Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion
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Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion David M. Ambrose1
· Jacob Woods1,2
Received: 27 October 2019 / Revised: 9 September 2020 / Accepted: 10 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative term. This generalizes work by the first author and Wright for third-order equations. In addition to inherent interest in fifth-order dispersive equations, this work is also motivated by a question from numerical analysis: finite difference schemes for third-order numerical equations can yield approximate solutions which effectively satisfy fifth-order equations. We find that such a fifth-order equation is well-posed if and only if the underlying third-order equation is ill-posed. Keywords Dispersion · Anti-diffusion · Well-posedness · Fifth-order dispersive equations
1 Introduction We study fifth-order linear constant-coefficient equations of the form u t = a(x, t)u x x x x x + b(x, t)u x x x x + c(x, t)u x x x + d(x, t)u x x + e(x, t)u x + f (x, t)u + h(x, t),
(1)
for given functions a, b, c, d, e, f , and h, subject to the initial condition u(x, 0) = u 0 (x).
(2)
We take the spatial variable x to be in the interval X = [0, M] for some M > 0, and we impose periodic boundary conditions at the ends of this interval. We assume that the
B
David M. Ambrose [email protected] Jacob Woods [email protected]
1
Department of Mathematics, Drexel University, Philadelphia, PA, USA
2
Present Address: Department of Mathematics, Temple University, Philadelphia, PA, USA
123
Journal of Dynamics and Differential Equations
coefficient functions satisfy periodic boundary conditions on X and that they are defined for all t ∈ [0, T ], for a given value T > 0. If the coefficient b(x, t) is ever positive, then we say that anti-diffusion or backwards diffusion is present. It is of course well known that backwards diffusion can cause catastrophic growth of solutions, leading to instability and ill-posedness. However, in some cases for third-order equations, dispersion has been shown to ameliorate the growth from backwards diffusion [1,2,4,7], suggesting that the initial value problem (1), (2) could be well-posed even in the presence of backwards diffusion. In this paper, we will demonstrate conditions under which the initial value problem is well-posed, as well as conditions under which the initial value problem is ill-posed. A number of authors have studied fifth-order dispersive equations, as they have been found to be useful as higher-order models in the theory of water waves [11], and the stronger dispersion
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