A sharp condition for the well-posedness of the linear KdV-type equation

Abstract

An initial value problem for a very general linear equation of KdVtype is considered. Assuming non-degeneracy of the third derivative coefficient, this problem is shown to be well-posed under a certain simple condition, which is an adaptation of the Mizohata-type condition from the Schrödinger equation to the context of KdV. When this condition is violated, ill-posedness is shown by an explicit construction. These results justify formal heuristics associated with dispersive problems and have applications to non-linear problems of KdVtype.

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

“…Actually their results even concern quasilinear version of (1.1). In [2], Akhunov proved that the associated linear equation is LWP under an assumption on the boundedness uniformly in time and space of the primitive function (t, x) → x 0 r(t, z)dz where r(•, •) is the ratio function r(t, z) = β(t, z)/α(t, z). He also showed some evidences on the sharpness of this assumption.…”

“…where b, c, d, e, f are real-valued smooth given functions with this time b ≥ 0. Note that this change of unknown is related to the gauge method that is used in similar contexts as in [2], [5], [8]. Actually, at this stage, to ensure that the coefficients e and f of the nonlinear terms are bounded we will require the boundedness from above uniformly in [0, T ] × R of − x 0 r 1 (t, z) dz where r 1 = β 1 /α is, roughly speaking, the ratio function between the positive part β 1 of β and α (see Hypothesis 3 in Section 3).…”

A conditional local existence result for the generalized nonlinear Kawahara equation

“…Up to our knowledge, problem has not yet been analyzed. A standard Picard iterative scheme (see Alinhac and Gérard for general details) have been used (see Israwi and Talhouk for example) to prove the local well‐posedness of a nonlinear KdV‐type equation (see other studies for different treatments). In this paper, we study the Cauchy problem for the general nonlinear higher order Kawahara equation that have a form of the KdV equation with an additional fifth order derivative term, given by …”

Higher Order Dissipative-Dispersive System and Application

“…This equation covers several important unidirectional models for the water wave problem at different regimes which take into account the variations of the bottom and the surface tension. We have in view in particular the example of the Camassa-Holm equation which was first derived by Camassa and Holm in [1] (see also [2], [3], [4]), which is more nonlinear than the KdV and BBM equations (see for instance [5] [17], [6,9], [10], [11], [12], [13], [14], [15], [16].). The presence of the fifth order derivative term is very important, so that the equation describes both nonlinear and dispersive effects as does the Camassa-Holm equation in the case of special tension surface values (see [8,18] page 230 the Kawahara approximation).…”