Consider then cubic defocusing nonlinear wave equation on three dimensional Euclidean space, with radial initial data. In this thesis, we construct a Gibbs measure for this system and prove its invariance under the flow of the NLW. In particular, we are in the infinite volume setting.

For the finite volume analogue, specifically on the unit ball with zero boundary values, an invariant Gibbs measure was constructed by Burq, Tvetkov, and de Suzzoni as a Borel measure on super-critical Sobolev spaces.

We first show that this finite volume Gibbs measure is supported on a space of weighted Holder continuous functions. Next, we show that the NLW is locally well-posed there, a counter-point to the Sobolev super-criticality noted by Burq and Tzvetkov. Furthermore, the flow of the NLW leaves this measure invariant.

We use a multi-time Feynman--Kac formula to construct the infinite volume limit measure by computing the asymptotics of the fundamental solution of an appropriate parabolic PDE. We use finite speed of propagation and results from descriptive set theory to establish invariance of the infinite volume measure.

We present several large-data results for nontranslation-invariant analogues of the energy-critical and mass-critical nonlinear Schrödinger equations, obtained by introducing external potentials or non-Euclidean geometries.

The main goal of this work is to prove a symplectic non-squeezing result for the Korteweg--de Vries (KdV) equation on the line $\R$. This is achieved via a finite-dimensional approximation argument. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough in the well-posedness theory of KdV in low regularity spaces \cite{KV18}, relying on its completely integrable structure. The employment of our methods also provides us with a new concise proof of symplectic non-squeezing for the same equation on the circle $\T$, recovering the result of \cite{CKSTT}.

This thesis studies scaling critical Strichartz estimates for the Schrödinger flow on compact symmetric spaces. A general scaling critical Strichartz estimate (with an ε-loss, respectively) is given conditional on a conjectured dispersive estimate (with an ε-loss, respectively) on general compact symmetric spaces. The dispersive estimate is then proved for the special case of connected compact Lie groups. Slightly more generally, for products of connected compact Lie groups and spheres of odd dimension, the dispersive estimate is proved with an ε-loss.

In this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces $H^s(\bb{T}^3)$. We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted $L^2$ spaces and the paucity of analogous results in $L^2(\bb{R}^d)$. To explain this, we show that the scattering problems for these equations are well-posed in weighted $L^2$ spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in $L^2$ in the sense that the scattering operators cannot be extended to all of $L^2$ without losing a positive (and, in the case of Hartree, infinite) amount of regularity.

We present several new results regarding the two and three dimensional energy-critical non- linear Schro ̈dinger equation in the presence of a second critical nonlinearity.