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arXiv · 2407.19971

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

Joachim Krieger·Tobias Schmid
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Abstract

We construct approximate solutions $ (ψ_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (λe^{i θ}W, λ^2 W^2)$ . To be precise for $ N \in \mathbb{Z}_+,\;N \gg1 $ we obtain a magnetic envelope/ion density pair of the form $$ ψ_*(t, x)= e^{iα(t)}λ(t) W(λ(t)x) + η(t, x), \;n_*(t,x) = λ^2(t) W^2(λ(t) x) + χ(t,x), $$ where $ W(x) = (1 + \frac{|x|^2}{8})^{-1}$, $α(t) = α_0 \log(t)$, $λ(t)= t^{-\frac{1}{2}-ν}$ with large $ν> 1 $ and further $$ i \partial_t ψ_* + Δψ_* + n_* ψ_* = \mathcal{O}(t^N),\; \Box n_* - Δ(|ψ_*|^2) = \mathcal{O}(t^N),\;\;η(t) \to η_0, χ(t) \to χ_0, $$ 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.

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Joachim Krieger, Tobias Schmid. 2024-07-29. Finite time blow up for the energy critical Zakharov system I: approximate solutions. https://arxiv.org/abs/2407.19971

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