Finite time blow up for the energy critical Zakharov system I: approximate solutions

Abstract

We construct approximate solutions $ (\psi_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (\lambda e^{i \theta}W, \lambda^2 W^2)$ . To be precise for $ N \in \mathbb{Z}_+,\;N \gg1 $ we obtain a magnetic envelope/ion density pair of the form $$ \psi_*(t, x)= e^{i\alpha(t)}\lambda(t) W(\lambda(t)x) + \eta(t, x), \;n_*(t,x) = \lambda^2(t) W^2(\lambda(t) x) + \chi(t,x), $$ where $ W(x) = (1 + \frac{|x|^2}{8})^{-1}$, $\alpha(t) = \alpha_0 \log(t)$, $\lambda(t)= t^{-\frac{1}{2}-\nu}$ with large $\nu > 1 $ and further $$ i \partial_t \psi_* + \Delta \psi_* + n_* \psi_* = \mathcal{O}(t^N),\; \Box n_* - \Delta (|\psi_*|^2) = \mathcal{O}(t^N),\;\;\eta(t) \to \eta_0, \chi(t) \to \chi_0, $$ as $ t \to 0^+$ in a suitable sense. The method of construction is inspired by matched asymptotic regions and approximation procedures in the context of blow up solutions introduced by the first author jointly with W. Schlag and D. Tataru, as well as the subsequently developed methods in the Schr\"odinger context by G. Perelman et al.