Jason Murphy

The Radial Defocusing Nonlinear Schrödinger Equation in Three Space Dimensions

We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting we employ a space-localized Lin–Strauss Morawetz inequality of Bourgain. In the intercritical regime we prove long-time Strichartz estimates and frequency-localized Lin–Strauss Morawetz inequalities.

Large data mass-subcritical NLS: critical weighted bounds imply scattering

We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity

The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions

s_c\in (\max \{-1,-\frac{d}{2}\},0)

The focusing cubic NLS with inverse-square potential in three space dimensions

sc∈(max{-1,-d2},0), we prove that any solution satisfying

The radial defocusing nonlinear Schr\"odinger equation in three space dimensions

\begin{aligned} \left\| \, |x|^{|s_c|}e^{-it\Delta } u\right\| _{L_t^\infty L_x^2} <\infty \end{aligned}

\dot{H}^{1/2}

|x||sc|e-itΔuLt∞Lx2<∞on its maximal interval of existence must be global and scatter.