Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin–Ono equations

Kenig, C.E.; Martel, Y.; Robbiano, L.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 853-887

We consider a family of dispersion generalized Benjamin–Ono equations (dgBO)

$$u_t-{\partial }_x{\left|D\right|}^{\alpha }u+{\left|u\right|}^{2\alpha }{\partial }_{x}u=0,(t,x)\in ℝ\times ℝ,$$

where $\widehat{{\left|D\right|}^{\alpha }u}={\left|\xi \right|}^{\alpha }\hat{u}$ and $1⩽\alpha ⩽2$. These equations are critical with respect to the $L^2$ norm and global existence and interpolate between the modified BO equation ($\alpha =1$) and the critical gKdV equation ($\alpha =2$).First, we prove local well-posedness in the energy space for $1<\alpha <2$, extending results in Kenig et al. (1991, 1993) [13,14] for the generalized KdV equations.Second, we address the blow-up problem in the spirit of Martel and Merle (2000) [19] and Merle (2001) [22] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space $H^{\frac{\alpha }{2}}$.The blow-up proof requires both extensions to dgBO of monotonicity results for local $L^2$ norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.

Nous considérons une famille dʼéquations de Benjamin–Ono à dispersion généralisée (dgBO)

$$u_t-{\partial }_x{\left|D\right|}^{\alpha }u+{\left|u\right|}^{2\alpha }{\partial }_{x}u=0,(t,x)\in ℝ\times ℝ,$$

où $\widehat{{\left|D\right|}^{\alpha }u}={\left|\xi \right|}^{\alpha }\hat{u}$ et $1⩽\alpha ⩽2$. Ces équations sont critiques par rapport à la norme $L^2$ et à lʼexistence globale et peuvent être vues comme des interpolations entre lʼéquation de Benjamin–Ono généralisée critique ($\alpha =1$) et lʼéquation de Korteweg–de Vries généralisée critique ($\alpha =2$).Dʼabord, nous montrons le caractère bien posé de ces équations dans lʼespace dʼénergie pour $1<\alpha <2$, étendant les résultats de Kenig et al. (1991, 1993) [13,14] pour les équations de Korteweg–de Vries généralisées.Ensuite, nous étudions le phénomène dʼexplosion dans lʼesprit de Martel et Merle (2000) [19] et Merle (2001) [22] concernant lʼéquation de gKdV critique, en étudiant les propriétés de rigidité du flot de dgBO dans un voisinage des solitons. Nous montrons que pour α proche de 2, les solutions dʼénergie négative proches des solitons explosent en temps fini ou infini dans lʼespace dʼénergie $H^{\frac{\alpha }{2}}$.La preuve de ce résultat dʼexplosion est basée dʼune part sur lʼadaptation à dgBO de résultats de monotonie de normes $L^2$ locales par des méthodes dʼopérateurs pseudo-differentiels et dʼautre part sur des arguments de perturbation pour obtenir des propriétés structurelles du flot linéarisé autour des solitons lorsque lʼéquation est proche de gKdV.

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