Mathematics Colloquia and Seminars

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The nonlinear Schrodinger equation
\label{1.1} i u_{t} + \Delta u = \mu |u|^{\frac{4}{d}} u
is said to be mass critical since the scaling $u(t,x) \mapsto \frac{1}{\lambda^{d/2}} u(\frac{t}{\lambda^{2}}, \frac{x}{\lambda})$ preserves the $L^{2}$ - norm, $\mu = \pm 1$. In this talk we will discuss the concentration compactness method, which is used to prove global well - posedness and scattering for $(\ref{1.1})$ for all initial data $u(0) \in L^{2}(\mathbf{R}^{d})$ when $\mu = +1$, and for $u(0)$ having $L^{2}$ norm below the ground state when $\mu = -1$. This result is sharp.

As time permits the talk will also discuss the energy - critical problem in $\mathbf{R}^{d} \setminus \Omega$,
\label{1.2} \aligned i u_{t} + \Delta u &= |u|^{\frac{4}{d - 2}} u, \\ u|_{\partial \Omega} &= 0, \endaligned
where $\Omega$ is a compact, convex obstacle.