Local well-posedness of the fifth-order KdV-type equations on the half-line

Abstract

This paper is a continuation of authors' previous work [6]. We extend the argument [6] to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the X s,b nonlinear estimates for b < 1 2 , which is almost optimal compared to the standard X s,b nonlinear estimates for b > 1 words and phrases. Fifth-order KdV-type equations, initial-boundary value problem, local wellposedness.C. Kwak is supported by FONDECYT de Postdoctorado 2017 Proyecto No. 3170067. 1 42 6. Proof of Theorems 1.3 -1.6 58 References 59 for a certain regularity swhere f is the space time Fourier coefficient (also denoted by F(f )) and · = (1 + | · | 2 ) 1 2 , initially introduced in its current form by Bourgain [3], and its various modifications, see below for a short explanation of spaces and Section 2.3 for precise definitions of spaces.The X s,b space is known to be an appropriate device to detect dispersive phenomena in the Fourier analysis. In other words, solutions to (1.1) have the dispersive smoothing effect which means that the (space-time) Fourier coefficients decay far away from the characteristic curve τ = ξ 5 , where ξ and τ are the Fourier variables corresponding to x and t, respectively. This property is naturally reflected in (1.3) as a weight τ − ξ 5 b . The choice of the exponent b < 1 2 in (1.3) is imposed in the study on IBVP due to the presence of the Duhamel boundary forcing operator, which reflects what role boundary conditions play in the solutions (see Section 4 for more details), while the standard X s,b space with b > 1 2 works well in the study on the initial value problem (IVP).The trade-off of choosing b < 1 2 causes the lack of τ -integrability in (1.2), when all functions are localized in the frequency support |ξ| ≤ 1. In order to resolve this problem, an additional low frequency localized space D α , α > 1 2 , is needed, i.e., (1.3) under the restriction |ξ| ≤ 1, precisely,On the other hand, time trace estimates of the Duhamel parts (Lemma 5.2) in X s,b -type spaces hold true for only positive regularities (see Remark 5.3), thus an additional introduction of the (time-adapted) Bourgain space Y s,b as an intermediate norm in the iteration process, which is defined similarly as the standard X s,b space but with a weight in terms of τ instead of ξ in the sense of ∂ t ∼ ∂ 5x in (1.1) (replacing ξ s by τ s 5 in (1.3)), precisely

Initial boundary value problems for some nonlinear dispersive models on the half-line: a review and open problems

“…Thus, the nonlinear estimates for b < 1 2 is the crucial point in this approach. To see other applications of the Colliander and Kenig approach we refer the results established by Cavalcante, Corcho, Holmer and Kwak [12,13,17,30,31].…”

“…• For the Kawahara equation (nonlinear term given by ∂ x (u 2 )) in [16] [17] was obtained local well-posedness for the IBVPs (3.1)-(3.6) and (3.2)-(3.7) for − 5 4 < s < 1 2 .…”

Low regularity for the Kawahara equation on the half‐line

“…Kwak 5 proved the LWP of () in on the half‐line, by using Duhamel boundary forcing operator, for more details, we can refer to Kwak 15 and Cavalcante and Kwak 16 …”

On the solutions for a Benney-Lin type equation

“…Moreover, in [55], the author deduced (8) to describe describing one-dimensional propagation of smal-amplitude long waves in various problems of fluid dynamics and plasma physics. Mathematical properties of (8) were studied recently in many detail, including the local and global well-posedness in Bourgain spaces [41,40,53,54,90], the local and global well-posedness in energy space [43,45,50,88,93,94], the existence of solitary wave solution [9,51], the stability of periodic traveling wave solutions see [2,1,73,87,96], the well-posedness of the initial-boundary value problem on a bounded domain [10,13,44,58], the initial-boundary value problem on the half-line [12,11], periodic solutions [6,79], and numerical solutions [80,4,42,52,72,91]. In [74], the authors prove that the solution of (8) converges to the solution of (7), while, following [26,19,20,27,66,82], in [23,24], the convergence of the solution of…”

“…In particular, if f ∈ C 2 , u is unique. Moreover, if u 1 and u 2 are two solutions of (1), we have (11), for some suitable C > 0, and every 0 ≤ t ≤ T .…”