Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

Soonsik Kwon; Tadahiro Oh; Haewon Yoon

doi:10.5802/afst.1643

Soonsik Kwon ¹ ; Tadahiro Oh ² ; Haewon Yoon ¹

¹ Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon 34141 (Republic of Korea)
² School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD (United Kingdom)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 649-720.

Résumé
Abstract

Dans cet article, nous revisitons le schéma d’itération infinie des réductions de forme normale, introduit par les premier et deuxième auteurs (avec Z. Guo), dans la construction des solutions des EDP dispersives non linéaires. Notre objectif principal est de présenter une approche simplifiée à cette méthode. Plus précisément, nous étudions les réductions de forme normale dans un cadre abstrait et nous réduisons les estimations multilinéaires de degrés arbitraires aux applications successives des estimations trilinéaires fondamentales. Comme application, nous montrons que des équations dispersives non linéaires canoniques sont inconditionnellement bien-posées sur la droite réelle. En particulier, nous implémentons cette approche simplifiée à l’itération infinie des réductions de forme normale dans le contexte de l’équation de Schrödinger non linéaire cubique (NLS) et de l’équation de KdV modifiée (mKdV) sur la droite réelle et nous prouvons qu’elles sont inconditionnellement bien posées dans $H^{s} (ℝ)$ avec (i) $s \geq \frac{1}{6}$ dans le cas pour NLS cubique et (ii) $s > \frac{1}{4}$ dans le cas pour mKdV. Notre approche de forme normale nous permet également de construire solutions faibles au NLS cubique dans $H^{s} (ℝ)$ , $0 \leq s < \frac{1}{6}$ , et solutions de distribution au mKdV dans $H^{\frac{1}{4}} (ℝ)$ (avec certaine forme d’unicité).

In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this simplified approach to an infinite iteration of normal form reductions in the context of the cubic nonlinear Schrödinger equation (NLS) and the modified KdV equation (mKdV) on the real line and prove unconditional well-posedness in $H^{s} (ℝ)$ with (i) $s \geq \frac{1}{6}$ for the cubic NLS and (ii) $s > \frac{1}{4}$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^{s} (ℝ)$ , $0 \leq s < \frac{1}{6}$ , and distributional solutions to the mKdV in $H^{\frac{1}{4}} (ℝ)$ (with some uniqueness statements).

Reçu le : 2018-03-16
Accepté le : 2018-11-06
Publié le : 2020-11-16

DOI : 10.5802/afst.1643

Classification : 35Q55, 35Q53
Mots clés : nonlinear Schrödinger equation, modified KdV equation, normal form reduction, unconditional well-posedness, unconditional uniqueness

Affiliations des auteurs :

Soonsik Kwon ¹ ; Tadahiro Oh ² ; Haewon Yoon ¹

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Normal form approach to unconditional well-posedness of nonlinear dispersive {PDEs} on the real line},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {649--720},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
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Soonsik Kwon; Tadahiro Oh; Haewon Yoon. Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 649-720. doi : 10.5802/afst.1643. https://afst.centre-mersenne.org/articles/10.5802/afst.1643/

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