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.
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This work explores the existence and behavior of solutions to the Korteweg–de Vries equation on the line for large perturbations of certain classical solutions. First, we show that given a suitable solution $V(t,x)$, KdV is globally well-posed for initial data $u(0,x) \in V (0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. In particular, we show that smooth periodic and step-like profiles $V(0,x)$ satisfy our hypotheses.
Our second main objective is to prove a variational characterization of KdV multisolitons. Maddocks and Sachs used that $n$-solitons are local constrained minimizers of the polynomial conserved quantities in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. We show that multisolitons are the unique global constrained minimizers for this problem. We then use this characterization to provide a new proof of the Maddocks–Sachs orbital stability result via concentration compactness.
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.