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 Schrödinger context by G. Perelman et al.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.19971
- arXiv:
- arXiv:2407.19971
- Bibcode:
- 2024arXiv240719971K
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 149 pages