Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations
By Carlos E. Kenig, Yvan Martel. 2010
82 citations recorded by the source corpus.
We consider a family of dispersion generalized Benjamin-Ono equations (dgBO) which are critical with respect to the L2 norm and interpolate between the critical modified (BO) equation and the critical generalized Korteweg-de Vries equation (gKdV). First, we prove local well-posedness in the energy space for these equations, extending results by Kenig, Ponce and Vega concerning the (gKdV) equations. Second, we address the blow up problem in the spirit of works of Martel and Merle on the critical (gKdV) equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of solitons. We prove that when the model is close to critical (gKdV), solutions of negative energy close to solitons blow up in finite or infinite time in the energy space. The blow up proof requires in particular extensions to (dgBO) of monotonicity results for localized versions of L2 norms by pseudo-differential operator tools.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/3749a8cc61ef101f67caf0dae8d20765d5ac4de1
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In Mathematical Physics
- Rupert L. Frank
- Youquan Zheng
- Guoyuan Chen
- E. Lenzmann
- Luis Silvestre
- Enrico Valdinoci
- Jinggang Tan
- Pablo Raúl Stinga
In Statistical and Nonlinear Physics
Citations in the corpus
- Uniqueness of Radial Solutions for the Fractional Laplacian (2013, background): “h results form a key constitute in the stability and blowup analysis for related time-dependent problems (e.g., generalized Benjamin–Ono equations, fractional Schr¨odinger equations etc.); see, e.g., [22, 23] for applications. Finally, we have the following uniqueness result for ground state solutio”
- Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations (2013, result): “which settled a conjecture by [46, 68] and generalized a classical result by Amick and Toland [5] on the uniqueness of solitary waves for the Benjamin-Ono equation.”
- Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, background): “, [31, 18, 3, 23, 6], where the nondegeneracy of L+ is imposed in terms of a spectral assumption, or proven for s close to 1 by perturbation arguments.”