Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

Mosincat, Razvan; Yoon, Haewon

doi:10.3934/dcds.2020003

Cited by 13 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In terms of H s (R) spaces, [31] proves local well-posedness in H 2 when σ ≥ 1 2 , local well-posedness in H 1 when σ ≥ 1, existence of weak solutions when σ < 1, and certain unconditional uniqueness results at high regularity. See [53] for more on unconditional uniqueness. The (gDNLS) equation with extremely rough nonlinearities 0 < σ < 1 2 is studied in [45,47], but not in standard Sobolev spaces H s .…”

Section: History On Well-posedness and Solitonsmentioning

confidence: 99%

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Pineau¹,

Taylor²

2021

Preprint

View full text Add to dashboard Cite

In this article we study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS)for small powers σ. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in H s when s ∈ [1, 4σ) and σ ∈ (2 , 1). Our result when s = 1 is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu, Simpson and Sulem (J. Nonlinear Sci. 2013) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable H 1 well-posedness theory.To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools developed in this article are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity wellposedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold s < 4σ is twice as high as one might naïvely expect, given that the function z → |z| 2σ is only C 1,2σ−1 Hölder continuous. Moreover, although we cannot prove H 1 well-posedness when σ ≤ √ 3 2 , we are able to establish H s well-posedness in the high regularity regime s ∈ (2 − σ, 4σ) for the full range of σ ∈ ( 1 2 , 1). This considerably improves the known local results, which had only been established in either H 2 or in weighted Sobolev spaces.

show abstract

Section: History On Well-posedness and Solitonsmentioning

confidence: 99%

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Pineau¹,

Taylor²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Such an issue was first proposed by Kato [36] in the context of Schrödinger equation. The unconditional uniqueness is referred to as the uniqueness in L ∞ T H s without the restriction of any auxiliary function space (for instance X The unconditional well-posedness of some dispersive equations have been studied (for instance [36,79,20,75,72,2,50,25,57,58,51,59] and references therein). Some of these uniqueness results employed some auxiliary function spaces (for example Strichartz spaces [72], X s,b -type [79,75,57,58]), which are designed to be large enough to contain C T H s such that the uniqueness of the solution holds.…”

Section: )mentioning

confidence: 99%

“…Some of these uniqueness results employed some auxiliary function spaces (for example Strichartz spaces [72], X s,b -type [79,75,57,58]), which are designed to be large enough to contain C T H s such that the uniqueness of the solution holds. On the other hand, a straightforward energy-type estimate via finite or infinite iteration scheme of the normal form reduction method is also available to prove the unconditional well-posedness in a certain class of C T H s [2,50,25,51,59]. Such an argument seems more natural and elementary since any other auxiliary function spaces does not be taken.…”

Section: )mentioning

confidence: 99%

Well-posedness issues on the periodic modified Kawahara equation

Kwak

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime [26]. We show in this paper some well-posedness results, mainly the global well-posedness in L 2 (T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [69,60], which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from L 2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in H s (T), s > 0, due to the lack of L 4 -Strichartz estimate for arbitrary L 2 data, a slight modification, thus, is needed to attain the local well-posedness in L 2 (T). This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the unconditional uniqueness in H s (T), s > 1 2 , and as a byproduct, we show the weak ill-posedness below H 1 2 (T), in the sense that the flow map fails to be uniformly continuous.2010 Mathematics Subject Classification. Primary 35Q53, 76B15, Secondary 35G25. 1 2 (T)). However, we do not take their argument for the proof of Theorem 1.5 in order to avoid abusing the normal form mechanism.Remark 1.6. Theorems 1.4 and 1.5 will be improved in the forthcoming work by developing the argument inspired by, for instance, Okamoto [64] and Molinet, Pilod and Vento [57], respectively. 3 The function space X s,b T is the time localization of the standard X s,b introduced by Bourgain [6]

show abstract

“…Since then the unconditional well-posedness for NLS was further improved, see [16,23,24,36,41] and studied for various other nonlinear dispersive PDEs, see e.g. [3,63] for KdV, [40,45,47,41] for the modified KdV equation, [13,50] for the derivative NLS equation, and [35] for the periodic modified Benjamin-Ono equation.…”

Section: Introductionmentioning

confidence: 99%

“…This technique of renormalizing the nonlinearity is akin to applying Poincaré-Dulac normal form reductions for ordinary differential equations. We refer to [58,51,3,40,11,12,23,41,13,50] for some applications to nonlinear dispersive equations, although the list is not exhaustive.…”

Section: Introductionmentioning

confidence: 99%

Unconditional uniqueness for the Benjamin-Ono equation

Mosincat¹,

Pilod²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in H s , both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold s = 1/6. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity.

show abstract

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

Cited by 13 publications

References 26 publications

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Well-posedness issues on the periodic modified Kawahara equation

Unconditional uniqueness for the Benjamin-Ono equation

Contact Info

Product

Resources

About