that, we have to prove that the variation of the mollified energy is small enough in order to construct a solution of which the mollified energy is smaller than the initial one by a nontrivial amount ( $(N^{s-1} \eta_{1} A^{-1})^{\alpha}$, see for example (\[Eqn:SepMolNrj\]) ), and this is why we introduce the parameter $\beta$: see (\[Eqn:DefBeta\]), (\[Eqn:MakeDecr\]) and (\[IneqChainNrj\]). The remainder of the argument is based upon a modification of arguments in [@nakanash1; @nakanash2], modulo errors.

$\textbf{Acknowledgements}:$ The second author would like to thank Kenji Nakanishi for providing him with explanations regarding [@nakanash1; @nakanash2] while he visited Courant Institute in March 2010. This work was done while the first author worked at Princeton University.

Proof of Theorem \[Thm:Scat\]
=============================

Let $\epsilon_{0}>0$ be a *fixed* constant (say $\epsilon_{0}=\frac{1}{1000}$).

In this section we prove Theorem \[Thm:Scat\], assuming that the following propositions are true [^5]. As it was pointed out in the introduction, we modify significantly an inductive argument ( see [@bourgjams; @nakanash1; @nakanash2] ). Unlike the energy space, we have to make a balance between the decay of the mollified energy at a given time and the growth of it, at each inductive step.

[**“ Separation of the localized mollified energy ”**]{} 