Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

Abstract

We construct a minimal mass blow up solution of the modified Benjamin–Ono equation (mBO) mBO $$\begin{aligned} u_{t}+(u^3-D^1 u)_{x}=0, \end{aligned}$$ 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 $$\begin{aligned} S(t)-\frac{1}{\lambda ^{\frac{1}{2}}(t)} Q\left( \frac{\cdot - x(t)}{\lambda (t)}\right) \rightarrow 0\quad \text{ in } H^{\frac{1}{2}}(\mathbb R) \text{ as } t\downarrow 0, \end{aligned}$$ where $$\begin{aligned} \lambda (t)\sim t,\quad x(t) \sim -|\ln t| \quad \hbox {and}\quad \Vert S(t)\Vert _{\dot{H}^{\frac{1}{2}}} \sim t^{-\frac{1}{2}}\Vert Q\Vert _{\dot{H}^{\frac{1}{2}}} \quad \hbox {as} t\downarrow 0. \end{aligned}$$ 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. Poincare 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.