Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 2 / 3

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- Solution 3

Solution

In three dimensions, the Hardy–Littlewood–Sobolev inequality applied to the Newtonian kernel gives

\int_{R^{3}} ∣\nabla ϕ ∣^{2} = - \int ϕ ∣ u ∣^{2} ≲ ∥∣ u ∣^{2} ∥_{6/5}^{2} = ∥ u ∥_{12/5}^{4} .

The Gagliardo-Nirenberg interpolation inequality then gives

∥ u ∥_{12/5}^{4} ≲ ∥ u ∥_{2}^{3} ∥\nabla u ∥_{2} .

X = ∥\nabla u ∥_{2}

and using the conserved mass, conservation of energy implies

E (u_{0}) \geq \frac{1}{2} X^{2} - C (u_{0}) X .

Thus

X

remains bounded. Together with the conserved

L^{2}

norm this bounds

∥ u (t) ∥_{H^{1}}

, and the supplied blowup criterion proves Global H1 solutions of the three-dimensional gravitational Hartree equation.

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