FINITE TIME BLOW UP FOR THE ENERGY CRITICAL ZAKHAROV SYSTEM II: EXACT SOLUTIONS

Based on the companion paper [23], we show that the 4D energy critical Zakharov system admits finite time type II blow up solutions, similar to the ones constructed in [33]. 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 [22], [21], [20]. 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.