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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.19972 |
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Table of Contents:
- Based on our companion paper [Krieger-Schmid, 2024], we show that the 4D energy critical Zakharov system admits finite time type II blow up solutions. The main new difficulty this work deals with is the appearance of a term in the linearization around the approximate solution, which is non-local with respect to both space and time. In particular this cannot be handled by straightforward adaptation of the methods developed in [Krieger-Schlag-Tataru, 2008/09]. The key new ingredients we use are a type of approximate modulation theory, taking advantage of frequency localisations, and the exploitation of an inhomogeneous wave equation with both a non-local, as well as a local potential term. These terms arise for the main non-perturbative component of the ion density $n$ and can be solved via inversion of a certain Fredholm type operator, as well as by using distorted Fourier methods. Our result relies on a number of numerical non-degeneracy assumptions.