Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 7 / 1 / e

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Solution

For the free transport equation, the mixed Lebesgue norm dispersion estimate is

∥ f (t) ∥_{L_{x}^{\infty} L_{v}^{1}} \leq ∣ t ∣^{- d} ∥ f_{in} ∥_{L_{x}^{1} L_{v}^{\infty}}, t \neq = 0.

(1)

Indeed, use the method of characteristics and then the change of variables

y = x - t v

\int ∣ f_{in} (x - t v, v) ∣ d v = ∣ t ∣^{- d} \int f_{in} (y, \frac{x - y}{t}) d y \leq ∣ t ∣^{- d} \int sup_{w} ∣ f_{in} (y, w) ∣ d y .

(2)

Taking the essential supremum over

x

proves the result. This decay measures the spreading of a velocity average; the total phase space Lp norm is conserved rather than decaying. The datum on the right is the initial datum, not the solution at the same time.

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 7 / 1 / e

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