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arXiv · 2407.19972

Finite time blow up for the energy critical Zakharov system II: exact solutions

Joachim Krieger·Tobias Schmid
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Abstract

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.

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Joachim Krieger, Tobias Schmid. 2024-07-29. Finite time blow up for the energy critical Zakharov system II: exact solutions. https://arxiv.org/abs/2407.19972

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