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

TL;DR

This paper demonstrates finite time blow-up solutions for the 4D energy critical Zakharov system, introducing new analytical techniques to handle non-local linearization terms and building on previous work.

Contribution

It develops a novel approach combining approximate modulation theory and inhomogeneous wave equations to analyze blow-up in the Zakharov system, addressing non-local linearization challenges.

Findings

Existence of finite time type II blow-up solutions.

Introduction of a new analytical framework for non-local linearization terms.

Reliance on numerical non-degeneracy assumptions.

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.