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- Title
Construction of a minimal mass blow up solution of the modified Benjamin-Ono equation.
- Authors
Martel, Yvan; Pilod, Didier
- Abstract
We construct a minimal mass blow up solution of the modified Benjamin-Ono equation (mBO) which is a standard mass critical dispersive model. Let $$Q\in H^{\frac{1}{2}}$$ , $$Q>0$$ , be the unique ground state solution of $$D^1 Q +Q=Q^3$$ , constructed using variational arguments by Weinstein (Commun. Part. Differ. Equations 12:1133-1173, 1987a; J. Differ. Equations 69:192-203, 1987b) and Albert et al. (Proc. R. Soc. Lond. A 453:1233-1260, 1997), and whose uniqueness was recently proved by Frank and Lenzmann (Acta Math. 210:261-318, 2013). We show the existence of a solution S of (mBO) satisfying $$\Vert S \Vert _{L^2}=\Vert Q\Vert _{L^2}$$ and where This existence result is analogous to the one obtained by Martel et al. (J. Eur. Math. Soc. 17:1855-1925, 2015) for the mass critical generalized Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for which the blow up problem is now well-understood in a neighborhood of the ground state, S is the first example of blow up solution for (mBO). The proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin-Ono type equations by Kenig et al. (Ann. Inst. H. Poincaré Anal. Non Lin. 28:853-887, 2011). Due to the lack of information on the (mBO) flow around the ground state, the energy estimates have to be considerably sharpened in the present paper.
- Publication
Mathematische Annalen, 2017, Vol 369, Issue 1/2, p153
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-016-1497-8