On continuation properties after blow-up time for $L^2$-critical gKdV equations

In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_{\gamma}(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-\gamma u|u|^{q-1})_x=0$ with the same initial data, where $\gamma>0$ and $q>5$. A standard argument shows that $u_{\gamma}(t)$ is always global in time and for all $t<T$, $u_{\gamma}(t)$ converges to $u(t)$ in $H^1$ as $\gamma\rightarrow0$. We prove in this paper that for all $t\geq T$, $u_{\gamma}(t)$ converges to some $v(t)$ as $\gamma\rightarrow0$, in a certain sense. This limiting function $v(t)$ is a weak solution to the unperturbed $L^2$-critical gKdV, hence can be viewed as a natural extension of $u(t)$ after the blow-up time.