The initial-boundary value problem for the Kawahara equation on the half-line

Cavalcante, Márcio; Kwak, Chulkwang

doi:10.48550/arxiv.1805.05229

Cited by 3 publications

(21 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We give L 2 -block estimates for the quadratic and cubic nonlinearities. The bi-and tri-linear L 2 -block estimates for the 5th order equations have already been introduced and used in several works, we refer to [14,13,22,30,42,41,11,12].…”

Section: Well-posedness Resultsmentioning

confidence: 99%

Global solutions and stability properties of the 5th order Gardner equation

Alejo

Kwak²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this work, we deal with the initial value problem of the 5th-order Gardner equation in R, presenting the local well-posedness result in H 2 (R). As a consequence of the local result, in addition to H 2 -energy conservation law, we are able to prove the global well-posedness result in H 2 (R). Finally, we present a stability result for 5th order Gardner breather solution in the Sobolev space H 2 (R).We also refer to [32,35,52,25,29] for the local well-posedness of for higher order KdV and mKdV equations.

show abstract

Section: Well-posedness Resultsmentioning

confidence: 99%

Global solutions and stability properties of the 5th order Gardner equation

Alejo

Kwak²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…

…”

mentioning

confidence: 75%

Section: Introductionmentioning

confidence: 82%

“…This paper is a continuation of authors' previous work [6]. In [6], the authors studied the Duhamel boundary forcing operator associated to the fifth-order linear operator, and established the local well-posedness of Kawahara equation posed on the right/left half-line. In this paper, we extend the previous study to the fifth-order KdV-type equations whose nonlinearities are different, in particular, do not satisfy the scaling symmetry.…”

Section: Introductionmentioning

confidence: 82%

“…Remark 1.2. Both (1.5) and (1.7) can be obtain in some positive regularity regime similarly as Proposition 5.2 in [6], but we, here, explore the nonlinear estimates for both F (u) = (1 − ∂ 2…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

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

Cavalcante

Kwak²

2019

Communications on Pure &Amp; Applied Analysis

Self Cite

View full text Add to dashboard Cite

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

show abstract

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

Cavalcante¹,

Kwak²

2018

Preprint

View full text Add to dashboard Cite

The initial-boundary value problem for the Kawahara equation on the half-line

Cited by 3 publications

References 36 publications

Global solutions and stability properties of the 5th order Gardner equation

Global solutions and stability properties of the 5th order Gardner equation

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

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

Contact Info

Product

Resources

About