Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 8 / 1 / f

Solution

For the globally invertible Hamiltonian flow from (a), the general characteristic formula is

f (t, z) = f_{0} (Φ (0, t, z)) + \int_{0}^{t} h (Φ (s, t, z)) d s .

Every

Φ (s, t)

preserves Lebesgue measure, by (b). Thus composition with it is an isometry of each Lp space. The Minkowski integral inequality gives

∥ f (t) ∥_{p} \leq ∥ f_{0} ∥_{p} + t ∥ h ∥_{p} .

Applying the Minkowski inequality also in time yields the finite-time Lp bound for Hamiltonian transport:

∥ f ∥_{L^{p} ([0, T] \times R^{2 d})} \leq T^{1/ p} ∥ f_{0} ∥_{p} + (\frac{T ^{p + 1}}{p + 1})^{1/ p} ∥ h ∥_{p} < \infty.

The smoothness assumptions allow the characteristic construction; the norm estimate itself only uses the Lp space data and volume preservation.

For a concrete failure on infinite time, choose

ω = 1

f_{0} = 0

, and

h (x, v) = e^{- (∣ x ∣^{2} + ∣ v ∣^{2}) /2} = e^{- H_{1} (x, v)} .

This is smooth, time independent and in every finite Lp space; it is invariant under the isotropic harmonic oscillator flow. Therefore the solution grows linearly:

f (t, x, v) = t h (x, v), \int_{0}^{\infty} ∥ f (t) ∥_{p}^{p} d t = ∥ h ∥_{p}^{p} \int_{0}^{\infty} t^{p} d t = \infty.

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