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

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- Solution 5

Solution

For

T_{2} > T_{1} \geq - 1

, the dual Strichartz estimate for the free Schrödinger equation gives

\int_{T_{1}}^{T_{2}} S (- s) (u_{η} ∣ u_{η} ∣^{2}) (s) d s_{2} ≲ ∥ u_{η} ∣ u_{η} ∣^{2} ∥_{L_{t, x}^{4/3} ([T_{1}, T_{2}])} = ∥ u_{η} ∥_{L_{t, x}^{4} ([T_{1}, T_{2}])}^{3} .

The assumed finite global

L_{t, x}^{4}

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_{η} (t) = S (1) u_{η} (- 1) + i \int_{- 1}^{t} S (- s) (u_{η} ∣ u_{η} ∣^{2}) (s) d s .

u_{η}^{\infty} = S (1) u_{η} (- 1) + i F_{\infty} \in L^{2} .

Since the free Schrödinger group is unitary,

∥ u_{η} (t) - S (t) u_{η}^{\infty} ∥_{2} = ∥ S (- t) u_{η} (t) - u_{η}^{\infty} ∥_{2} ⟶ 0.

This is scattering from a finite Strichartz norm.

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