$$E_N(t)-E_N(0)\leq C\epsilon^{3/4}\int_0^t(1+s)^{-1}E_N(s)\,ds,\qquad t\in[0,T],$$ which shows that $E_N(t)\leq E_N(0)(1+t)^\delta$, for any $t\in[0,T]$. Therefore $U\in X^N_T$ and $$\|U\|_{X^N_T}\leq 2\epsilon^{3/4}.$$ We can apply now Proposition \[Norm\] to conclude that $$\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)-U(0)\|_{Y^{N_0}}\lesssim \epsilon^{3/2}.$$ In other words, if $U$ satisfies $\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\leq \epsilon^{3/4}$ as in then $U$ satisfies the stronger inequality $\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\leq \|U(0)\|_{Y^{N_0}}+C\epsilon^{3/2}$. In view of the continuity of the norm $Y^{N_0}$ (see Proposition \[LocExLemma\] (iii)), it follows that $$\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\leq 2\|U(0)\|_{Y^{N_0}}$$ for any solution $U\in C([0,T]:H^N)$ of the initial-value problem with $\|U_0\|_{H^N}+\|U_0\|_{Y^{N_0}}\leq\overline{\varepsilon}$. The global regularity part of the theorem and the bound follow, using again the energy estimate .

Proof of Proposition \[LocExLemma\] {#LocalExistence}
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In this section we prove Proposition \[LocExLemma\], using the energy method. We regularize the equation by parabolic regularization, get uniform estimates and then pass to the limit. The method is, of course, well known. We present all the details here for two reasons: to justify the key energy estimate , with the somewhat unusual $Z'$ norm in the right-hand side of the inequality, and to justify the claim in Proposition \[LocExLemma\] that the solution defines a continuous flow in the $Z$ space. These facts are important in passing to the global result.

We start with the main lemma.

\[energylem1\] With $K=50$, there 