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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2012

Paper 7

For the <free transport equation>, the <mixed Lebesgue norm> dispersion estimate is
$$
\boxed{\|f(t)\|_{L_x^\infty L_v^1}
\leq |t|^{-d}\|f_{\mathrm{in}}\|_{L_x^1L_v^\infty},\qquad t\ne0.}
$$
Indeed, use the <method of characteristics> and then the <change of variables> $y=x-tv$:
$$
\int |f_{\mathrm{in}}(x-tv,v)|\,dv
=|t|^{-d}\int
\left|f_{\mathrm{in}}\left(y,\frac{x-y}{t}\right)\right|dy
\leq |t|^{-d}\int\sup_w|f_{\mathrm{in}}(y,w)|\,dy.
$$
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.

Solution

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

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

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

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