For $T_2>T_1\geq-1$, the dual <Strichartz estimate for the free Schrödinger equation> gives
$$
\left\|
\int_{T_1}^{T_2}S(-s)(u_\eta|u_\eta|^2)(s)\,ds
\right\|_2
\lesssim
\|u_\eta|u_\eta|^2\|_{L^{4/3}_{t,x}([T_1,T_2])}
=\|u_\eta\|_{L^4_{t,x}([T_1,T_2])}^3.
$$
The assumed finite global $L^4_{t,x}$ norm makes the right side tend to zero as $T_1,T_2\to\infty$. The displayed family is therefore a <Cauchy sequence> in the complete space $L^2$ and has a strong limit $F_\infty$.

The <Duhamel principle> gives
$$
S(-t)u_\eta(t)
=S(1)u_\eta(-1)
+i\int_{-1}^tS(-s)(u_\eta|u_\eta|^2)(s)\,ds.
$$
Define
$$
u_\eta^\infty=S(1)u_\eta(-1)+iF_\infty\in L^2.
$$
Since the free Schrödinger group is <unitary>,
$$
\|u_\eta(t)-S(t)u_\eta^\infty\|_2
=\|S(-t)u_\eta(t)-u_\eta^\infty\|_2\longrightarrow0.
$$
This is <scattering from a finite Strichartz norm>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 3 / 5 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-154/3/5/solution