Blow up for the critical gKdV equation. II: Minimal mass dynamics

Abstract

Abstract. We consider the mass critical (gKdV) equation ut + (uxx + u 5 )x = 0 for initial data in H 1 . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [33].

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

“…as n → ∞ for any γ > 1 and ξ 0 ∈ R. This implies (43) for J = 2 with the help of (46). Repeat this argument and construct ψ j ∈ V(r j−1 ) and G j n ∈ G, inductively.…”

“…and define η(P ) := sup φ∈V(P ) (φ). By definition, η(P ) = 0 implies that we may not find any weak limit from a sequence {P n } n even modulo the orbit by deformations G. Conversely, if η(P ) > 0 we can find a non-zero weak limit modulo G. The main result of this section is decomposition with a smallness of remainder with respect to η. G j n ψ j + r l n (42) for all l 1 with η(r l ) → 0 as l → ∞. Further, a decoupling inequality for any j.…”

“…The important thing in decoupling inequality (43) is that the coefficient of each term in the right hand is equal to one. This is the reason why we work not with · M α 2,σ but with (·).…”

“…Hence, G j n is orthogonal to G k n for 1 k j − 1. Then, by (42) and by Lemma 5.3, we have ψ j ∈ V(u), from which boundedness (44) and (45) follow.…”

“…To conclude the proof, we shall show lim j→∞ η(r j ) = 0 and (43). Notice that the inductive construction gives us (ψ j+1 ) 1 2 η(r j ) for all j 1 and…”