On long time behavior of solutions to nonlinear dispersive equations

In part I, we address the issue of existence of solutions for Cauchy problems involving nonlinear hyperbolic equations for initial data in Sobolev spaces with scaling subcritical regularity. In particular, we analyse nonlinear estimates for null-forms in the context of wave Sobolev spaces $H^{s,b}$, first in a flat background, then we generalize to more general curved backgrounds. We provide the foundations to show that the Yang-Mills equation in $\mathbb{R}^{1+3}$ are globally well-posedness for small weighted $H^{3/4+}\times H^{-1/4+}$ initial data, matching the minimal regularity obtained by Tao \cite{tao03}. Our method, inspired from \cite{dasgupta15}, combines the classical Penrose compactification of Minkowski space-time with a null-form estimates for second order hyperbolic operators with variable coefficients. The proof of the null-form appearing in the Yang-Mills equation will be provided in a subsequent work. As a consequence of our argument, we shall obtain sharp pointwise decay bounds. In part II, we show that the finite time type II blow-up solutions for the energy critical nonlinear wave equation $\Box u = -u^5$ on $\R^{3+1}$ constructed in \cite{krieger2009slow}, \cite{krieger2009renormalization} are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $\lambda(t) = t^{-1-\nu}$ is sufficiently close to the self-similar rate, i. e. $\nu>0$ is sufficiently small. This result is qualitatively optimal in light of the result of \cite{krieger2015center}, it builds on the analysis of \cite{krieger2017stability} and it is joint work with my thesis advisor Prof. J. Krieger.