UC Berkeley

The objective of this dissertation is to study the long time dynamics of a scalar quasilinear wave equation \begin{align*}g^{\alpha\beta}(u)\partial_\alpha\partial_\beta u=0,\hspace{2em}\text{in }\mathbb{R}^{1+3}_{t,x}.\end{align*} This equation satisfies the weak null condition introduced by Lindblad and Rodnianski \cite{lindrodn,lindrodn2}. Lindblad \cite{lind} proved that, for small and localized initial data, this equation has a global solution. In the present work, we establish a modified scattering theory for the above equation. Such a modified scattering theory provides an accurate description of asymptotic behavior of the global solutions.

To study modified scattering, we first identify a notion of asymptotic profile and an associated notion of scattering data. One candidate for the asymptotic profile is given by the asymptotic PDE\begin{align*}2U_{sq}+G(\omega)UU_{qq}=0\end{align*} which was derived by H\"{o}rmander \cite{horm2,horm,horm3}. In Chapter 2, we derive a new reduced system, called the \emph{geometric reduced system}, by modifying H\"{o}rmander's method. In our derivation, we make use of the optical function, i.e.\ a solution to the eikonal equation. In this setting, the scattering data is the initial data for our geometric reduced system, and it is chosen in a way such that the global solution to the quasilinear wave equation and the exact solution to the reduced system match at infinite time. One may infer, from this dissertation, that this new system is more accurate, in that it both describes the long time evolution and contains full information about it.

In Chapter 3, we prove the existence of the modified wave operators for the scalar quasilinear wave equation. Fixing a scattering data which is the initial data for the geometric reduced system, we can first construct an approximate solution to the model equation. Then, by studying a backward Cauchy problem, we show that there exists a global solution to the scalar quasilinear wave equation which matches the approximate solution at infinite time.

In Chapter 4, we prove the asymptotic completeness for the same equation. Given a global solution to the scalar quasilinear wave equation, we rigorously derive the geometric reduced system with error terms. These allow us to recover the scattering data, as well as to construct a matching exact solution to the reduced system.