carried out for KdV-like equations in [12] (and see also Akhunov [13]) gives some clues to the present situation. It is found in references [12,13] that the Kato smoothing effect from the dispersive terms must be strong enough to counteract the growth inherent in the anti-diffusion. In the shallow-water KdV case, there is a loss of one derivative from the anti-diffusion (the anti-diffusion comes from a second derivative term of indeterminate sign, leading to terms in the energy estimates with one more derivative than can be controlled through a naive estimate), but the Kato smoothing effect from the leading-order, dispersive term is also one derivative.…”

“…The issue of whether or not surface tension can counteract the ill-posedness we see in the WW2 and WW3 models is somewhat subtle. A study of the competition between dispersion and antidiffusion that the first author and Wright have carried out for KdV-like equations in [12] (and see also Akhunov [13]) gives some clues to the present situation. It is found in references [12,13] that the Kato smoothing effect from the dispersive terms must be strong enough to counteract the growth inherent in the anti-diffusion.…”

Well-posedness, ill-posedness, and traveling waves for models of pulsatile flow in viscoelastic vessels

“…These models are simpler, and reduce to a single equation for η. We consider the contrast in the bidirectional case between well-posedness when r 0 is constant and likely ill-posedeness when r 0 is non-constant to be an interesting feature of the present work; this constrast is not present in the unidirectional models, as (relying on results such as those of [1], [2], [6], or [12]) the unidirectional models can be shown to be well-posed in either case. As the bidirectional models are therefore more interesting, we restrict our studies to them.…”

Well-posedness of fully nonlinear KdV-type evolution equations

“…The conditions that we presently require on f are similar to the conditions assumed by the authors in [Akh14] and [AW13], adapted to the fully nonlinear evolution equation (1), and allowing for as much generality as possible. These conditions will be specified more technically in what follows, but they are, roughly: (a) the function f is sufficiently smooth, (b) the partial derivative of f with respect to u xxx does not vanish, so that the dispersion does not degenerate, and (c) the "modified diffusion ratio," to be defined, but which balances the effects of dispersion and backwards diffusion, must either be integrable or be the derivative of a smooth function (this condition is closely related to the "Mizohata" condition needed in [HG15a]).…”