Sharp asymptotics for the minimal mass blow up solution of critical gKdV equation

Let $S$ be a minimal mass blow up solution of the critical generalized KdV equation as constructed by Martel, Merle and Rapha\"el in arXiv:1204.4624. We prove both time and space sharp asymptotics for $S$ close to the blow up time. Let $Q$ be the unique ground state of (gKdV), satisfying $Q"+Q^5=Q$. First, we show that there exist universal smooth profiles $Q_k\in\mathcal{S}(\mathbb{R})$ (with $Q_0=Q$) and a constant $c_0\in\mathbb{R}$ such that, fixing the blow up time at $t=0$ and appropriate scaling and translation parameters, $S$ satisfies, for any $m\geqslant 0$, $$\partial_x^m S(t) - \sum_{k=0}^{[m/2]} \frac 1{t^{\frac 12+m-2k}} Q_k^{(m-k)}\left(\frac{\cdot+ \frac1t}{t}+c_0\right)\to 0\quad \mbox{in}\ L^2 \mbox{as}\ t\downarrow 0.$$ Second, we prove that, for $0<t\ll 1$, $x\leqslant -\frac 1t -1$, $$S(t,x) \sim - \frac 12 \|Q\|_{L^1} |x|^{-3/2},$$ and related bounds for the derivatives of $S(t)$ of any order. We also prove $\int_{\mathbb{R}} S(t,x)\,dx=0$.