Well-posedness of fully nonlinear KdV-type evolution equations

Abstract

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, ut = f [u], where f may depend on up to the first three spatial derivatives of u. We make three primary assumptions about the form of f : a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of u is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space H 7 (R). The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.

On well-posedness for some Korteweg-De Vries type equations with variable coefficients

“…He also showed some evidences on the sharpness of this assumption. Adaptation of the LWP in high regularity Sobolev spaces under this hypothesis for quasilinear and fully nonlinear generalizations of (1.1) can be found in respectively [1] and [3]. In [8], Israwi and the second author proved the LWP of (1.1) in H s (R), s > 3/2, under the same type of integrability assumption on the ratio function r(t, x).…”

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

“…This implies that (η t , u t ) is uniformly bounded with respect to in L ∞ × L ∞ as well, when s ≥ 2. This implies that the sequence (η , u ) is an equicontinuous family, and thus by the Arzela-Ascoli theorem there exists a subsequence (which we do not relabel) (η , u ) which converges uniformly to some (η, u) ∈ (C([0, 2π] × [0, T ])) 2 . We now establish regularity of this (η, u) and that (η, u) is a solution of the non-regularized initial value problem.…”

“…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.…”

A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

“…There is a lot of research about KdV equation. Some research focuses on the well-posedness and regularity of the solution [4][5][6]. More generally, the general exact solution is hard to obtain and it can just be given in some special forms, therefore, numerical solutions and the corresponding analysis are very important in applications.…”